Rings, Modules, Algebras, and Abelian Groups

did. We begin by proving the following well known result, the proof of which we shall use in the proposition below. Lemma 1. Let T be a Noetherian ring'with a : T —> T a ring endomorphism. If a is a surjection, then a is an injection. 55

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J. W. Brewer

Proof. Since a is surjective, cr(T) = T and it follows by induction that crl(T) = T for i > 1. Now consider the sequence ker a C ker a2 C • • •

of ideals of T. Since T is Noetherian, the sequence stabilizes, say ker an+1 = ker a". Let x € kercr. Since an(T) = T, there exists an element y £ T such that <Jn(y) = x. Then an+1(y) =
X = b0 + bif + ••• + bn

n

We first treat the case when R := D is an integral domain. As noted earlier, we may assume that 60 — 0. After expanding the equality above, we equate coefficients. The coefficient of Xmn on the left hand side of the equality is 0 and, on the right hand side, it is bno^- Now, n > 1 and, if m > 2, then mn > 2 and hence am = 0. Thus, m = I , from which it follows that n = 1 as well. But then, bi(a0 + a\X) — X and ai is a unit of D. Consequently, in the case of an integral domain, with / = CQ + a\X + • • • + amXm, iff is an automorphism if (and, as noted earlier, only if) a\ is a unit and QJ = 0 for i > 2. That is, / = a0 + a\X with ai a unit. This can easily be extended to give necessary conditions on / in the case of general -R. If P is an arbitrary prime ideal of 'R, the map p : R[X] —> (R/P)[X\ given by reducing coefficients modulo F is a surjective homomorphism with kernel P[-X"], the ideal of R[X] consisting of those polynomials having all their coefficients in P.

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If R[X] = R[f], then (R/P)[X] = (R/P)[f], where "/" denotes the image of / in (R/P)[X\. By the integral domain case, a\ is a unit of R/P and 0^ = 0 for i > 1. Since P was arbitrary, ai is in no prime ideal of R and a,, for i > 2, is in every prime ideal of R. Thus, ai is a unit of R and a^, for i > 2, is nilpotent. We can now give Gilmer's main result, Theorem 3 of [4]. The proof is essentially verbatim from the original. Theorem 3. Let / = ao + a\X + • —h amXm e .HpsT]. The ring R-endomorphism (ff : R[X] —> R[X] determined by ipf(X) = f is an automorphism if and only if a\ is a unit and a^ is nilpotent for i > 2. Proof. We have proved the necessity of the conditions. It remains to prove the sufficiency. By Proposition 2, we have only to prove that (ff is a surjection. We use induction on k, the order of nilpotence of the ideal cz(f) — (°2i a 3i • • • i am)- F°r any / for which k — 1, each a* = 0 for i > 1 and ipf is surjective. We assume the truth of the theorem for all elements / of R[X] of the desired form for which the order k of nilpotence of the associated ideal c 2 (/) is less than r, where r > 2, and we consider an element h = h0 + hiX H ---- + hvXv of R[x], where hi is a unit of R and the order of nilpotence of c^Cn) = (h%, . . . , hv) is r. Since R[h] = R[h!(h - ho)}, it suffices to consider the case in which hi = 1 and h0 = 0. We then consider, Ui

for i > 2, hih1. Now, h1 has the form X1 + ^3

u

ij^ > where each u^ G C2(h).

J=i+l n

Hence, hih1 — h^X1 +

i

]P v^X^ , where each Vij & [c2(h)}2. Therefore, if g =

h — /i2/i2 — h3h3 --- • — hvhv, then g has the form X + g^X^ H ----- h gmXm, where each 9i e [c2(/i)]2. It follows that [c2(5)]r-1 C [c2(h')}^r-1\ and

because 2(r — 1) > r. Consequently, the order of nilpotence of c 2 (
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American Mathematical Society, and after receiving no correspondence concerning the results for several months, we concluded at last that if Theorem 3 is known it certainly is not well known, and that such a basic result ought to appear in some widely read journal." 3. Polynomial rings over von Neumann regular rings We begin by recalling the definition of a von Neumann regular ring. As there are different versions, we list several of them in the following result. Proposition 4. Let R be a commutative ring The following are equivalent: 1. If r £ R, there exists an idempotent element e S R and a unit u e R such that r = ue. 2. If r & R, there exists an element s € R such that r = sr2. 3. Ifr e R, then rR = r2R. 4. The Krull dimension of R is zero and R is reduced, that is, R has no nonzero nilpotent elements. 5. For each prime ideal M of R., RM is " field. Proof. (1) => (2): Let r e R and write r = ue, where u is a unit and e is an idempotent. Then r 2 = u2e2 = u2e. Thus, u~lr2 = ue = r. (2) => (1): Let r £ R. Then there exists an element 5 £ R such that r — sr2. Now, (sr)2 = s2r2 = s(sr2) = sr. Thus, sr is idempotent. Set e := sr and let u = 1 — e + r. An easy computation shows that u(l - e + se) = 1 and hence that u is a unit. Finally, ue = (I — e + r)e = e — e + re = re = rsr = sr2 = r

(3) & (2): This is obvious. (1) =>• (4): Let x e R and write x = ue, where u is a unit and e is an idempotent. Then x2 = u2e2 = u2e. Thus, xn = une for n > 1. If xn = 0, then e = 0 and so x = 0. It follows that 0 is the only nilpotent element of R. Let P be a prime ideal of R. If a; e P, then there exists an element y S R such that x = yx2 and so, x(l — yx) = 0. Thus, in RP, I — yx is a unit and so, x = 0. Since PRp = (0) and PRp is the unique maximal ideal of Rp, it follows that P is a maximal ideal of R. Hence, the Krull dimension of R is zero, (4) =>• (5): Recall that if a ring T is reduced, then so is any localization T$. (If (j) n = 0 in Ts, there exists an element s € S such that san — 0. But then snan = (sa)n = 0 and so sa = 0. Thus, ( f ) = 0.) In our case, Rp is reduced for each prime ideal P of R. But, R is zero-dimensional and so PRp is both the nilradical and the unique maximal ideal of -Rp. Therefore, PRp = (0) and Rp is a field. (5) => (3): We have to show that for each element x & R, xR = x2R. Well, 2 x R C xR and for P a prime ideal of R, x £ P if and only if x2 £ P. If x ^ P, then x2Rp = Rp = xRp. If x e P, then xRP = (0) = x2Rp. In either case, xRp = x2Rp at each prime ideal of R and so xR = x2R. D

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If one, and hence all, of these conditions is satisfied, then R is called a von Neumann regular ring. Remark 5. Since it follows easily from condition (5) that each module over a von Neumann regular ring must be flat, these are also the "absolutely flat rings" of Bourbaki. Remark 6. It follows from (2) that each principal ideal of R is generated by an idempotent element. In fact, any finitely generated ideal is generated by an idempotent. (In the case of two idempotent elements e\and e% of R, e\R + e-^R = (BI + 62 — e^e^R. The general case follows by induction.) The paper [5] is a note following up the paper [9] by Paul McCarthy. The principal result of Paul's was: If R is a von Neumann regular ring, then the ring of polynomials R[X] in a single indeterminate X is a semihereditary ring. This oft-quoted result is now referred to as "McCarthy's Theorem." Robert notes that a useful lemma for McCarthy was: If f ( X ) = OQ + aiX + • • • + anXn is in R[X] and if e» is an idempotent generator of the ideal a,iR for each i, then the annihilator of f ( X ) is the principal ideal of R[X] generated by (1 - e 0 )(l — e i ) . . . ( l -en). The purpose of Robert's paper was to extend this result to the case of finitely many polynomials in the ring .R[{Xx}], where {X\} is a family of indeterminates over the von Neumann regular ring R. With hindsight, this was a straightforward effort. In fact, the essence of McCarthy's lemma is simply the fact noted above that a finitely generated ideal in a von Neumann regular ring is generated by an idempotent. McCarthy's Theorem itself is an interesting result and here is a short proof using a lovely result of Endo [3]. Theorem 7 (McCarthy). // R is a von Neumann regular ring, then R[X] is semihereditary. Proof. Endo's Theorem says the following: A commutative ring S is semihereditary if and only if the total quotient ring of S is von Neumann regular and SM is a valuation domain for each maximal ideal M of S. We shall show that the hypotheses of Endo's theorem are satisfied by R[X}. Since the nilradical TV of R is zero and since the nilradical of R[X] = N[X], R[X] has no nonzero nilpotent elements. It follows that the same is true for T(R[X]), the total quotient ring of R[X}. If / is a regular element of R[X], let c(f) denote the content ideal of /, that is, the ideal of R[X] generated by the coefficients of /. Since R is a von Neumann regular ring, c(f) = eR, for some nonzero idempotent element e e R. If 1 ^ e, then (1 - e) ^ 0 and (1 - e) • / = 0. Thus, c(f) = R and it follows that a polynomial / is a regular element of R[X] if and only if / has unit content. Hence, T(R[X}) = (R[X])S = R(X), where S is the set of polynomials in R[X] having unit

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content. It is well known [6] Proposition 33.1 that the map M —> MR(X) is always a homeomorphism from the maximal spectrum of R to the maximal spectrum of R(X). Consequently, T(R[X\) is zero-dimensional. It follows that T(R[X]) is a von Neumann regular ring. If OT is a maximal ideal of R[X], then (R[X])m is a localization of RfmnniX] at a nonzero prime ideal. Since -R is a von Neumann regular ring, RmnR is a field and so (J?[J\r])jrn is a valuation domain. Thus, the hypotheses of Endo's theorem are satisfied and R[X] is a semihereditary ring. D Gilmer, in collaboration with T. Parker [7], made an additional contribution to the topic by proving the following nice result. We give their proof virtually unchanged. Theorem 8. If R is a von Neumann regular ring, then R[X] is a Bezout ring. Proof. We have to show that every finitely generated ideal of R[X] is principal and for that, it suffices to prove that if / and g are nonzero elements of -R[X], then (/, g) is principal. We assume that deg/, the degree of /, is less than or equal to the degree of g, and we use induction on deg/. If deg/ = 0, that is, if / € R, then we can assume that / is an idempotent of R. Then (/, g) = (/i), where h = f + (1 — f ) g , because / = fh and g = (1 — / — f g ) h . We assume that (/, g) is principal if deg/ < n, and we consider the case where deg/ = n + 1. Let t be the leading coefficient of / and let e be an idempotent generator of the ideal tR of R. Then R = eR <£ (1 - e)R, R[X] = eR[X] ® (1 - e)R[X], and (/, 9) = (e/, eg) © ((1 - e)f, (I - e)g)). The ideal ((1 - e)f, (1 - e)g)) is principal by the induction hypothesis. And, as an element of e.R[X], the leading coefficient et of ef is a unit, since et • eR = eR. Hence, eg = q • ef + r for some q, r S eJR[.X"] with r = 0 or degr < dege/. It follows that (ef,eg) = ( e f , r ) , and again the induction hypothesis implies that (ef, eg) is principal. Since each of the summands of the ideal (/, g) in the decomposition

is principal, it follows that (/, g) is also principal.

D

Motivated by altogether different considerations, J. Brewer, D. Katz, and W. Ullery [2] strengthened this result by proving that R[X] is an elementary divisor ring. Recall from [8] that a ring R is called an elementary divisor ring if each matrix A over R admits a diagonal reduction— that is, there exist invertible matrices P and Q such that PAQ is diagonal. Proposition 9. If R is a von Neumann regular ring, then R[X] is an elementary divisor ring. Proof. Let A be a matrix over f?[JT]. Set / = {a £ R \ A admits a diagonal reduction over _R a [X]}, where Ra denotes the localization of R at the element a, and let /* be the ideal of R generated by /. For each maximal ideal M of R, RM[X]

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61

is a principal ideal domain and so A admits a diagonal reduction over RM [X]. It follows easily from this that /* = R so there exist elements o,j £ I and r^ e R with riai H

\-rnan = 1.

Since .R is a von Neumann regular ring, we may assume that each a^ is idempotent. Replacing ai, a 2 , . . . , an by
i1 — a •x—\-b, ?, yc

2

2

c

and so, ab = ab • 1 = a • (6 • f) + b • (a • *) = a2(3x + b2ay.

D

Proposition 11. If R[X] is a Bezout ring, then R is a von Neumann regular ring. Proof. Let r e R. The ideal (r, X} of R[X] is principal, generated by a non-zerodivisor / of R[X]. Then (r,X)2 = (r2,X2) = (/ 2 ) and so, rX = or2 + /3X2 for a, /3 € -R[J^]. Since the degree of /3X2 is greater than or equal to 2, the coefficient of X on the right hand side is r2a, where a is the coefficient of X in the polynomial a. Thus, r = r2a and R is von Neumann regular. D Remark 12. The proof actually shows that if R is a ring and if r £ R, then there exists an element a & R such that r = r2a if and only if the ideal (r, X) of R[X] is such that (r, X)2 = (r2,X2). Before stating and proving the main result of this section, we need to make two additional remarks. Remark 13. If S is a commutative ring of Krull dimension n, and if X is an indeterminate, then n + 1 < dimS'piT] < In + 1. Hence, if R is a von Neumann regular ring, dimR[X] = 1. And, as noted above, the nilradical extends to the nilradical of R[X]. Thus, R is a von Neumann regular ring if (and only if), R[X] is one-dimensional and reduced.

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Remark 14. The usual proof that a projective ideal in an integral domain is invertible can be used to prove that, over any ring, a projective ideal which contains a non-zero-divisor must be invertible. Thus, projective ideals containing non-zerodivisors in semihereditary rings must be invertible. Putting all this together, we have the following lovely result. Theorem 15. Let R be a commutative ring with X an indeterminate. The following are equivalent: 1. R is a von Neumann regular ring. 2. R[X] is one-dimensional and reduced. 3. R[X] is an elementary divisor ring. 4. R[X] is a Bezout ring. 5. R[X] is a semihereditary ring. 6. Each ideal I of R[X] generated by two elements is invertible. 7. //*P is a prime ideal ofR[X], then (R[X])

The Picard Group of the Ring of Integer-valued Polynomials on a Valuation Domain Jean-Luc Chabert Department of Mathematics, Universite de Picardie, 80039 Amiens, France, LAMFA CNRS-UMR 6140 jean-luc.chabertfiu-picardie.fr Abstract. We study the class group of invertible ideals, or Picard group, of the ring lni(E, D) of generalized integer-valued polynomials, that is, {/ 6 K[X] f ( E ) C D] where D is a domain with quotient field K and E is a subset of D. A good description is known in the classical case where E = D and D is a Dedekind domain with finite residue fields. We focus here on a local study where D is any valuation domain and where E is any precompact subset of D. We show that the Picard group is still generated by the classes of the maximal ideals lying over the zero ideal of D and we describe the relations between these generators. We also characterize the case where the Picard group is generated by the classes of prime ideals lying over the maximal ideal of D. Finally, we globalize this last result to the case where D is a Dedekind domain with finite residue fields.

Introduction Let D be an integral domain with quotient field K. The fractional invertible ideals of D form an Abelian group J(D) and the principal fractional ideals of D form a subgroup P(D) of J(D}. Obviously, if D x denotes the group of units of D, then T(D)~K*/DX. The class group of J(D) with respect to P(D) is called the Picard group of D and is denoted by Pic(Z?):

Pic(Z3) = J(D)/P(D). 1991 Mathematics Subject Classification. Primary 13F20; Secondary 13A15, 13C20. Key words and phrases. Integer-valued polynomial, Picard group, valuation domain.

63

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J.-L. Chabert

Let E be a subset of D. We consider the ring of integer-valued polynomials on E with respect to D, that is, We are interested here in the description of the Picard group of the integral domain Int(E,D): Pic(IntCE,D)) = J(Iat(E,D))/P(Iat(E,D)). Note that, for every domain D and every subset E of D, one has: P(Iut(E,D))~K(X)*/Dx. There are two steps in this study: — to what extent can we reduce the global study to a local study? — in the local case, can we describe Pic(Int(E, £))) by generators and relations? In the case where E = D and where D is a Dedekind domain, the first step is solved in [11, Theorem 1] and the second in [3, Proposition 4.1] (see also [4, Theorem VIII. 1.9 and VIII. 5. 8]). In the case where D = 1 and E = P is the set of prime numbers, we find an extensive study of Pic(Int(P, Z)) in [8]. With respect to the first problem, extending the results of [11], one may prove [4, Exercise VIII. 4] that, if Z? is a one-dimensional Noetherian domain, then there is a natural short exact sequence: 0 -> Pic(D) -> Pic(Int(JB, D}) -> ® m€ma x(D) Pic(Int(£, D m )) -+ 0. In the non-Noetherian case, the determination of hypotheses on D and on E that allow to consider such an exact sequence is still an open problem. Here, we are going to focus our study on the local case where -D is a valuation domain and E is a precompact subset of D. Our main results are the following: — Theorem 3.2 establishes an isomorphism between the Picard group and a class group of continuous functions. — Theorem 5.1 states that the Picard group is generated by classes of maximal ideals lying over the ideal (0) of D. — Theorem 6.5 shows that the Picard group is generated by classes of maximal ideals lying over the maximal ideal of D if and only if E has at most one accumulation point. — Proposition 7.7 says that, when the valuation is discrete, the Picard group is always a free Abelian group. — Theorem 8.9 globalizes these two last results to Dedekind domains with finite residue fields. We begin with some general remarks. 1. General results Notation. For every invertible ideal 3 of Int(£;, D), let us denote by [3] the class of 3inPic(Int(.E,L>)). Let us recall a technical notion which will be very useful to study the Picard group:

The Picard group

65

Definition 1.1. An ideal 3 of Int(E,D) is said unitary if 3 is an integral ideal containing nonzero constants, that is, 3 C lm,(E, D] and 3 n D ^ (Q). Remark 1.2. In each class p] € Pic(Int(J5,D)) there is a unitary ideal 30, that is, there exists g e K[X] such that 30 = ^3 is a unitary ideal of lnt(E, D}. Notation. Denote by Ju(lnt(E, D)) the set of unitary invertible ideals of lnt(E, D). This is a subset of the group J(lut(E, D)) which is stable by product and which contains the unit element Int(£', D): endowed with the induced law, Ju(lnt(Ey D)) is a submonoid of J(lnt(E, D)). Proposition 1.3. There is a short exact sequence of natural morphisms: 0 -> P(D) -> Ju(lnt(E, D}) -+ Pic(Int(E, D)) -> 0.

Pic(Int(E,I>)) ~

Ju(IrA(E,D])/T(D}.

The surjectivity results from Remark 1.2. We just have to check the exactness in Ju(Int(E,D)), that is that a unitary principal ideal of lnt(E,D] is necessarily generated by a constant polynomial. To study Pic(Int(.E, D)) we then may most often consider unitary ideals. Remark 1.4. For all 3i,32 6 J'u(Int(.E,.D)), one has: (Pi] + Pa] = 0} «=> {3a 6 D \ {0} such that d^ = a!nt(E, D)}. Notation. Let 3 be an integral ideal of Int(.E, D). We consider: 1. The ideal of values of 3 at a for each a € E:

3(a) = {/(a) | / e J}. 2. The i>aZwe function vj of 3 defined on E by:

113: a H-> 3(a). This leads us to the following: Notation. Consider I? as a topological ring such that the nonzero ideals of D form a fundamental system of neighborhoods of 0. Let C(E, J(D}) be the group formed by the locally constant functions from E to J(D) where E is endowed with the induced topology. Proposition 1.5. The map 3 H-> vy induces by restriction and quotient two injective morphisms: v.3& Ju(Ir&(E,D}) ^v^e C(E,J(D)) and v:

Pic(Int(E,D))-*C(E,J(D)}/P(D)

where P(D) is identified with the subgroup of constant maps from E to P(D).

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Proof. Note first that, if the unitary ideal 3 is finitely generated, then there is a nonzero ideal a of D such that:

Namely, let b — 3 n £>, let ( f i , . . . , f r ) be a system of generators of 3, and let d 6 .D \ {0} be such that d/j e D[X] for i = 1, . . . , r. Then, we may choose a = db: x - y 6 a implies d(fi(x) - fi(y)) e (x - y) C db, and hence, /;(z) - /j(j/) e b; consequently,

Moreover, if the unitary ideal 3 is invertible, then 3(a) is invertible for each a G E. This is an obvious consequence of Remark 1.4. Consequently, v is well defined. It is then obvious that f is a morphism. Finally, we prove that v is injective thanks to Proposition 1.6 below which says in particular that a unitary invertible ideal 3 is characterized by the function vj. On the other hand, v is well defined and injective because the morphism from Ju(lut(E, D)) to C(E, J(D}}/P(D) obtained by composition of v with the quotient map has a kernel isomorphic to P(D). D Proposition 1.6 ([5, Lemma 3.7]). 7/3 is a unitary divisorial ideal of!nt(E,D) and if f G Int(£1, D) is such that f ( a ) € 3(a) for each a G E, then f belongs to 3. Corollary 1.7. Let 3 i , . . . ,3S € Ju(Int(E,V)) s

and fci,..., ks e Z. Then:

s

s

2. Hypothesis, notation, and known results Throughout, V denotes the ring of a valuation v on a field K, with maximal ideal m and value group F = v(K*). Let E be a subset ofV. Then K is a topological field where the ideals

37 = {x e K v(x) > 7}

(7 e F)

form a fundamental system of neighborhoods of 0 [2]. We denote respectively by V, K, m, and E the completions of V, K, m and E (but we simply denote by v the extension of the valuation to K). Let C(E, V) denote the ring of continuous functions from E to V. We assume that E is precompact, that is, that E is compact. Such a situation has been studied in [6]. So we first recall the results that will be useful. One knows that: — If y contains an infinite precompact subset F, then the topology of K is metrizable. — A subset F of V is precompact if and only if, for each positive 7 G F, F meets at most finitely many cosets of V modulo 3T.

The Picard group

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The corner stone for most properties of the ring Int(J5, V) is the following p-adic Stone- Weierstrass theorem and its corollary. Proposition 2.1 ([6, Thm 1.10]). The ring Int(E, V) is dense in the ring C(E, V) for the uniform convergence topology. Corollary 2.2. Let U\, . . . , Ur be disjoint open subsets covering E. Let oti,. . . ,ar be arbitrarily chosen elements of the value group F. Then there exists a polynomial h 6 K[X] such that: Vi <E {!,..., r}, Vz £ UiC]E,

v(h(x)) = on.

It follows from this theorem that: Proposition 2.3 ([6, Prop. 2.2 and 2.3, Thm 4.1]). The ring Int(£1, V) is a Priifer domain. Its Krull dimension is equal to l + dim(V). There are two kinds of nonzero prime ideals of!nt(E,V): 1. The prime ideals above (0); they are in one-to-one correspondence with the monic polynomials q which are irreducible in K[X]: q i—> *pg = gA'[Ar] n Ini(E, V). 2. The prime ideals above p, where p is a nonzero prime ideal ofV; they are in one-to-one correspondence with the elements a of E:

Moreover, for each non-zero prime ideal p of V, each a e E, and each monic polynomial q irreducible in K[X]:

<$q C pa «=> q(a) = 0. Corollary 2.4. The localization oflnt(E,V)

with respect to:

1. the prime ideal tyg is the valuation domain K[X}^ corresponding to the q-adic valuation of K(X). 2. the prime ideal pa is the valuation domain {(p e K(X) 0}. We denote by C(E,F) (resp., C(E,T+)) the set of continuous functions from E endowed with the induced topology to F (resp., F + ) endowed with the discrete topology.

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Proposition 3.1. There is a natural isomorphism of monoids:

v. 3 e Ju(liit(E, V)) ^ v3 e C(E,r+). Proof. For each 3 G j7u(Int(-B, V)), the value function 1*3 is a priori a continuous function from E to F+. Since this function is uniformly continuous, it can be extended to a continuous function from E to F+. In fact, for x £ E, v-j(x) = mff£3v(f(x)~). The morphism v defined in Proposition 1.5 is injective and the extended morphism is also injective. Let us prove that it is onto. Let w e C(E,F+). Since E is compact and F+ is discrete, w takes only finitely many values 71, . . . , 7r. It follows from Corollary 2.2 that there exists h € Int(£, V) such that v(h(x)) = w(x) for every x € E. Let t 6 V be such that t;(i) > sup^^u^z)) = sup 1 < l < r 7 i . Let 3 be the ideal generated by t and h. Then 3 € Ju(Int(E, V)) and u3 =~u;~ D We easily deduce the following: Theorem 3.2. There is a natural isomorphism of groups:

where the group F is identified with the constant functions

ofC(E,F).

Corollary 3.3. The natural map 3 e Ju(Int(£, V}) » 3Int(E, V) € Ju(Int(E, V")) is an isomorphism of monoids. Consequently, Pic(Int(E, V)) ~ Pic(Int(E, V)). Remark 3.4. Note that, to define the previous isomorphism between the Picard groups, we may choose in each class of Pic(Int(£, V)) a unitary ideal 3 of Int(.E, V), extend it to lnt(E,V), and then, take its class in Pic(Int(.E, V)), because two unitary ideals are in the same class if and only if they differ by a multiplicative constant. But, this is no more true for non-unitary ideals: For instance, if (3 G V and f3 $. V(JE, then ty(X-0) is an invertible ideal of Int(E, V) (cf. Proposition 4.3 below) and is not extended from an ideal of Int(.E, V). Nevertheless, as soon as b e V is such that v(b — p ) > supxeBv(x — /3), then b ^ E, 5p(x-b) is invertible, and one has the following equality in Pic(Int(.§, V)}: [V(x~/3)} = [^(x-b)] (cf. Corollary 7.2 below) . Corollary 3.5. Pic(Int(.E, V)) has no torsion. Proof. Let 3 6 Ju(Iut(E, V)). If there is an integer n > 0 such that 3" is principal, then the function v^ is constant, and hence, vy also, that is, 3 is principal. D Corollary 3.6. For each 3 e Ju(Int(£', V ) ) , there exists h e 3 such that (h(a)) = 3(a) for every a € E. For such a polynomial h and for every c e 3 n V, c ^ O , one has 3 = (c, /i).

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Proof. Let w = vj G C(E,T+). We have seen in the proof of Proposition 3.1 that there is h G Int(.E, V) such that v(h(x)) = w(x) = vy(x) for every x € E ; in particular, (h(a)) — 3(a) for every a&E. Let c e 3 n V and let ^ = (c, ft). Then, v^1 = w = vj, and hence, 3i = 3. D Proposition 3.7. 77ie domain Int(E, V) has the Steinitz property, that is, for each pair of invertible ideals 21 and 03 o/Int(E, V), one /IBS tfie following isomorphism: Thanks to Corollary 3.6 we may easily extend the proof given in the case where E = V and V is a one-dimensional Noetherian domain with finite residue fields [4, Proposition VIII. 4. 13]. 4. The invertible primes Now we determine the invertible prime ideals of Int(.E, V). Proposition 4.1. A unitary prime ideal oflnt(E, V) is invertible if and only if it is of the form ma where the maximal ideal m of V is principal and a is isolated in E. Proof. Assume that the unitary prime ideal pa is invertible, then it is generated by finitely many polynomials gi, . . . ,gs. By continuity, if a G E is close enough to a, then 5»(a) G p for each i = I , . . . , s and pa — (9i>- • • ,9s) ^ Pa- Proposition 2.3 implies a = a, and hence, a G E and a is isolated in E. On the other hand, since a £ E, p = p a (a) = (51(0:), . . . ,gs(a)), thus p is finitely generated. But, a nonzero prime ideal in a valuation domain is finitely generated if and' only if it is maximal and principal. Conversely, assume that m is principal generated by IT and that a is isolated in E. By the Stone- Weierstrass theorem, the characteristic function of E \ {a} in E may be approximated modulo in by a polynomial / G lnt(E, V). Let 3 — (TT, /). For each maximal ideal 9Jt of Int(E,V) distinct from m a , 3 is not contained in EOT. Moreover, 3ma C (m a ) ma , but (m a ) ma = -nlnt(E,V)ma by Corollary 2.4 and 7rInt(jB, F)ma C 3 ma ; consequently, 3ma = (m a ) ma , and hence ma = 3 is finitely generated. D Remark 4.2. There is a contradiction between the previous proposition and [6, Corollary 3.9]. In fact, Proposition 3.7 of [6] is not correctly written unless F* is replaced by T. Both Corollaries 3.8 and 3.9 in [6] are no longer consequences of this proposition, moreover they are false. Proposition 4.3. Let q G V[X] be irreducible in K[X]. are equivalent: 1. ?fiq is invertible 2. Cpg is finitely generated, 3. ^?q is maximal, 4. q has no root E.

The following assertions

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Note that we no more assume that q is monic (cf. Proposition 2.3), because it is more convenient to consider polynomials with coefficients in V. But, we may still assume that they are irreducible in Proof. The equivalence 1 «-> 2 results from the fact that lut(E, V) is a Priifer domain. The equivalence 3 <-> 4 results from Proposition 2.3. Let us prove that 3 —» 2. Assume that ^3g is maximal. Then, for each a G E, there exists ga G ^[^] such that qga G ^Jg and 9(0)50(0) G" m. By continuity and compactness, there is a finite covering of E with clopen subsets U\ , . . . , Ur and there are polynomials gi,. . . ,gr G K[X] such that qgi G lnt(E, V) and q(x)gi(x) £ m for every a; G f/j. The ideal 21 = (q, qgi, . . . , g 3, we are going to show that 2 and not 3 lead to a contradiction. Assume that (<7i(a)). Then, for every f ( X ) = (X — a)g(X) e ^3g, one has v(g(a)) > m. But, it follows from the corollary of the Stone- Weierstrass theorem that there exists h G K[X] such that v(h(a)) < m and h(E \ {a}) C V. Of course, (X - a)h(X) G ^3, and we have a contradiction. D Proposition 4.3 shows that there always exist non finitely generated prime ideals. Next proposition gives a description of the generators of the primes tyq when they are invertible. More generally, for any polynomial / G liA(E, V), we consider the ideal Proposition 4.4. Let f G V[X] without any root in E. Then the ideal Q5/ is invertible and there are polynomials
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that (h(a)) = 3 (a) for every a e E and, for such an h, one has: 3 = (c, h) because c £ 3 n V. Then, for gf = £, one has
5. Generators Theorem 5.1. T/ie group Pic(Int(.B, V)) is generated by the classes of the ideals tyq which are maximal. Proof. Let 3 e Ju(Int(E, V}} and let c e 3 n V, c ^ 0. There exists h e 3 such that 3(a) = (h(a)) for every a & E (Corollary 3.6). In particular, v(c) > v(h(a)) for every a £ E, v(h(a)) is bounded on E by v(c), and hence, h has no root in E. Let us consider the decomposition of h in where d 6 K* and gi, . . . , qr are irreducible in V[.X"]. The qi's do not have any root in E, and hence, the prime ideals ^39i are invertible. Now we consider the fractional invertible ideal 21 = x 3 x
In particular, 21 is an integral ideal of Int(.E, VQ. Moreover, since 21 is finitely generated, by continuity we obtain that 3L(a)V = V for every a £ E. Finally, 21 is unitary and cannot be contained in any maximal ideal of Int(.E, V). Thus, 21 = Int(.E, V) and J x «p x . . . x «P = Mnt(E, V). D More precisely, we have proved the following algorithm: Algorithm. Let 3 e Ju(Int(E, V)). By the Stone-Weierstrass theorem, there exists h 6 3 such that 3(a) = /i(a)V for every a & E. Let /i = dg"1 • • • q™r be the decomposition of h in V[X] (d € K* and the q^s are irreducible in V[X]). Then, in Pic(Int(E,y)), one has:

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Remark 5.2. Fontana and Gabelli characterized the Prufer domains D whose Picard group Pic(D) is generated by the classes of invertible maximal ideals: they are exactly the Prufer domains D such that some overing S(D) is a Bezout domain [9, Theorem 1.7]. This overring S(D] is the intersection of the localizations Dp where p is either a non-invertible maximal ideal of D, or the intersection n£L0n" where n is an invertible maximal ideal of D. Let us describe the overring S(D) in the case where D = lnt(E,V). Denote by E0 the set formed by the isolated elements a of E and by q the prime ideal n£L0mn. The prime ideals p we have to consider are of 3 types: either p = tnQ where a & E \ EQ, or p = n n m™ = q a where a £ EQ, or p = *P9 where q has a root in E. Notice that we don't have to consider nn£p™ when q has no root in E because this intersection is (0). It then follows from Corollary 2.4 that:

S(Int(E, V)) = {? € K(X)

Proof. We adapt the proof given in [4, Theorem X.4.1] replacing the assumption that V is a discrete valuation domain by the compactness of E. We first note that, for every

Av = {v(ip(x)) | x e E} n {7 e r 1 7 > 0} is not empty, then it contains a smallest element 70: writing tp = *- with / and g relatively prime in V[X], the function vv : v(ip(x)) = v(f(x]} - v(g(x)) from E to F U {00} is continuous except for the x's which are roots of / or g; consequently, if 5 denotes some element of A^,, then vv is continuous on the compact {x & E \ 0 < Vip (x) < 6} , and hence, takes only finitely many values. Now, let r, s € K(X) \ {0}. Consider

If v((p(x)} > 0, then v(ip(x)} = 0. If v((p(x)} < 0, then v(ip(x)) > 0. Thus ip 6 Int R (^, V), in particular, ip e lntR(E, V). If r,s £ Int (E,V), then T == ipr + s belongs to the ideal generated by r and s in Int (E,V). Conversely, v(r(x)) = mf(v(r(x)),v(s(x))') shows that r, s 6 TlntR(E,V). Consequently, every finitely generated ideal of Int^-E, V) is principal. D

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6. Unitary generators

In the previous section the Picard group was shown to be generated by the classes of maximal invertible ideals above (0) . In this section we determine when the Picard group is generated by the classes of unitary invertible primes. We begin with the finite case. Proposition 6.1. Assume E is finite. Then Pic(Int(.E, V)) is generated by the classes of unitary invertible primes if and only if V is a discrete valuation domain. Moreover, in this case, if Card(E) = r then: Ju(IiA(E,V))

~ Nr and Pic(Int(.E, V)) ~ F~l.

Proof. Let E = {ai,...,a r }, let / = Yli
an

d, for 1 < J < r, let

Ini(E, V) = fK[X] the unitary ideals are of the form:

where the a/s are any nonzero ideals of V (and then, 3 (a,-) = a,) and the unitary invertible ideals are of the form:

where the t j ' s are any nonzero elements of V (see [12] or [4, Exercise IV. 1]). If V is a discrete valuation domain, every unitary invertible ideal is then of the following form: <-r

i=i On the other hand, if V is not a discrete valuation domain, we do not always have such a formula for the unitary invertible ideals because either m is not principal and the m a 's are not invertible, or because the rank of v is > 1 and the tj's may belong to q = n n m n . Assume that V is a discrete valuation domain: the monoid i7u(Int(.E, V)) is a free Abelian monoid with basis the m a 's (a e E). Consequently, Ju(lni(E, V)) ~ N r . In fact, we obviously have: C(E,T] ~ Z r . Considering classes, we have: ^2 ka[ma} = 0 <^> 37 6 T such that ^ kavma = 7. Such a relation happens if and only if all the fca's are equal to 7: the relation results then from n a eE m a =TOInt(.E,V).'Consequently,the group Pic(Int(.E, V)) is the class group of the free Abelian group generated by the [ma]'s with respect to the relation 5Ta6£;[tna] = 0. It is then isomorphic to Zr~l. D

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Corollary 6.2. If V is a discrete valuation domain and E is finite, lnt(E, V) is a Bezout domain if and only if r = 1, that is, E — {a}. Note that, in this case, Int({a), V) — (X — a)K[X] + V and this is a Bezout domain (see [10, § 13, Ex. 5]). The previous proposition may be generalized. Notation. Denote by EQ the set formed by the isolated points of E. Remark 6.3. The subset EQ is finite or countable. This is a consequence of the fact that E is compact: if E is not finite, there is a sequence (7 n )neN of elements of F such that, for each 7 G F, there is an integer n such that 7 < 7^ and, for each 7n, E intersects at most finitely many classes modulo /„ = {x 6 V v(x) > 7n} [6, § 1.2]. One could also say that, since E is precompact, there is a ^-sequence of elements of E [7, Proposition 2.3]. Obviously, such a ^-sequence is dense in E, and then EQ is contained in the countable set formed by the elements of the u-sequence. Proposition 6.4. Assume V is a discrete valuation domain. Then: 1. The ideals ma where a £ EQ generate in J~(IrA,(E,V)) a free Abelian group isomorphic to J^E°\ 2. The classes [ma] (a £ E0) generate in Pic(Int(.E, V)) a free Abelian group isomorphic to Zr-1 if Card(E) — r is finite and to Z^ N ' if E is infinite. Proof. 1. Let G be the subgroup of J(Lr&(E, V}} generated by the m a 's (a G EQ). The image of G by v is clearly the subgroup H of C(E, Z) formed by the functions with finite support contained in EQ since vma (a) = 1 and vma (x) = 0 for x G E\{a}. One knows that v is injective: G and H are isomorphic. 2. If E is finite, this is the previous proposition. If E is infinite, then the [m0]'s form a free basis (consider the morphisms v). Moreover, EQ is countable (Remark 6.3). D Theorem 6.5. Assume E is infinite. Then Pic(Int(.E, V)) is generated by the classes of unitary invertible primes if and only ifV is a discrete valuation domain and E has exactly one accumulation point. In this case,

Proof. Assume first that Pic(Int(£', V)} is generated by the classes of unitary invertible primes. Since there are unitary invertible primes, the ideal m is principal (Proposition 4.1); let ?r be a generator of m. By hypothesis, for each unitary invertible ideal 3, one has:

where the sum is finite, the fcj's are integers and the ttj's are isolated points of E. Equivalently, the value function v? of 3 is constant, except for at most finitely many points (the a^'s), that is: there exists 7 G F such that vj(x) = 7 for x ^ ai and 1*3(0..;) = 7 + kiv(n).

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75

Let c 6 V \ {0}, let e ^ / € E, and consider the unitary invertible ideal 3 = (c(e - f ) , X - /). One has 3(e) = (e - /) and J(/) = c(e - /). If v3 has the previous form, then v?(f) — vy(e] = v(c) must belong to v(n}Z. This implies that v(V \ {0}) = Z, that is, V is a discrete valuation domain. Since E is infinite, there is at least one accumulation point in E. Assume there are at least two accumulation points a and /3. Let k = v(a — /3), let a and b e E be such that v(a — a) > k and i>(6 — /?) > k, and consider the polynomial: f ( X ) = (a- b)5

X -a b —a

i

X -b a —b

- 7T

By construction, the polynomial / has no root in E. If v(x — a) > k, then v(f(x)) = 5k + I = 7 and, if v(x - b) > k, then v(f(x}) = 5fc + 2 = 5. The function w/(x) = — v(/(x)) takes infinitely many times both values -7 and —8. Corollary 4.5 shows that the class of the invertible ideal 25 / is not a product of classes of the form [mQ]. This is a contradiction. Conversely, assume that V is a discrete valuation domain and that a is the only accumulation point in E. Let q be any irreducible polynomial in V[X] without any root in E. There is an open neighborhood U of a in E such that v(q(x)) = v(q(a)) — 7 for x G U. By compactness, E \ U is finite and then the function wq(x) — —v(q(x)) for x e -E is almost constant. It follows from Corollary 4.5 that: a£E\U

Consequently, all the generators [tyg] of Pic(Int(.E, V)) are generated by the [ma]'s. The last assertion of the proposition results from the Proposition 6.4. D Example 6.6. Let p be a prime number, V = Z( p ) and E = {pn \ n e N}. Then E0 = E and E = E U {0}. The group Pic(Int(B, Z (p) )) is the free Abelian group with basis formed by the classes of the ideals mn = {/ e Int(£,Z (p) ) | f ( p n ) e pZp}. Proposition 6.7. Assume that V is a discrete valuation domain and that there are exactly s (s > 1) accumulation points in E. Then there are s — 1 polynomials f i , . . . , / s _i e lnt(E, V] such that Pic(Int(£", V)) is a free Abelian group with basis formed by the classes [tnn] (a € EQ) and by the classes [25/J (1 < i < s). Proof. Let ai, . . . , as be the accumulation points. Then E = EQ U {ai, . . . , as}. We first describe a basis of the Z-module C(E,Z). Obviously, there are functions Pj € C(E,N) such that ^-(ccj) = (5jj for 1 < i,j < s. For each a € E0, let \a be the characteristic function of the subset {a} of E. Now, let w € C(E,Z) and let W0 = W —

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By continuity of u>o and compactness of E, wo(a) ^ 0 for at most finitely many a £ EQ, that is,

where the sum is finite. Thus, the Xa's and the (pj generate C(£',Z). Moreover, these generators are linearly independent. Assume that

aaXaa + yj / _ hjtfj = 0 where ka and hj € Z.

/

Considering the values at a,-, we see that hj = 0 for each j. Then, considering the values at a e EO, we see that fca = 0 for each a. In other words, one could say that the choice of the functions tf>j leads to an isomorphism of groups:

given by

w i It follows from Proposition 3.1 that: — there are unitary ideals 2li,..., 2ls e Ju(lnt(E, V)} such that u
/

kaXa + ^ hjiflj = 1^0 where / e Z.

Then, considering the values at ay, we see that hj = I for each j, and, considering the values at a & EQ, we see that ka = lk'a for each a. Consequently,

Clearly, such a relation exists. It then follows from Theorem 3.2 that the classes [2lj] (1 < j < s) and [tn0] (a 6 E0) form a basis of the Abelian group Pic(Int(E, V ) ) . Instead of the unitary ideals 2l^ one could consider non-unitary ideals 03 j : it results from the Stone- Weierstrass theorem that there are polynomials /i, . . . , fs-i in Int(E, V) such that, for each x € E and each j € {1, . . . , s— 1}, one has v(fj(x}) = V(
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Remarks 6.8. 1. The previous proposition shows again that if Pic(Int(.E, V)) is generated by the classes of unitary invertible primes then E has at most one accumulation point. 2. The polynomial / constructed in the proof of Theorem 6.5 gives an explicit ideal 25 / in the case where there are exactly two accumulation points. 7. Relations We consider the relations between the generators [tyq]. Proposition 7.1. Let qi,... ,qs be polynomials which are irreducible in V[X] and let ki,..., ks 6 Z. Then, s

s

^fcippgj= 0 <=> 37 e F such that ^fc-Xft^))= 7 Vx e B. Equivalently, there exists d € K such that the rational function r = rfHi=i 9i* takes its values on E in V \ m. Proof. Let us write the relation with positive coefficients: i

j

This means that there are /, g € K[X\ such that:

In particular, in K[X] this equality becomes:

xg J] q^ =/Y[ r^ where x<=K*. i

0

Thus, / = xh Y[ qf* and g = h JJrJ* where /i e A"[J5C] and x e AT*. » j For each q, let a, e V \ {0} be such that: 3 J g -—^ 9 3 -fg is an integral (unitary and invertible) ideal. The relation becomes: l\_3%=y]lty where yeK*. i

j

Corollary 1.7 shows that this is equivalent to the existence of an element 7 e F such that: ,

It follows from Corollary 4.5 that this is equivalent to:

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D Corollary 7.2. Let q and r be irreducible polynomials in V[X}.

Then,

[ 37 6 T such that v(q(x)) - v(r(x)) = 7 Vz e E. In particular: — If d is such that v(d) > supxeEv(q(x}) andifg-r 6 dV[X],then [y$g] = [*$r]— Spg is principal if and only if there exists d G K* such that dq(E) C V \ m, that is, v(q(a)) is constant for a & E. Example 7.3. Assume that m is principal generated by TT. Let e i , . . . , es be a set of representatives of the classes of E modulo m. For 1 < i < s, the polynomial qi = (X — ei}2 + TT is irreducible in -K"[X]. Let / = Hi=i <7t- F°r eacn a e -^> one has v(f(d)) = V(TT), and hence, Hi ^Pg» ^s principal, that is, ^i=i [^PgJ — 0Example 7.4. Let *$q be an invertible prime ideal. Then, there is gq e K[X] such that v(q(x)gq(x)) = 0. If gg = xq*1 • • • q*' where the q^s are irreducible in V[X] and x £ K*, then 5

i [^39i] = 0 where fcj € N. The previous example shows that one may only consider relations between the [$}g]'s with positive coefficients. Proposition 7.5. The group Pic(Int(£, V)) is the quotient of the free Abelian group generated by the [tyq] 's by relations of the form: s

ki [q3g.] = 0 /

with

ki e N. \

Such a relation holds if and only if v ((q^ • • • q*")(a)) is constant on E. Example 7.6. When the ideal ma is invertible, it follows from Theorem 5.1 that the class [ma] is a linear combination of the ppg]'s. For instance, let V = Z(2) and E = {2} U {3 + 4fc | k e Z}. Then, 2 is isolated in E and m2 is invertible. The polynomial f ( X ) = (X~1)2(X~2) + 2 belongs to Int^,^); moreover, /(x)Z (2 ) = m 2 (x) for every x e E. The polynomial 2/ = X2 - 3X + 6 is irreducible in QLX] and has no root in E = (2}U{3 + 4Z(2)}. Thus, we have [m0] = — PPx 2 -3X+e] (see the algorithm following Theorem 5.1). In fact, generalizing particular results of the previous section (Propositions 6.1 and 6.7), we are going to see that Pic(Int(S, V)) is free as soon as v is discrete. Proposition 7.7. If V is a discrete valuation domain, then the Abelian group Pic(Int(£, V)) is free. Proof. We generalize a proof given in [11]. We may assume that E is not finite. Note first that C(E,Z) is countable: since E is compact, for each element (p of C(.E, Z) there is a smallest h e N such that v(x — y) > h implies (p(x) = (p(y). For

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a fixed h, E meets k(h) classes modulo mh, so that j}j£j is a basis of C(E, Z), then 1 = ^i) -> Pic(Int(E,D)) -+ © m6max(D) Pic(Int(£, Dm)) -> 0. For instance, it follows from Proposition 7.7 that: Proposition 8.1. The group Pic(Int(J?, £>)) is isomorphic to the direct sum of Pic(£>) and of a free Abelian group. If D is the ring of integers of a number field, Pic(J9) is a torsion group, and hence: Corollary 8.2. If D is the ring of integers of a number field, then Pic(Int(£1, D}) is a free Abelian group if and only if D is a principal ideal domain. We now focus our study on the unitary primes. Since, for each maximal ideal m of D, one has Int(.E,D) m = Int(J5, An), the unitary prime ideals of lut(E,D) are of the form: mQ = {/ e Int(£, D) \ f(a] e

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where m G max(D), a G Em and Em denotes the completion of E in the m-adic topology. Moreover, ma = rag if and only if a = j3. Proposition 8.3. The unitary prime ideals oflnt(E, D) that are finitely generated are the maximal ideals

ma = {/ G Int(E,D) | /(a) G mA7} where m is a maximal ideal of D and a is an element of E which is isolated in E in the m-adic topology. Notation. Let -Eo.m denote the subset of E formed by the elements of E that are isolated in E in the m-adic topology. Proof. If the ideal ma of Int(.E, A) is finitely generated, then the ideal (m a ) m = {/ G Int(£, An) | /(a) G mDm} of lnt(E,Dm) is also finitely generated, and hence, it follows from Proposition 4.1 that a belongs to J5o,m- Conversely, if a belongs to Eb.mi then it still follows from Proposition 4.1 that the ideal (m a ) m of Int(jE/, An) is generated by finitely many elements /i, . . . ,/ r that can be chosen in m a . Then, the ideal ma of lnt(E, D} is generated by m and / i , . . . , / r : let 21 = (m, /i, . . . , / r ) li\t(E, D); by construction, one has (m a ) m = 2lm and, for each maximal ideal n of D distinct from m, (m a ) n = Iut(E,Dn) = 2ln; consequently, ma = 21. D Proposition 8.4. In the group J'(Int(E, D)) of invertible ideals oflnt(E,D), the unitary invertible prime ideals ma generate a free Abelian subgroup and the ideals ma (m G max(A), a G -Eo,m) form a basis of this subgroup.

Proof. Assume that we have: I]

( II m£

mGmax(D) \i€.Eo,m

where ka^m G Z and all but finitely many of the ka,m's are equal to zero. By localization with respect to some m we obtain: [] m*-™ Int(E, An) = !nt(E, D It follows from Proposition 6.4 that all the fca|IT1's are equal to zero.

D

We now distinguish the case where E is finite and the case where it is not. Proposition 8.5. If E is finite, then: 1. The monoid ,Ju(liit(E , D)) is a free Abelian monoid with basis formed by the ideals ma where m G max(.D) and a £ E. 2. The group Pic(Int(J5, £>)) is generated by the classes [ma] (m G max(D), a&E).

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Proof. The second assertion is a straightforward consequence of the first one. One knows [4, Exercise IV. 1] that the unitary invertible ideals of lnt(E,D) are of the form:

where

and the o,-'s are nonzero ideals of D. Obviously, if a-,- = Hm™^"1 where the m's are maximal ideals of D, then 3 = Ylm IIj=i ma^'m , and hence, the ma's generate Ju(Int(E , D)) . That the ma's form a basis results from the previous proposition. D

Remark 8.6. If E is finite, one easily see that the image of the elements of Pic(D) in Pic(IiLt(E , D}) are linear combinations of the classes [ma] since Yla&Ema = m!nt(E,D), and hence,

Corollary 8.7. If E is finite and D is a principal ideal domain (with finite residue fields'), we obtain a basis of the free Abelian group Pic(Int(.E, £>)) in the following way: let ao be a fixed element of E and let E\ = E \ {ao}, then the classes [ma] (m € max(£>) and a & EI) form a basis of Pic(lnt(E , D)). Proof. Since D is a principal ideal domain, one has: Pic(Int(£, £»)) = e memax( D) Pic(Int(£, £» m )) and Proposition 6.1 says that Pic(Int(.E; -Dm)) admits for basis the classes [(m a ) m ] where a e E\. D Analogously, the proof of Proposition 6.4 shows the following: Proposition 8.8. If E is infinite, then the classes [ma] (m e max(D), a 6 Eo,m) generate a free Abelian subgroup o/Pic(Int(£', D)). Finally, by globalizing Theorem 6.5 we obtain: Theorem 8.9. Let D be a Dedekind domain with finite residue fields and let E be an infinite subset of D. The Picard group oflnt(E,D) is generated by the classes of unitary invertible prime ideals ma (m e max(D), a £ Eo>m) if and only if: 1. D is a principal ideal domain, 2. for each m £ max(D), E has exactly one accumulation point in the m-adic topology. In this case, Pic(Int(.E, D)) is a free Abelian group with basis formed by the [ma] 's.

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Proof. Assume that Pic(Int(E, £>)) is generated by the [ma]'s. Then Pic(Int(E, D)) is a free Abelian group (Proposition 8.8), and hence, Pic(D) is also a free Abelian group (Proposition 8.1). If D is the ring of integers of a number field, it is then easy to see that D is a principal ideal domain. But this is still true in the general case. Let a be a nonzero ideal of D and suppose that fca,m[m0 mSmax(r') a€£o,m

that is,

where / e K(X)* . Localizing with respect to some m, we obtain: aInt(E,D m ) = / J] m*-" lnt(E,Dm). Since alnt(E,Dm) is principal, we may write (in Pic(Int(£;,D m )j):

Proposition 6.4 shows that all the fcaim's are equal to zero. Finally, [a lnt(E, D)} = 0, aInt(£',D) is a principal ideal of Int(E, D) and a is a principal ideal of D. This is the first condition. Moreover, if the [tna]'s generate Pic(Int(.E, £>)) then, for each maximal ideal m of D, the [(m0)m]'s generate Pic(Int(.E, £> m )), and then, it follows from Theorem 6.5 that E has exactly one accumulation point in the m-adic topology. Conversely, if D is a principal ideal domain, then Pic(Int(£, D ) ) ~ © memax(D) Pic(Int(£, £> m )). It follows still from Theorem 6.5 that the [(m 0 ) m ]'s generate Pic(Int(.E, An)), and hence, the [m0]'s (m e max(.D), a € Eotm) generate Pic(Int(£', D)). D Examples 8.10. Both conditions of Theorem 8.9 are satisfied for D = Z and for each of the following subsets E: 1) Let pn be the nth prime number, let an = Tl'k=iPJk~k^ and let E = {an \ neN}, 2) E = {n! n e N}, 3) E = {(nl)s n € N} where (nl)s is the generalized factorial associated to the subset S of Z [1]. References [1] M. BHARGAVA, P-orderings and polynomial functions on arbitrary subsets of Dedekind rings, J. Reine Angew. Math. 490 (1997), 101-127. [2] N. BOURBAKI, Commutative Algebra, Chapter VI, English transl., Springer-Verlag, Berlin, Heidelberg, New York, 1984. [3] P.-J. CAHEN, Polynomes a valeurs eniieres, These, Universite Paris XI, Orsay, 1973. [4] P.-J. CAHEN AND J.-L. CHABERT, Integer-Valued Polynomials, Amer. Math. Soc. Surveys and Monographs, 48, Providence, 1997.

The Picard group

83

[5] P.-J. CAHEN AND J.-L. CHABERT, Skolem Properties and Integer-Valued Polynomials: A Survey, in Advances in Commutative Ring Theory, 175-195, Lecture Notes in Pure and Appl. Math. 205, Dekker, New York, 1999. [6] P.-J. CAHEN, J.-L. CHABERT, AND THOMAS K. A. LOPER, High dimension Priifer domains of integer-valued polynomials, J. Korean Math. Soc. 38 (2001), 915-935. [7] J-L. CHABERT, Generalized factorial ideals, The Arabian Journal for Science and Engineering, t. 26 (2001), 51-68. [8] J.-L. CHABERT, S. CHAPMAN AND W. SMITH, Algebraic Properties of the Ring of IntegerValued Polynomials on Prime Numbers, Comm. algebra 25 (1997), 1945-1959. [9] M. FONTANA AND S. GABELLI, Priifer domains with class group generated by the classes of the invertible maximal ideals, Comm. Algebra 25 (1997), 3993-4008. [10] R. GILMER, Multiplicative Ideal Theory, Dekker, New York, 1972; rep. Queen's Papers in Pure and Applied Mathematics, vol. 90, Queen's University, Kingston, Ontario, 1992. [11] R. GILMER, W. HEINZER, D. LANTZ AND W. SMITH, The ring of integer-valued polynomials of a Dedekind domain, Proc. Amer. Math. Soc. 108 (1990), 673-681. [12] D. L. McQuiLLAN, Rings of integer-valued polynomials determined by finite sets, Proc. Roy. Irish Acad. Sect. A 85 (1985), 177-184. [13] E. SPECKER, Additive Gruppen von Folgen ganzer Zahlen, Portugaliae Math. 9 (1950). 131140.

A Note on Cotilting Modules and Generalized Morita Duality Robert R. Colby Department of Mathematics, The University of Iowa, Iowa City, IA 52242 U.S.A Riccardo Colpi Dipartimento di Matematica Pura e Applicata, Universita di Padova, Via Belzoni 7, 35131 Padova, Italy Kent R. Fuller Department of Mathematics, The University of Iowa, Iowa City, IA 52242 U.S.A

Let $UR be a bimodule, and let A(/R = Hom/j(_, U) and As{/ = Hom,5(_, U) to obtain a pair of contravariant functors At/ R : Mod-R ^ 5-Mod : As!7, both of which we shall, when convenient, simply denote by Aj/. A right .R-module or a left S'-module M is Ay-reflexive if the evaluation map SM '• M —> A^M is an isomorphism, M is U-torsionless if there is an embedding M <—> UA of M in a product of copies of [7, and M is U-torsion if Ay(M) = 0. Also we let TUR — Ext^j(_, U) and Tsu = Ext;j (_,{/) to obtain another pair of contravariant functors TUR : Uod-R <=± 5-Mod : TsU both of which we often refer to as TU- In certain cases (see [9], [15]) there are natural isomorphisms 7M : r^-M —> M for M in a particular subcategory of Mod-JJ or S'-Mod. When this occurs, we shall say that M is Tu-reflexive. Following [10], and [8 , [9], [11], UR is a cotilting module whenever a right -R-module M is Utorsionless if and only if TjjR(M) — 0. Let K be the center of the artin algebra R, let CK = E(K/J(K)), and let D : Mod-R ^ .R-Mod : D denote the artin algebra dual, so that DM = B.omK(M,C). Given a tilting bimodule nVs and setting sUn = DV, it follows from Proposition 2.2 and Theorem 3.3 in [2] that UR is a typical finitely generated cotilting module Colby wishes to express his gratitude for the hospitality of the University of Iowa where he is an independent scholar. 85

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with S = End([/R). Then as pointed out in [4] the Tilting Theorem [3], [13], [6] dualizes for artin algebras to include the facts that a finitely generated right R- or left S-module M satisfies Tv o Af/(M ) — 0 = AC/ ° IV (M), is Ay-reflexive if it is [/-torsionless, and is IV -reflexive if it is [/-torsion. Here we shall prove that, in either sense, these are the only reflexive modules. In [4] and [5] Colby introduced the notion of a generalized Morita duality (GMD) as a duality induced by a faithfully balanced bimodule S£/R such that the [/-reflexive modules in Mod-/? and S'-Mod are closed under submodules and extensions. In [5, Corollary 10] he proved that, if R and S are artin algebras and sUfi induces a GMD, then UR is a (finitely generated) cotilting module; and he speculated that, conversely, if UR is a finitely generated cotilting module over an artin algebra R and S — End ([/#), then $UR induces a GMD. That this is indeed the case is a consequence of our theorem. Theorem 1. Let R be an artin algebra and let UR be a finitely generated cotilting module with S = End ([/#). If M is either a right R-module or a left S -module, then 1. M is A £/ -reflexive if and only if M is finitely generated and U -torsionless; 2. M is Tu-reflexive if and only if M is finitely generated and U -torsion. Proof. In particular the conditions in each part of the theorem are sufficient. To see that the conditions are each necessary let R\VR = D(RRR), so that RWR is a balanced two-sided injective cogenerator. Also, let H = HomR(V; _) : Gen( R l/) «=i Ker Torf (V, .) :(V®S-)=T be the equivalence induced by the tilting bimodule RVs (see [6], for example). Now we have isomorphisms Homfl(V, W) = Hornby, RomK(R, C)) K((R®RV),C)

= SUR. Let AaH : Mod-R ^ S'-Mod : AsU denote the sC/^-dual, and let AW : Mod-R ^ .R-Mod : A H / denote the ^W^j-dual. Then, since S&R is a cotilting module we know that every finitely generated [/-torsionless module is [/-reflexive, and since R is artinian and RWR induces a Morita duality, the W-reflexive modules are the finitely generated ^-modules. Moreover, by adjointness we have natural isomorphisms

(1)

A

for each M e Mod-R, and

(2)

&

A Note on Cotilting Modules and Generalized Morita Duality

87

for each N e 5-Mod, as observed in [11, page 284]. Now according to [14, Proposition 3.8] UR is product complete, so if MR is t/-torsionless, then there is an embedding 0 -> M -* U(A) of M into a direct sum of copies of U. But then we have an epimorphism A

RV

* HomR(!7, W}A -> A WR (M) - 0,

and since RVA e Gen(/jVr) [12, Lemma, page 408] so is A M / R (M) whenever M# is [/-torsionless. Thus for any such MR we finally have A sU o AUR (M) * AR;y oToHo Aw/R (M) since, by the Tilting Theorem, To H ~ IcenUV)- Thus MR if is A j/- reflexive, then MR is W-reflexive and U- torsionless, or equivalently, MR is finitely generated and J7-torsionless. (2) Now from Equation (1) we see that AV^R (M) € Ker if whenever A[/R (M) = 0. Also, as discussed in [11, page 284], if H' = ExtJj(V,_) and T" = Torf (V,_), there are natural isomorphisms

(3)

I

for M 6 Mod-JR and

(4)

I

for AT e S-Mod. But now we have, for any M e Mod-J?,

A

W

o T' o

since, by the Tilting Theorem T'oH' ^ lKerH- Thus if M 6 Mod-.R is T-reflexive, then M is VF-reflexive, and hence finitely generated. But then IV (M) is finitely generated and AyM = A[/r^(M) = 0. Since RVs is a tilting bimodule, by symmetry the same results hold for right S'-modules. D Corollary 2. Lei .R fee an artin algebra and let UR be a finitely generated module with S — End(C/n). Then UR is a cotilting module if and only if sUR induces a generalized Morita duality. Proof. If UR is a cotilting module then the finitely generated torsionless modules are closed under (submodules and) extensions by [7, Lemma 4.6]. D Given any cotilting bimodule SUR (in the sense of [9]), there exist largest full abelian subcategories of CR of Mod-/? and sC of S-Mod admitting torsion theories such that any module M in CR or, respectively, in gC has a torsion part that is IVreflexive and a torsion-free part that is Ay-reflexive (see, for example [8, Theorem 1.5]). In this context we also have the following

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Corollary 3. Let R be an artin algebra and let UR be a finitely generated cotilting module with S = End(£/#). Then CR and sC consist exactly of the finitely generated right R- and left S-modules. Proof. The class CR contains all finitely presented modules, as asserted in [8, Theorem 1.5 d)]. Conversely, from [8, Theorem 1.5 b), c)], it follows that any module M in CR (or, respectively in gC) is an extension of a IV-reflexive module r^(M) by a Ay-reflexive module A^(M). Since they are both finitely generated by Theorem 1, M is too. D References [1] F. W. Anderson and K. R. Fuller. Rings and Categories of Modules. Springer-Verlag, Inc., New York, Heidelberg, Berlin, second edition, 1992. [2] L. Angeleri Hugel. Finitely cotilting modules. Comm. Algebra, 28(4): 2147-2172, 2000. [3] S. Brenner and M. C. R. Butler. Generalizations of the Bernstein-Gelfand—Ponomarev reflection functors. Springer-Verlag LNM, 832: 103-170, 1980. [4] R. R. Colby. A generalization of Morita duality and the tilting theorem. Comm. Algebra, 17(7): 1709-1722, 1989. [5] R. R. Colby. A cotilting theorem for rings. In Methods in Module Theory, pages 33-37. M. Dekker, New York, 1993. [6] R. R. Colby and K. R. Fuller. Tilting, cotilting, and serially tilted rings. Comm. Algebra, 18: 1585-1615, 1990. [7] R. R. Colby and K. R. Fuller. Weak Morita duality. Comm. Algebra, to appear. [8] R. Colpi. Dualities induced by cotilting bimodules. Algebra Conference Venezia 2002 Proceedings, to appear. [9] R. Colpi. Cotilting bimodules and their dualities. Interactions between ring theory and representations of algebras (Murcia). In Lecture Notes in Pure and Appl. Math., volume 210, pages 81-93. Marcel Dekker, New York, 2000. [10] R. Colpi, G. D'ljste and A. Tonolo. Quasi-tilting modules and counter equivalences. J. Algebra, 191: 461-494, 1997. [11] R. Colpi and K. R. Fuller. Cotilting modules and bimodules. Pacific J. Math., 192: 275-291, 2000. [12] R. Colpi and C. Menini. On the structure of *-modules. J. Algebra, 158: 400-419, 1993. [13] D. Happel and C. M. Ringel. Tilted algebras. Trans. Amer. Math. Soc., 274: 399-443, 1982. [14] H. Krause and M. Saorin. On minimal approximations of modules. Contemp. Math., 299: 227-236, 1998. [15] A. Tonolo. Generalizing Morita duality: a homological approach. J. Algebra, 232: 282-298, 2000.

Dualities Induced by Cotilting Bimodules Riccardo Colpi Dipartimento di Matematica Pura e Applicata, Universita di Padova, Via Belzoni 7, 35131 Padova, Italy colpiSmath.unipd.it Abstract. The theory of cotilting bimodules can be considered as a far reaching generalization of Morita dualities, by means of a (local) dual form of the celebrated Brenner and Butler Theorem for tilting modules. More precisely, given arbitrary rings R and S, a cotilting bimodule sUR gives rise to two torsion pairs (T&, FR) in Mod-fJ, (sT, s3~) in S-Mod and to a pair of dualities

TR 3 yR ^ y c SF, A s

TRDXR^±r SXC ST

such that the subcategory CR of Mod-R (resp. sC of 5-Mod) consisting of modules with torsion-free part in 3^H and torsion part in XR (resp. in sy and sX) is quite representative in Mod-fi (resp. in S-Mod). In this paper we first discuss some notions of linear compactness related to the torsion theory cogenerated by U: these notions on one hand guarantee that the subcategories CR and sC are large enough (i.e., they contain all finitely generated modules), on the other hand they serve to characterize the reflexive torsion-free modules. Then we introduce a natural way to obtain cotilting (bi)modules from tilting ones, showing that frequently we obtained desirable compactness conditions for the ring(s) and the (bi)module U.

1. The Cotilting Theorem Given two arbitrary associative rings with unit R and S, we denote by Mod-.R and 5-Mod the category of all unitary right R- and left S'-modules, respectively. Similarly, mod-/? (resp. S'-mod) and fpmod-R (resp. S'-mod) will denote the finitely generated and the finitely presented right R-modules (resp. left S'-modules). All the classes of modules that we introduce are to be considered as full subcategories of modules closed under isomorphisms, and all the functors are additive functors. Given a module U, we denote by Cogen(C7) the class of all modules cogenerated by

89

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R. Colpi

U, that is all the modules M such that there exists an exact sequence 0 —> M —> Ua, for some cardinal a. We denote by Rej;y(—) the reject radical, defined by the position Rej(7(M) = n{Ker(/) | / e Homfi(M, U)}, i.e., the least submodule L of M such that M/L belongs to Cogen(fJ). The torsion theories that we consider are generally non-hereditary. For further notation, we refer to [1], [28] and [30]. Definition 1.1. A cotilting module UR is a right R-module which satisfies the condition Cogen(t/R) = KerExtJj(—, [/R) or, equivalently, the conditions (see [14, Proposition 1.7]) (i) injdim(£7 B ) S'-Mod and A,F: S-Mod ->• Mod-/?, defined as follows: A = Horn? (-,
and

F = Ext}(-, SUR)

where ? = -R or S. For any right .R-module (respectively, left S'-module) M, the evaluation map SM is defined by (5M: M-> A 2 (M),

x

„ [£ „ £(x)l for all £ e A(Af).

Since Ker SM = Rejy(M), the morphism SM is monic if and only if M € Cogen(fJ). In this case M is called U-torsionless. If SM is an isomorphism, M is called Ureflexive. Note that the evaluation maps are the two units of the adjunction on the right of A with A. Unfortunately, the functors F are not in general adjoint to each other, i.e., there is no canonical map from F 2 (M) to M which takes the role of 5. In other words it is not possible, in general, to introduce a natural notion of F-reflexivity.

Dualities Induced by Cotilting Bimodules

91

Definition 1.3. Given a bimodule sf/R, we denote by 5^ and 6^ respectively the 0-th and 1-st left derived of the evaluation map 5. Then we introduce the classes of right fl-modules C°R = [M <E Mod-.R | M ^ L/N for some [/-reflexive L, N 6 Mod-ft} and

C]j = {M e Mod-.R 6M and 5M are isomorphisms}. The classes of left 5-modules sC° and sCl are denned similarly. The problem to find a "maximal" subclass of modules where the functors F are mutually adjoint on the left was investigated by A. Tonolo in [31], where he proved the following result: Proposition 1.4. Let $UR be a cotilting bimodule. Then the functors F restricted to C^ and sC1 are adjoint on the left to each other via natural maps 7 : F2 —> 1. Collecting together the results [13, Theorem 6], [23, Theorem 11 and Proposition 12] and [31, Corollary 2.10], we can state the following "Cotilting Theorem": Theorem 1.5. Let $UR be a cotilting bimodule. Then there are full abelian subcategories CR C Mod-f? and $C C 5-Mod such that, setting XR = CR n Ker A and yR = CR fl KerF (and similarly for $X and 5^), we have: (a) the two pairs (XR, VR) and (s%, sV) are torsion theories in CR and $C respectively; (b) for any M <E CR (resp. in SC), A(M) G SV and F(M) € SX (resp. A(M) £ yR and F(M) G XR), and the short sequence 0 -» F 2 (M) -^ M -^U A 2 (M) -» 0

is exact;

A r (c) A f ^ and FfA" are exact functors, defining dualities yR ^^ 5^ and XR ^=± A r s*\ (d) fpmod-J? C C°R C C]j C Cfl and S'-fpmod C SC° C sC1 C SC; moreover the classes C^, C^, $C° and $Cl are closed under finite direct sums and kernels and cokernels of morphisms between modules belonging to them.

Examples 1.6. a) Morita bimodules sUn are exactly the injective cotilting bimodules. In this case (Ker A, Ker F) = (0,Mod-.R), and the classes C = y consist of the linearly compact modules (see [27]). In particular RR, $S, UR and $U are all linearly compact, and U is finitely cogenerated on both sides. b) (G. D'Este [18]) The bimodule of the form RRR - where R is the algebra ( o \ ) ' f°r an arbitrary infinite dimensional k- vector space V - is a cotilting bimodule. Here R is neither linearly compact nor finitely cogenerated, and CR is neither closed under submodules or factors ;nor contains the finitely generated modules. Moreover CR = fp mod-.R and ^ consists of the finitely generated projective right ^-modules.

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R. Colpi

c) (communicated by K. Fuller, 1999) If R is any ring which is hereditary and perfect on both sides, then RRR is a cotilting bimodule. Note that the previous example b) is of this type. d) It is well known that the finitely generated cotilting (bi)modules over an Artin algebra are exactly the duals of the finitely generated tilting (bi)modules. Recently, in [8]) they are proved to coincide with those modules which induce (over an Artin algebra) a generalized Morita duality in the sense of [7]. In this case CR = mod-.R. e) ([15, Remark 3.8]) Any cotilting triple (S,U,R) in the sense of [6, §2] over rings of finite representation type S and R is a cotilting bimodule. f) ([15, §3]) Any linearly compact, Noetherian serial ring R has a Morita selfduality. If R WR induces such a duality, then for any finitely generated tilting module RV, with S = Eind(nV), the S-R bimodule Hom#(F, W) is a cotilting bimodule, which is generally infinitely generated. Next, as an application of Theorem 1.2, we show that similarly to the case of a Morita duality, no infinite sum of reflexive modules can be reflexive. The following result is an adaptation of an argument due to B. Osofsky (see, for instance, [28, Proposizione 1.11, page 308]): Lemma 1.7. Let $UR be a bimodule and let {M\ : A € A} be an infinite family of non-zero U -reflexive left S-modules. //Hom fl (n AM A /© AM A , U) ^ 0, then ®M\ is not U -reflexive. Proof. First remark that A(©M A ) = H AMA canonically. Let now 0 ^ V e Homfi(n AM A / © AM A , U). By composition with the canonical projection we obtain a map 0 ±

= <5®MA (x) for some element 0 ^ x = (o: A ) Ae A in ©MA. Let 0 ^ XM G MM. As M^ is reflexive by assumption, we get SM^XH) ¥" 0, so that there exists £ 6 AMM < ©AM\ such that 0 ^ ( ( a X O = £(zi = £ z — ^eMO^XO = v ( > a contradiction. D Proposition 1.8. Let $UR be a cotilting bimodule. Then no infinite direct sum of non-zero reflexive modules is reflexive. Proof. Thanks to the previous Lemma, it is enough to show that, given an infinite family of non-zero [/-reflexive left ^-modules {MA : A 6 A}, we have Homjj(FJ AM A / © AM A , U) ^ 0. Since ©AMA is a pure submodule of FJ AMA and UR is pure-injective, F(FJ AM A /© AM A ) embeds into F(FJ AM A ) = 0. Therefore F(I] AM A / © AM A ) is zero. On the other hand, FJ AM A /© AMA is clearly different from zero, and so the thesis follows from the condition 1.1 iii) D

Dualities Induced by Cotilting Bimodules

93

2. Linear Compactness If $UR is a Morita bimodule, then Theorem 1.5 assumes a much simpler form. Indeed, as we have already remarked in Example 1.6 a), the functor F vanishes, therefore KerF is respectively Mod-fl and S'-Mod and Ker A = 0. In this case both the classes CR and sC, consisting of the linearly compact modules, are closed under extensions, submodules, factor modules and they contain any finitely generated module. In particular gUp. is linearly compact and finitely cogenerated on both sides. In general cotilting bimodules are neither finitely cogenerated nor linearly compact, and the class C is neither closed under submodules nor factors, nor does it contain every finitely generated module: this is for instance the case quoted as Example 1.6 b). In this section we introduce three different notions of linear compactness connected with the (bi)module U, and we relate them with the classes C and y. Recall that a module M is linearly compact, briefly Ic, if for every inverse system of epimorphisms {M -^-> MX}, the map p = limp\ is as well an epimorphism. To any cotilting (bi)module U is associated a torsion theory (Ker A, Ker F), with a "large" torsion-free class KerF = Cogen([/) (in fact, it contains all the projectives) and a "negligible" torsion class Ker A. A map M —> N between two torsion-free modules is called almost epic if its cokernel is negligible, i.e., Coker/ e Ker A. So, there are at least three natural ways to generalize linear compactness: Definition 2.1. Let U be a (bi)module, and let M 6 Cogen([/). a) M is called U-linearly compact (see [20]), briefly U-lc, if for every inverse system of epimorphisms {M —^-> MA}, with each M\ e Cogen([/), the map p — limp\ is as well an epimorphism. b) M is called U-torsionless linearly compact (see [21]), briefly U-tl-lc, if for every inverse system of almost epics {M -^ M\}, with each M\ e Cogen([/), the map p = limp\ is as well almost epic. c) M is called U-reflexive linearly compact, briefly U-refl-lc, if for every inverse system of epimorphisms {M -^-» MA}, such that every KerpA is [/-reflexive, the map p = limp A is as well an epimorphism. Of course if M is [/-torsionless and linearly compact, then it is both [/-linearly compact and [/-reflexive linearly compact. Moreover, all these notions coincide with the classical linear compactness, whenever U is a Morita bimodule. Similarly, in case of a finitely generated cotilting module over an Artin algebra (see Corollary 2.6 and Example 2.7 b)) all these linear compactness conditions are equivalent to the finite length condition. Nevertheless, even if $UR is a cotilting bimodule, [/-torsionless linear compactness and usual linear compactness are in general two independent notions. For instance the ring R — U described in the Example 1.6 b) admits exactly two indecomposable projective modules: the first one is a simple, hence linearly compact,

94

R. Colpi

module which is not [/-torsionless linearly compact; the second one, conversely, is a [/-torsionless linearly compact module which has infinite socle, hence it is not linearly compact (see [18]). Moreover, considering M = U = R = S again in Example 1.6 b), we get a [/-reflexive module M which is neither linearly compact nor [/-torsionless linearly compact. Nevertheless, any [/-reflexive module is both [/-linearly compact and [/-reflexive linearly compact, thanks to the next two results. The first one is a combination of Theorem 1.2 and [25, Theorem 2.11 and Proposition 2.12]: Theorem 2.2. Let S&R be a cotilting bimodule. equivalent for any module MR £ KerF:

The following conditions are

(a) MR is U -reflexive; (b) MR is U-lc and every finitely U-cogenerated factor of M is U -reflexive. Once more, using Theorem 1.2 we can prove the following: Proposition 2.3. Let sUR be a cotilting bimodule. Then every U-reflexive module is U-refl-lc.

Proof. Let M be a [/-reflexive right .R-module, and let {M —^> MA) be an inverse system of epimorphisms, where K\ = Ker f\ is [/-reflexive for each A. Applying the functor lim to the exact diagram

0

> AM.

>• AM

s> AJG,

s- FMX

>0

we get the two exact sequences (1)

0 -> limCA -* Um AK\ -> \JmTMx -> 0

(2)

0 -> lim AM* -» AM -»lim CA -> 0

where limFMA € Ker A and limAi^AS limCA 6 KerF, because U is pure-injective (see Theorem 1.2) and so KerF is closed under direct limits (see [25, Corollary 1.4]). Therefore, applying Horns(—,[/) to (1) and (2), we get the commutative diagram

Dualities Induced by Cotilting Bimodules

95

with exact rows

0

0

0

• lim A2KA

»• lim ACA

> limF 2 M A =* r(limFM A )

• lim /iTA

>M

> lim MA

*• lim AC*

s> A 2 M

> lim A2MA

lim/A

0

>0

2

where limF M A = F(limFM A ) since U is pure-injective (see [3, Proposition 10.1]), and the third column in the diagram is obtained applying the functor lim to the exact sequence 0 -> T2MA -4 MA -> A2MA -> 0. Finally, we see that lim/A is an_epimorphism by a diagram chasing.

D

Conversely, [7-torsionless linear compactness implies J7-reflexivity. Actually, in [15, Theorem 1.8] the following result is proved: Theorem 2.4. Let sUn be a cotilting bimodule. The following conditions are equivalent for any module MR S KerF: (a) MR is U-tl-lc; (b) MR is U-reflexive and AF./V = 0 for any factor N of AM; (c) all the submodules of A(M) are U-reflexive. Combining 2.3 with 2.4 we immediately get the following Corollary 2.5. Let $UR be a cotilting bimodule, and let MR be U-tl-lc. Then AM is Ic. As an application of the previous results, we consider the case of a cotilting bimodule over arbitrary rings which induces a generalized Morita duality in the sense of [7], i.e., such that the class of (7-renexive modules is closed under submodules. The following corollary can be considered a completion of [15, Corollary 1.9] and [25, Corollary 2.14]: Corollary 2.6. Let $UR be a cotilting bimodule and lei y^ (resp. $y] be the class of the U-reflexive right R- (resp. left S-) modules. Then yR is closed under submodules if and only if sy = {-W G 5*-Mod M is U-tl-lc}. In this case the following conditions are equivalent for any M € Mod-.R:

(a) M e yR, (b) M is U-lc, (c) M is U-torsionless and Ic.

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Proof. The first statement is an immediate application of Theorem 2.4. Now, the assumption that yR is closed under submodules implies that every finitely URcogenerated module is {/-reflexive. Therefore the equivalence "(a) <=> (b)" is a consequence of Theorem 2.2. Since "(c) =>• (b)" is straightforward, it remains to show that "(a) => (c)". Assuming that M is [/-reflexive, we have that AM is [/-reflexive too, hence [7-tl-lc. Finally, M = A 2 M is linearly compact, thanks to Corollary 2.5. D Examples 2.7. a) In 24], F. Mantese studied cotilting modules satisfying the further condition ~E~x.tl(E(U),U) = 0. These are exactly the cotilting modules cogenerating a hereditary torsion theory: for that reason they axe called hereditary cotilting modules. In [24, Proposition 1.12] it is proved that any hereditary cotilting bimodule satisfies the hypothesis of Corollary 2.6. b) The previous Example 1.6 d)—i.e., a finitely generated cotilting (bi)module over an Artin algebra—satisfies the hypothesis of Corollary 2.6 on both sides. In this case the [/-reflexives right R- and left S-modules are exactly the finitely generated [/-torsionless modules. 3. When finitely generated modules play a role A Morita Duality is characterized from the fact that the two mutually dual subcategories CR of Mod-/J and $C of S'-Mod are both finitely closed (i.e., closed under submodules, factor modules and extensions) and generating (i.e., they contains all finitely generated modules). This guarantees that the categories CR and sC are "large" abelian fuh1 subcategories of the whole categories of modules. Looking at the more general case of a cotilting bimodule, as described in Theorem 1.5, one immediately see that the abelian categories CR and gC are quite mysterious. There is no explicit description of them. They are "approximated from below" by means of the abelian full subcategories C^ and C^ of Mod-R, and sC° and sC1 of S'-Mod, which contain all the finitely presented modules. This is probably one of the reasons for which cotilting bimodules first appeared in contexts where the finitely presented modules play an important role, such as finite dimensional algebras or, more generally, Noetherian rings. Indeed, in case of Noetherian rings both the classes CR and gC contain the classes of all submodules of finitely generated modules, and these, as in case of Morita dualities, represent finitely closed and generating subcategories of the whole categories of modules. In this section, see Theorem 3.3, we discuss when this occurs for arbitrary rings. The following result is proved in [15, Proposition 1.6]: Lemma 3.1. Let UR be a cotilting module and let S = End([/#). Then UR is U-tl-lc if and only if AF vanishes on every cyclic left S-module. Lemma 3.2. Let UR be a cotilting module, S = End([/#), and assume that mjdim.sU < I . Then the subclass KerAF of S'-Mod is closed under submodules and extensions.

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Proof. I f O — > L —» M —> N —> 0 is any exact sequence in 5-Mod, since inj dim g?7 < 1 we get the exact sequence TN —>• FM —> TL —> 0. We can conclude, using the fact that Ker A is, by assumption, a torsion class in Mod-/?. D Theorem 3.3. Let sUR be a cotilting bimodule. The following conditions are equivalent: (a) UR is U-tl-lc; (b) every left ideal of S is U-reflexive; (c) AFM = 0 for every M e S-mod (resp. for every cyclic M e S-Mod); (d) sC° contains all the submodules of finitely generated modules; (e) S-mod C gC. If these conditions hold true, then $S is linearly compact. Proof, (a) <=> (b). It follows from Theorem 2.4 applied to UR. (a) <=> (c). It is a consequence of Lemma 3.2 and Lemma 3.1. (c) =3- (d). Since any submodule of a finitely generated module is obviously of the form N = M/L, for some L < M < sSn, it is enough to verify that any submodule of sSn is {/-reflexive. Let / be any left S-submodule of Sn. By assumption, AF(5"/J) = 0. Since Sn = A(t/£) and U% is fj-reflexive, from Theorem 2.4 we obtain (d). (d) => (e). Straightforward. (e) ==> (c). Prom Theorem 1.5. b) we have that AFM = 0 for every M € gC. The last assertion is a consequence of Corollary 2.5. D Combining 3.3 with Theorem 1.5 we immediately get the following Corollary 3.4. Let sUR be a cotilting bimodule. If S is left coherent, then UR is U-tl-lc and 38 is linearly compact. Theorem 3.3 shows that gC is large enough to contain a finitely closed and generating subcategory of 5-Mod exactly when UR is [7-tl-lc. This corresponds to the fact that every Morita bimodule is linearly compact. Since the existence of a Morita bimodule forces the rings to be linearly compact, it seems quite natural to analyze cotilting bimodules U over [7-tl-lc rings. Not surprisingly, in this case we get a nice description of [/-reflexive modules: Proposition 3.5. Let sUn be a cotilting bimodule, and assume that sS is Utorsionless linearly compact. Let M e Mod-/?. Then M is U-reflexive if and only if M is U-linearly compact. Moreover the following conditions are equivalent for MR 6 Ker F: (a) M is linearly compact, (b) every submodule of M is U-reflexive, (c) AM is U-torsionless linearly compact. In particular UR is linearly compact. Proof. Since $S is t/-torsionless linearly compact, applying Theorem 2.4 (a) => (b) to SS we see that AF vanishes on every factor of pU. Thanks to Lemma 3.2 applied to gU, we easily see that AF vanishes on every factor of RUn too. Therefore, using

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Theorem 2.4 (b) => (a) we get that sSn is [/-torsionless linearly compact too. Hence we can apply Theorem 2.4 (a) => (c) to $Sn, getting that any finitely URcogenerated module is [/-reflexive. Finally, from Theorem 2.2 we see that a right .R-module is {/-reflexive if and only if it is [/-linearly compact. This proves the first assertion. (a) =4> (b). If M is linearly compact, then all its submodules are ([/-)lineariy compact, hence reflexive. (b) =^ (c). If every submodule of M is [/-reflexive, then Theorem 2.4 guarantees that AM is [7-tl-lc, so that Corollary 2.5 proves that M = A 2 M is linearly compact. (c) => (a). From Corollary 2.5 we see that A 2 M is linearly compact. Since M embeds in A 2 M, it is linearly compact too. The last assertion is now straightforward. D From Corollary 2.5 we immediately have the following Corollary 3.6. If §UR is a cotilting bimodule such that UR, sU, RR and $S are U-tl-lc, then UR, sU, RR and sS are linearly compact. Example 3.7. Suppose that $UR is a cotilting bimodule such that UR is finitely generated and sU is [7-tl-lc. Then from Theorems 3.3 and 2.4 we see that $S is U-tl-lc as well. In particular, again thanks to Theorem 3.3, given Noetherian rings R and S, any finitely generated cotilting bimodule §UR is always U-tl-lc on both sides, as well as RR and sS. This gives an example of the setting described in Corollary 3.6. In the next section we will see how, constructing a cotilting bimodule U as "dual" of a tilting one, we often come up with UR, sU, RR and $S all [/-torsionless linearly compact. • This will confirm that [/-torsionless linear compactness is, in cotilting theory, a natural additional finiteness condition. 4. Duals of tilting modules The notion of a tilting module arises from representation theory of finite dimensional algebras, and goes back to the fundamental work of S. Brenner and M. Butler [5]. Tilting modules over arbitrary rings where first introduced by Y. Miyashita in [26]. They can be defined dually to Definition 1.1. So, following [17], a left Rmodule RV is tilting if it satisfies the condition Gen( fi F) = KerExtJj(V, -) (where Gen(RV) denotes the class of the modules generated by RV) or, equivalently, the three conditions (i) proj dim(jjV) < 1, (ii) Extk(V, V^) = 0 for any cardinal a, (iii) there is an exact sequence 0 — > / ? — > V —» V" —> 0, where V', V" are direct summands of V^ for some cardinal 0. As expected, a tilting bimodule is just a faithfully balanced bimodule RV§ such that both RV and Vs are tilting modules. Similarly to the case of projective generators, the symmetry and finiteness join in tilting bimodules, i.e.:

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(a) if RV is a finitely generated tilting module and S = End(.RV), then R = End(V"s) and Vg is a finitely generated tilting module (see [9, Proposition 1.1]); (b) if pVs is a tilting bimodule, then it is finitely generated on both sides (see [17, Proposition 2.5]). Tilting bimodules satisfy to a "Tilting Theorem" which generalizes Morita equivalences in the same way Theorem 1.5 generalizes Morita dualities. The main difference here is that all modules are involved in the two equivalences, by means of their torsion and torsion-free parts, respectively. In other words, the classes corresponding to CR and sC are, in the covariant case, the whole categories .R-Mod and 5-Mod, respectively. For a general formulation of this theorem, and for an exhaustive list of references, we refer to [9], [12] and [22]. We are now going to prove that any dual of a finitely generated tilting module is a cotilting module. Lemma 4.1. Let RMg be a bimodule, Qs an injective cogenerator and let —* = Hom,s(—, Qs)- Then for every i £ N we have: (a) Extn(-,Afp) = Torf (-,flM)* canonically in Mod-R; (b) if nM admits a finitely generated projective resolution, then there is a canonical isomorphism Torf (-*, RM) = ExtlR(RM, -)* in R-Mod. Proof, a) is [29, Theorem 11.54] and b) is contained in the proof of [29, Theorem 9.51]. D Proposition 4.2. Let RV be a finitely generated tilting module. Let S —> 'End(RV) be any ring homomorphism and let Qs be cm injective cogenerator in Mod-5. Then VR = Horns(V, Q) is a cotilting module. Proof. From Lemma 4.1 a) we see that, for any right .R-module L, Ext}j(L, V*) — 0 if and only if Torf (L, V")* = 0 if and only if Torf (L, V) = 0. Hence it will suffice to show that Cogen(V^) = KerTorf (-, V). Let us prove the inclusion Cogen(V^) C KerTorf (—, V). Since Ext^(l/, V) = 0 by assumption, applying Lemma 4.1 b) to RV we at once see that VR belongs to KerTor-L (—, V). Moreover KerTorf ( — , V ) is closed under submodules, since the weak dimension of RV is at most one. Finally, arguing as in the proof of [16, Proposition 2.8 d)], from the existence of an exact sequence 0 —> F' —> F" —> RV —> 0, with finitely generated projectives F' and F", we directly see that the functor Tora (—, V) commutes with arbitrary direct products. In particular, KerTorf (—, V) is closed under products. This proves the first inclusion. Conversely, let MR e KerTorf (-,V). Let us prove that MR € Cogen(V*) by showing that Rejy, (M) = 0. First note that Lemma 4.1 a) provides an isomorphism Hom R (M,F*) ^ (M ®R V)*, which gives the identity Rej v ,(M) = Ann v (M) = {x e M \ x ® v = 0 Vu e V}. Now, by hypothesis, there is an exact sequence of the form 0—> R •?* Vn —> V —> 0, where V is a direct summand of a finite direct sum of copies of V. From that we derive an exact sequence Torf (M, V'} —> M =U M ®R Vn. Since Torf(M, V) = 0 by assumption, we see that M j is monic,

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i.e., for every x 6 M, from x <8> j(l) = (M ® j)(z) = 0 we get x = 0. In particular Amv(M) - 0, and so Rej^, (M) = 0. D Remark 4.3. The previous result was already known in case S = End(Vft) (see [16, Example 2.3. 3)]). Combining this with [19, Lemma 1.3], we get an alternative proof of Proposition 4.2. If we specialize 4.2 to the case 5 = End(Vft), we obtain the cotilting torsionfree class KerF, cogenerated by UR = VR, which is equivalent to the tilting torsion class Gen(Vs) via Horns (.R^S,— ) (see, for instance, [12, Theorem 4.1)]). In this case, there are reasonable conditions which guarantee the existence of enough Utorsionless linearly compact modules, thanks to the following result, due to E. Gregorio: Proposition 4.4. Let RV$ be a tilting bimodule, Qs an injective cogenerator and UR = Homs(y,Q). If M is a linearly compact right S-module, then Homs(Vr, M) is a U-torsionless linearly compact right R-module. Proof. Since Homs(V, M) = Horns (V, M') for some submodule M' of M which is generated by V$, we can assume that M & Gen(Vs)- Let us denote by H the full and faithful functor Homs(V, — ) : Gen(Vs) —> KerF, and by T its inverse functor — ®fl V. Let {HM -^-> L\} be an inverse system of almost-epic maps in KerF. Since CokerpA G Ker A for every A, and Ker A — KerT by Lemma 4.1 a), we get a new inverse system of epimorphisms {M = THM —^ TL\} in Gen(Vs). Now linear compactness of M applies, providing the short exact sequence ITl

lira TpA

0 -> K -> THM —

> limTLA -» 0,

and the commutative exact diagram HTHM

Hl

™Tpx ; HmHTL X

> Ext^(V, K)

>0

HM -—-s- lim x L

which shows that CokerlimpA — Ext^V,^). Finally Ext^V,^) e Ker A, since Homfl(Ex4(V, K), U) ^Ext^V, K) ®R V)* = 0 (see [12, Theorem 4.1 c)]). D Corollary 4.5. Let ^Vs be a tilting bimodule, Qs an injective cogenerator and UR = -Roms(V,Q). Then: (a) if Qs is Ic, then UR is U-tl-lc; (b) ifVs is Ic, then RR is U-tl-lc; (c) if Qs induces a Morita duality, then both UR and RR are U-tl-lc. Recall that any commutative linearly compact ring has a Morita self-duality (see [2]). The following result generalizes the well known fact that a dual of a finitely generated tilting module over an Artin algebra is always a cotilting bimodule:

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Theorem 4.6. Let A be a linearly compact commutative ring, R an A-algebra and fiV a tilting module which is linearly compact as an A-module. If S = End(fiV) and AW A is a Morita bimodule, then the S-R bimodule V* = Hom^(flVs, W) is a cotilting bimodule. Moreover V£, sV*, RR and sS are V*-torsionless linearly compact. Proof. Both Vft and $V* are cotilting modules, thanks to Proposition 4.2. Moreover the functor —* = Hom^(—,W) gives a duality between the linearly compact Amodules. The same functor induces a duality between, respectively, the right and left ^-modules which are linearly compact as A-modules. By assumptions, RV is reflexive, so that S = End(flV) = End(V^). Replacing R with S, we analogously see that R = End(Vs) = End( s ^*). This proves that SVR is faithfully balanced. To finish the proof, first remark that considering the injective cogenerator Sg = Homyi(sS<, W) of Mod-51, we have a natural isomorphism V£ = Homs(j?V"s, S$). Moreover, since 5 embeds into some Vn by assumption, we derive that S is a linearly compact A-module. Therefore S* is linearly compact as A-module and so, a fortiori, as a right 5-module. For the same reason V is linearly compact as a rigut j_f-rnouuie. j-'iorn v^oronary <±.o we at once see tnat v^ anu ±\R are v torsionless linearly compact. In the same way one can prove that sV* and sS are y-torsionless linearly compact. D Acknowledgments I wish to thank my colleagues and friends Kent Fuller and Enrico Gregorio for the useful discussions and suggestions which improved the paper. References 1. F. W. Anderson and K. R. Fuller, Rings and categories of modules, second ed., Graduate Texts in Mathematics, vol. 13, Springer-Verlag, New York, 1992. 2. P. N. Anh, Morita duality for commutative rings, Comm. Algebra 18 (1990), no. 6, 1781-1788. 3. M. Auslander, Functors and morphisms determined by objects, Representation theory of algebras (Proc. Conf., Temple Univ., Philadelphia, Pa., 1976), Dekker, New York, 1978, pp. 1-244. Lecture Notes in Pure Appl. Math., Vol. 37. 4. S. Bazzoni, Cotilting modules are pure-injective, to appear on Proc. Amer. Math. Soc. 5. S. Brenner and M. C. R. Butler, Generalizations of the Bernstein-Gel'fand-Ponomarev reflection functors, Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Math., vol. 832, Springer, Berlin, 1980, pp. 103-169. MR83e:16031 6. R. R. Colby, A generalization of Morita duality and the tilting theorem, Comm. Algebra 17 (1989), no. 7, 1709-1722. MR 90i:16021 7. , A cotilting theorem for rings, Methods in module theory (Colorado Springs, CO, 1991), Lecture Notes in Pure and Appl. Math., vol. 140, Dekker, New York, 1993, pp. 33-37. MR 94e: 16011 8. R. R. Colby, R. Colpi, and K. R. Fuller, A note on cotilting modules and generalized Morita duality, these Proceedings. 9. R. R. Colby and K. R. Fuller, Tilting, cotilting, and serially tilted rings, Comm. Algebra 18 (1990), no. 5, 1585-1615. MR 91h:16011 10. , Costar modules, J. Algebra 242 (2001), no. 1, 146-159. MR 2002k:16006 11. , Weak Morita duality, to appear on Comm. Algebra, 2002.

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12. R. Colpi, Tilting in Grothendieck categories, Forum Math. 11 (1999), no. 6, 735-759. MR 2000h:18018 13. , Cotilting bimodules and their dualities, Interactions between ring theory and representations of algebras (Murcia), Lecture Notes in Pure and Appl. Math., vol. 210, Dekker, New York, 2000, pp. 81-93. MR 2001f: 16015 14. R. Colpi, G. D'Este, and A. Tonolo, Quasi-tilting modules and counter equivalences, J. Algebra 191 (1997), no. 2, 461-494. MR 98g: 16003 15. R. Colpi and K. R. Fuller, Cotilting modules and bimodules, Pacific J. Math. 192 (2000), no. 2, 275-291. MR 20015:16014 16. R. Colpi, A. Tonolo, and J. Trlifaj, Partial cotilting modules and the lattices induced by them, Comm. Algebra 25 (1997), no. 10, 3225-3237. MR 981:16003 17. R. Colpi and J. Trlifaj, Tilting modules and tilting torsion theories, J. Algebra 178 (1995), no. 2, 614-634. MR 97e: 16003 18. G. D'Este, Reflexive modules are not closed under submodules, Representations of algebras (Sao Paulo, 1999), Lecture Notes in Pure and Appl. Math., vol. 224, Dekker, New York, 2002, pp. 53-64. MR 1 884 806 19. K. R. Fuller, Ring extensions and duality, Algebra and its applications (Athens, OH, 1999), Contemp. Math., vol. 259, Amer. Math. Soc., Providence, RI, 2000, pp. 213-222. MR 2001h:16005 20. J. L. Gomez Pardo, Counterinjective modules and duality, J. Pure Appl. Algebra 61 (1989), no. 2, 165-179. MR 90k: 16026 21. J. L. Gomez Pardo, P. A. Guil Asensio, and R. Wisbauer, Morita dualities induced by the M-dual functors, Comm. Algebra 22 (1994), no. 14, 5903-5934. MR 96g:16007 22. E. Gregorio, Tilting equivalences for Grothendieck categories, J. Algebra 232 (2000), no. 2, 541-563. MR 2002e:18013 23. F. Mantese, Generalizing cotilting dualities, J. Algebra 236 (2001), no. 2, 630-644. MR 2001m: 16007 24. , Hereditary cotilting modules, J. Algebra 238 (2001), no. 2, 462-478. MR 2002b:16011 25. F. Mantese, P. Ruzicka, and A. Tonolo, Cotilting versus pure-injective modules, to appear on Pacific J. Math. 26. Y. Miyashita, Tilting modules of finite projective dimension, Math. Z. 193 (1986), no. 1, 113-146. MR 87m:16055 27. B. J. Miiller, Linear compactness and Morita duality, J. Algebra 16 (1970), 60-66. MR 41 #8474 28. A. Orsatti, Una introduzione alia teoria dei moduli, Aracne, Roma, 1995 (Italian). 29. J. J. Rotman, An introduction to homological algebra, Pure and Applied Mathematics, vol. 85, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1979. MR 80k:18001 30. B. Stenstrom, Rings of quotients, Springer-Verlag, New York, 1975, Die Grundlehren der Mathematischen Wissenschaften, Band 217, An introduction to methods of ring theory. MR 52 #10782 31. A. Tonolo, Generalizing Morita duality: a homological approach, J. Algebra 232 (2000), no. 1, 282-298. MR 2002a:16008

Symmetries and Asymmetries for Cotilting Bimodules Gabriella D'Este Dipartimento di Matematica, Universita di Milano, Via Saldini 50, 20133 Milano, Italy [email protected] Abstract. We investigate the different behaviour of the two underlying modules of a cotilting bimodule with respect to injective envelopes, their possible bimodule structure, torsion and torsion free modules.

Introduction and notation In this note we collect some properties of cotilting bimodules that are more or less close to Morita bimodules, i.e., of both "hereditary" and "non hereditary" cotilting bimodules in the sense of Mantese [Ml]. By dealing with finite dimensional bimodules and by cqmparing finite Auslander-Reiten quivers, we show that the results of [Ml] are as precise as possible. Roughly speaking, on the one hand, very large modules show up as injective envelopes of small cotilting modules. On the other hand, almost all symmetries between left and right modules seem to vanish. Before we describe the results proved in the sequel, we recall some definitions and we fix the notation. Throughout the paper, we say that an A-module C is a cotilting module, if KerExt^(—,C) is the class CogenC of all modules cogenerated by C. On the other hand, we say that a bimodule sCn is a cotilting bimodule, if S&R is faithfully balanced and sC and CR are cotilting modules. Given a bimodule sUn, we often denote by A (resp. F) both the contravariant functors AS = Honi5(—, U) and AR = Homfi(-, U) (resp. Fs = Ext^-, U] and TR = Ext]j(-, U)). Following [Ml], we say that a cotilting module C is hereditary, if the torsion theory (KerA,KerF) cogenerated by C [C, Lemma 2] is hereditary. We also say that a cotilting bimodule sCR is hereditary, if both $C and CR are hereditary cotilting modules. Moreover, 2000 Mathematics Subject Classification. 16D20, 16G20. This work was partially supported by G.N.S.A.G.A., Istituto Nazionale di Alta Matematica "Francesco Severi", Italy.

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given a cotilting bimodule S^R, we say that a left S'-module (resp. right .R-module) Mis • A-reflexive if M is C-reflexive [AF], i.e., if the evaluation map SM '• M —> A 2 (M), denned by the formula x i-> [£ i-> £(:r)] for every x £ M and £ 6 A(M), is an isomorphism; • T-reflexive if M £ Ker A and there exist two A-reflexive modules Y and Y' such that M = Y/Y'. Throughout the paper, K denotes an algebraically closed field, while the ring S (resp. R) is almost always a finite dimensional JC-algebra A, given by a quiver Q and defined according to [R]. In particular, for every vertex i of Q, we denote by 6i the primitive idempotent of A corresponding to i. Moreover, all useful S — R bimodules are finite dimensional K-vector spaces V, equipped with a basis B formed by normed elements [Z] of both gV and VR. Hence, for every x € B, there exist two primitive idempotents &i e S and BJ £ R such that e^x — x and xej = x. For brevity, a picture of the form jD;- will describe an element x of B with the above property. With terminology suggested by [GSZ, page 2917], we may say that the basis B of S^R consists of left-right uniform elements. Similarly, given a basis B of the underlying K-vector space of sV (resp. VR), a picture of the form jD (resp. D f ) will describe an element x of B such that e%x = x (resp. xej — x) for some primitive idempotent e, € S (resp. BJ € R). Under the same hypotheses, straight (resp. wavy) lines will describe how left (resp. right) multiplications by elements of S (resp. R) act on the vectors of B. In this way, we will often replace modules and bimodules by some obvious picture describing their structure. On the other hand, (xi,..., xm} denotes the K-vectoi space generated by the elements xi,... ,xm. Finally, we always identify indecomposable modules and their isomorphism classes. For unexplained representation theoretic terminology, we refer to [AuRS] and [R]. This paper is organized as follows. In section 1, we recall some useful results proved in [Ml]. In section 2, we sum up the main reasons why several cotilting bimodules S^R have quite different behaviour with respect to • the injective envelopes E$(C) and ER(C) of $C and CR respectively, and their possible bimodule structure (Proposition 2 (ii); Proposition 3 (iii)). • F-reflexive modules and possible dualities between them (Propositions 2 and 3 (i)). • torsion theories and simple modules (Proposition 1 and Corollary 4). Finally, in section 3 we construct some reasonably small cotilting bimodules, and we collect all the proofs. 1. Preliminaries The following results of [Ml] indicate that the gap between a cotilting module C and its injective envelope E(C) is neither very large nor very small. (1) socE(C)/C e Ker A [Ml, Lemma 2.2]. (2) If C is hereditary, then Ker A C CogenE(C)/C [Ml, Proposition 2.1].

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(3) If sCn is a hereditary cotilting bimodule, then ImFs C Ker A#, ImF^ C Ker A s [Ml, Proposition 1.7], and r?(M) is isomorphic to Hom?(M,£?(C')/C') for every M e Ker A?, where ? = S,R [Ml, Theorem 1.17]. On the one hand, in view of the so called Cotilting Theorems (see e.g., [Cl], [C2], [CF] and [Ml]), the last result points out that the duality induced by FS and F# between left and right F-reflexive modules, say the T-duality for short, behaves nicely if the cotilting bimodule sCn is hereditary. On the other hand, the same result says that the factor modules Es(C)/C and En(C)/C must be "complicated enough", to describe the F-duality. (For generalizations of the preceding definitions of cotilting modules and bimodules, and for other results on Morita type dualities, see e.g., [An], [CbF], [M2], [T] and the references therein.) In the sequel, we recall some old and new properties of cotilting modules. On the one hand, by ([CDT1, Proposition 1.7]; see also [CDT2]), an ^-module C has the property that Cogen C = Ker Ext^(—, C) if and only if the following conditions hold: (i) The injective dimension of C is at most 1. (ii) ExtlA(CQ, C) = 0 for any cardinal a. (iii) KerHom A (-,C')nKerExt]t(- ) C') = 0. On the other hand, by [AnTT, Proposition 2.3], we may replace (3) by the following condition on exactly one module. (iii') There is an exact sequence of the form 0 —> C" —» C" —> Q —> 0, where Q is an injective cogenerator for the category of all A-modules, while C" and C" are direct summands of products of copies of C. A very important theorem, recently proved by S. Bazzoni ([Ba, Theorem 2.8]), establishes that every cotilting module is pure-injective. Moreover, by the new results of Buan and Krause [BK, Corollary 1.10], a pureinjective module M, of injective dimension at most one and defined over an artinian ring A, is product selforthogonal, i.e., Ext^(M tt , M) = 0 for every cardinal a, if and only if Ext A (M,M) = 0. As we shall see, in all useful examples considered in this note, S (resp. R) is a ^sT-algebra A of finite representation type, while sC (resp. CR) is a multiplicity-free [HR] A-module satisfying a definition of cotilting module often used in Representation Theory [R, page 169]. In other words, C satisfies the following conditions: • The injective dimension of C is at most 1. « Ext^C^C) =0. • C is the direct sum of n pairwise non-isomorphic indecomposable modules, where n is the number of simple A-modules. 2. Outline of the results One of the first asymmetries of a cotilting bimodule sCn is the relationship between the factor modules Es(C}/C and En(C}/C and the classes of left and right torsion modules. For instance, the proof of the next statements shows that exactly one of

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the modules ER(C}/€ and Es(C)/C may be semisimple, hence a torsion module [Ml, Lemma 2.2], and that both these modules may be mixed. Proposition 1. There exist cotilting bimodules sCR satisfying one of the following conditions: (i) Ker AR = CogenER(C)/C and Ker As C CogenEs(C)/C. (ii) Ker A? C CogenE? (C)/C for t=R,S. (iii) Cogent?(C)/C C Ker A? for ? = R, S. (iv) CogenEs(C)/C CKei&s and GogenER(C)/C g Ker A R . (v) ImFs C KerA.R, ImF^ C Ker AS and the modules $C and CR are not hereditary. As we shall see, several one-sided properties of the two underlying modules of a cotilting bimodule appear, when one deals with hereditary cotilting bimodules admitting few F-reflexive modules. Proposition 2. There exist hereditary cotilting bimodules sCR such that every T-refiexive module is semisimple, and the following facts hold: (i) FS (resp. TR) is represented by an S — R bimodule Z (resp. Z') such that SZ ~ ES(C)/C (resp. Z'R ~ ER(C)/C). (ii) At least one of the modules E$(C) and ER(C) fails to be the support of an S — R bimodule containing sCR. Moreover, we may choose C with the property that the bimodules Z and Z' in (i) satisfy one of the following conditions: (1) sZ'R C SZR and Homs(Z, Z') = 0. (2) Hom?(Z, Z') ^ 0 and Hom?(Z', Z} + 0 for ? = R, S. (3) Roms(Z, Z') = 0 and RomR(Z', Z) = 0. The next proposition shows also that the underlying modules of non hereditary cotilting bimodules, that are quite close to Morita bimodules, have a different behaviour with respect to injective envelopes. Proposition 3. There exist non hereditary cotilting bimodules sCR such that Es(C)/C is semisimple, and the following facts hold: (i) FS (resp. TR) is not representable by an S - R bimodule. (ii) ER(C)/C is (resp. is not) semisimple. (iii) Any (resp. Exactly one) of the modules Eg(C) and ER(C) is the support of an S — R bimodule containing sCR. In dealing with well behaved cotilting bimodules, it is natural to compare the left and the right socle of certain bimodules. However, there is no two-sided version of the result mentioned in (1) of Section 1, even in very special cases. Corollary 4. There exist a cotilting bimodule sCR and S-R bimodules U and W, containing SCR, such that SU ~ ES(C)» WR ~ ER(C), s(U/C) and (W/C)R are semisimple, and the following facts hold: • (i) soc(U/C)R is a simple module cogenerated by CR. (ii) socs(W/C) is a simple module cogenerated by $C.

Symmetries and Asymmetries for Cotilting Bimodules

107

u/c (iii) Eoms(U/C,W/C) = 0, HomR(W/C,U/C) = 0, but the bimodules ->c(U/C) R and

W soc /w/c)

are

isomorphic and simple on both sides.

The following question naturally arises from our first constructions of possible big bimodules with prescribed underlying left of right module (Examples A, B and D below). (+) Are there non hereditary rings A with the property that a cotilting bimodule A_XB, with XB non injective, may be embedded in a bimodule A%-B with XB~E(XB}? Our positive answer to (+) also shows that the structure of torsion modules does not influence the existence of possible big bimodules. In fact, we have the following Corollary 5. There exist cotilting bimodules S&R and A^-H with the following properties: (i) CR and XB are not injective, (ii) The torsion classes Ker AS and KerHorn,^— , X ) are equivalent. (iii) ER(C) is not (resp. EB(X) is) the support of a bimodule containing S&R (resp. AXB)(iv) The rings S and A are not hereditary. 3. Examples and proofs We begin with two examples of hereditary cotilting bimodules of very small dimension over K, defined over /C-algebras admitting only uniserial indecomposable modules. Example A There is a hereditary cotilting bimodule S@R satisfying the following conditions: (a) There is a unique duality between A-reflexive (resp. F-reflexive) modules. (b) FS (resp. F.R) is represented by a bimodule S%R (resp. S^'R) such that SZ ~ ES(C}/C (resp. Z'R ~ ER(C)/C), with the property that SZ £ Ker A s (resp. Z'R is a simple module and Ker A^ = CogenZ^j). (c) Es(C) is not the support of an SR bimodule containing sCR. (d) There is a bimodule sWn such that WR ~ ER(C) and sW/j contains sCR. Construction a

Let R (resp. S) denote the /f-algebra given by the quiver * c

ba = 0 (resp. *

b

*

* with

d

*

* ), and let $CR denote the cotilting bimodule sat-

isfying CR ~ ? ® 2 © 2 . Then we have $C ~ 6 © 5 ® 4, and (a) is an immediate 6 consequence of the following Auslander-Reiten quivers of the algebras R and 5.

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G. D'Este

Cogen CR

CogensC

KerAs

Since ES(C)/C ~ £, ER(C}/C ~ 3, while KerA fl and KerA s are the homogeneous components [J, page 121], determined by the simple modules 3 and 5 respectively, we conclude that the bimodules S%R and sZ'R satisfying (V) are of the form

and 5

3

'

3

respectively. Moreover, a bimodule sW# satisfying (d) is described by the following picture, where {l,m,n,p, q} (resp. {l,m, n,p, q, v}) denotes a basis of C (resp. W).

Symmetries and Asymmetries for Cotilting Bimodules

109

Finally, to prove (c), let {I, m,n,p,q,x,y} denote a basis of a module sU such that 50 C SU and sU ~ Es(C). Then $U is described by the following picture.

Assume, by contradiction, 5^7 is the support of an S-R bimodule containing . Since x = 642, y = e^y, the structure of $U implies that

(1)

xb € 64 [/ = (x,p, 1)

and

yb e 65 C7 = (y, m) .

Hence there exist ki,k% £ K such that yb = kiy + k^m. This implies that n = qb = (dy)b = d(yb) = k\q + k2n, and so we obtain (2)

yb = m.

Finally, by (1), there exist k\, fc2, k3 £ K such that xb = k\x + k2p + k$l. Consequently, we deduce from (2) that m = yb = (cx)b = c(xb) = kiy + k^m. This means that (3)

xb = p + k^l for some fca 6 K.

Therefore, we have (xb)a = (p + k^V)a = I ^ 0. This contradiction shows that (c) holds. Example B There is a hereditary cotilting bimodule S&R satisfying the following conditions: (a) There is a unique duality between A-reflexive modules, and there are exactly two dualities between F-refiexive modules. (b) TS (resp. TR) is represented by a bimodule S%R (resp. sZ'R) such that SZ ~ ES(C}/C (resp. Z'R ~ ER(C)/C) with the property that SZ £ Ker A s , Cogen^ = Ker A fi (resp. Z'R <£ Ker'A^, Cogen5Z' = Ker A5). (c) There is a bimodule sVn such that ^VR contains $CR, both sF and VR are isomorphic to maximal submodules of Es(C) and ER(C] respectively, and

G. D'Este

110

we have Cogens(V/C) = KerA 5 , Cogen(V/C)R = KerA fl and r?(M) ~ Hom?(M, V/C) for every M e Ker A? for ? = R,S. (d) Es(C) (resp. ER(C}) is not the support of an S — R bimodule containing s^RConstruction

Let R (resp. S) denote the K-algebra given by the quiver • with relation cb — 0 (resp.

d

e

f

with relation ed = 0), and

let sCn denote the cotilting bimodule satisfying CR ~ 1 ® 2 © 3 © | . In other words, we have Cogen C"fi = Cogen EH, and the dimension of CR over K is as small as possible. Therefore, we have gC ~ 8 © 7 ® 6 ® | , and (a) is an immediate 8 consequence of the following Auslander-Reiten quivers. Cogen CR

Cogen CR

kerA

kerAs

Moreover, we clearly have Es(C}/C ~ f ® 5, ER(C}/C ~ | ® 4, and it is easy to check that Ts(7) ~ 4. Consequently, the bimodule S%R (resp. 5^) described by the picture

(resp.

satisfies condition (b). On the other harid, the next picture describes both a bimodule sVft satisfying condition (c), and the cotilting bimodule $CR generated by the vectors jDj such that ( i , j ) (£ {(5, 2), (7, 4)}.

Symmetries and Asymmetries for Cotilting Bimodules

111

Finally, contrary to (d), assume Es(C) (resp. ER(C}) is the support of a bimodule $CR such that S&R C S^R. Then $V (resp. VR) is isomorphic to a maximal submodule M of SC (resp. CR), and we have S(C/M) ~ 6 (resp. (C/M)R ~ 3). In particular, it is easy to see that there is an element x € e&C\M (resp. x 6 Cea \M) such that xc (resp. dz) is a generator of the one dimensional subspace e^Ce^. Hence, we obtain (xc}b ^ 0 (resp. e(dx) ^ 0), a contradiction to the hypothesis cb = 0 (resp. ed = 0). Thus also (d) holds. Remark C Some of the hypotheses of Example B, i.e., conditions (b), (c) and (d), hold also for less complicated cotilting bimodules with a more symmetric structure. For instance, let A be the JC-algebra given by the quiver

with relation ab = 0, and

let A be the cotilting bimodule of dimension 4 over K. In other words, let (resp. CA) be the module 2 ® 1. Then AC A is of the form

G. D'Este

112

and the structure of the indecomposable left (resp. right) A-modules is described by the following Auslander-Reiten quiver, with identification along the vertical dashed lines.

Cogen C

Thus there is exactly one duality between A-reflexive (resp. F-renexive) modules, and ACA satisfies the requirements of (b), (c) and (d) of Example B. In particular, the F-duality is represented by a bimodule A%A (resp. AZ'A), such that AZ = E(AC)/C (resp. Z'A = E(CA)/C), of the form

(resp.

All the hereditary cotilting bimodules constructed up to now have the property that (Ker AS, Cogens C) is not splitting. On the other hand, in the next example we construct a non hereditary cotilting bimodule sCR such that both gC and CR cogenerate a splitting torsion theory. Example D

There exists a non hereditary cotilting bimodule sCR with the following properties: (a) There is a unique duality between A-reflexive (resp. F-reflexive modules), and FS (resp. TR) is not representable by an S — R bimodule. (b) There is a bimodule SUR such that SCR C SUR, SU ~ ES(C), Gens(U/C) = Cogeus(U/C) C Ker AS and Gen(U/C)R = Ker AR. (c) There is a bimodule SWR such that SCR C SWR, WR ~ ER(C), Gen(W/C)R = Cogen(W/C)R C Ker AR and Gens(W/C) = Ker As. (d) Under the above hypotheses on U and W, assume sVR and sVR are bimodules such that SCR C SVR C SUR and S,CR C SVR C SWR. Then V and V are uniquely determined, the bimodules U/V and W/V are isomorphic, while the 5-modules (resp. ^-modules) V/C and V'/C are simple but not isomorphic. (e) ImFs C Ker A fi and lmTR C Ker A s .

113

Symmetries and Asymmetries for Cotilting Bimodules

Construction

Let R and S denote the /^-algebras given by the quivers * *

^*

i

*

^ * respectively. Next, let sCn denote the cotilting bimodule such 1

that CR ~ o , ® 2 ® 2. Since SC ~ 5 © 5 © 5, we have 6

(1)

and Ker A s = Gen £ .

Ker A fi = Gen |

Moreover, the Auslander-Reiten quivers

Cogent Cogen5 C

show that (a) holds. Next, let {x, y, z, t, u, v} be a basis of C over X, and let i = ii + ^. Then the bimodules $UR and sWjj described by the following pictures contain Ctt, and we

4

2 / 5

1 X 4

3

4

5

2 / 6

1

3

5

6

5

0

2

6

2

/

5

1

2 / 6

1

m

2

Hence, the bimodules U/C and W/C have the form

and (2)

4

3

5

1

114

G. D'Este

respectively. Putting (1) and (2) together, we conclude that sUR and sWR satisfy (b), (c) and (d). Finally, let M be a left S-module (resp. right ^-module). Then F(M) ~ F(L) for some L € Ker A. Since any L € Ker A is isomorphic to a direct sum of copies of the F-reflexive modules \ and 4 (resp. 3 and 1), it follows from [Au, Chap. 1, Proposition 10.1] that F(L) is isomorphic to a direct product of copies of the modules 1 and 3 (resp. 4 and |). Consequently, we have F(M) ~ F(L) e Ker A, as claimed in (e). Roughly speaking, the previous cotilting bimodule is, in some sense, a "minimal" non hereditary cotilting bimodule. Indeed, there is exactly one indecomposable torsion-free left ^-module M such that E(M) is not torsion-free. On the other hand, in the next example we investigate a more complicated cotilting bimodule, very far from being hereditary. Indeed, every indecomposable non injective torsionfree right .R-module M has the property that E(M) is not torsion-free. Example E There exists a non hereditary cotilting bimodule S&R with the following properties: (a) There are exactly two equivalences -between A-reflexive (resp. F-refLexive modules), and TS (resp. TR) is not representable by an S — R bimodule.

(b) There is a bimodule SUR such that SCR C SUR, SU ~ ES(C), Gens(U/C) = Cogens(U/C) C Ker As and Gen(U/C)R = KerAR. (c) ER(C) is not the support of an S - R bimodule containing sCR. (d) The bimodule U satisfying (b) has the following properties:

HomR(U/C, ER(C)/C) + 0 and

HomR(ER(C)/C, U/C) = 0.

(e) A maximal submodule of ER(C) is the support of a bimodule sDR such that sCR C SDR7 Cogen(D/C)R = CogenER(C)/C £ KerA f l and KerA s C Construction Let R (resp. S) be the /("-algebra given by the quiver •

•

(resp.

with relations fd = 0 and fe = 0), and let sCR be the cotilting bimodule such that C t f C ± l © 2 ® 3 ©4 and A f i (l) ~> f . Then we have S C ~ | © 7 ® f 0 f , and condition (a) is an immediate consequence of the following Auslander-Reiten quivers.

Symmetries and Asymmetries for Cotilting Bimodules

115

CogenCj

On the other hand, the next picture describes both a bimodule $UR satisfying condition (b), and the bimodule S^R, generated by the vectors p, q, r, s, u. v, w.

7

1

Consequently, the module ER(C) has the following shape.

G. D'Este

116

Assume now, contrary to (c), that ER(C) is the underlying module of a bimodule sWu such that sCR Q sWp.. Since ex, dt, v £ e-jV\re^ our hypotheses of ER(C) and sCR enable us to write

ex = with xi,ti,vi e ( x , y ) with z 2 , t2jv2 e ( z , t ) . Moreover, we clearly have (2)

vca = q = ep = e(xca),

vcb = s = dr = d(tcb),

xcb = 0 = tea.

Putting (1) and (2) together, we obtain (3)

v = y + z,

ex = y, and dt = z.

Therefore, we have fv = f(ex+dt) = 0, a contradiction to the hypothesis fv = w ^ 0. Hence condition (c) holds. Since (U/C)R ~ f ® \ and ER(C}/C ~ | 0 | © |, we conclude that also (d) holds. Finally, keeping all the above notation, let DR denote the submodule of ER(C) generated by the elements x,y,z,tc, and assume z = v - y. Then DR is the underlying module of the following bimodule S^R, and we obviously have S^R £ SDR.

1

7

Since s(D/C) ~ 576 ® f and (D/C)R ~ | © | ® 3, condition (e) follows from the structure of Ker AS and Ker A#. We can now prove the assertions of section 2. Proof of Proposition 1. (i) (resp. (ii)) This follows from condition (b) of Example A (resp. Example B). (iii) (resp. (iv)) This follows from conditions (b) and (c) of Example D (resp. (b) and (e) of Example E). (v) See condition (e) in Example D. D

117

Symmetries and Asymmetries for Cotilting Bimodules

Proof of Proposition 2. Condition (b) of Example A (resp. Example B) guarantee that (i) holds. On the other hand, (ii) follows from conditions (c) and (d) of Example A (resp. condition (d) of Example B). Finally, the bimodules Z and Z' considered in Example A, Example B and Remark C clearly satisfy all the assertions in (1), (2) and (3) respectively. D Proof of Proposition 3. Conditions (a), (b), (c) of Example D (resp. Example E) guarantee that Es(C)/C is semisimple, and that (i) and (iii) hold. Finally, in the same example, the factor module En(C)/C is a direct sum of copies of the indecomposable module 1 (resp. 3). Hence also (ii) holds. D Proof of Corollary 4- Let C, U and W be the bimodules used in the construction of Example D. Then conditions (i), (ii) and (iii) immediately follow from the structure of the bimodules U/C and W/C described in (2). D Proof of Corollary 5. We claim that the bimodule S&R used in Example E has the desired properties. Indeed, let B (resp. A) denote the J^-algebra given by the quiver (resp.

with relation ca = d). Next, let A^B denote the cotilting bimodule such that XB — 2 4 ® 2 ® 1 ® 4 - Since A% ~ 7 ® | ® ^ © 7, it follows that 576 generates any 8 torsion left A-module. Consequently, (i), (ii), (iv) and the first part of (iii) follow from our assumptions on sCR. Finally, let A^-B denote the bimodule described by the following picture.

7 7

2

3

7

3 7

4

G. D'Este

118

Then the vectors I, m, n, p, q, r + s, t, u generate a bimodule isomorphic to Since XB — EB(X), this observation completes the proof of (iii).

D

References [AF] [An] [AnTT] [Au] [AuRS] [Ba] [BK] [Cl] [C2] [CbF] [CDT1] [CDT2] [CF] [GSZ] [HR] [J] [Ml] [M2] [R] [T] [Z]

F. W. Anderson and K. R. Fuller, Rings and categories of modules, GTM 13, SpringerVerlag, 1992. L. Angeleri Hiigel, Finitely cotilting modules, Comm. Algebra 28 (2000), 2147-2172. L. Angeleri Hiigel, A. Tonolo and J. Trlifaj, Tilting preenvelopes and cotilting precovers, Algebras and Representation Theory 4 (2001), 155-170. M. Auslander, Functors and morphisms determined by objects, Proc. Conf. on Representation Theory, Philadelphia, LNPAM 37, M. Dekker (1978), 1-244. M. Auslander, I. Reiten and S. O. Smal0, Representation theory of artin algebras, Cambridge University Press, 1995. S. Bazzoni, Cotilting modules are pure-injective, to appear in Proc. Amer. Math. Soc. A. B. Buan and H. Krause, Cotilting modules over tame hereditary algebras, to appear in Pac. J. Math. R. Colpi, Cotilting bimodules and their dualities, Proc. Euroconf. Murcia '98, LNPAM 210, M. Dekker (2000), 81-93. R. Colpi, Dualities induced by cotilting bimodules, to appear in Proc. Algebra Conf. Venice 2002, LNPAM, M. Dekker. R. R. Colby and K. R. Fuller, Weak Morita duality, to appear in Comm. Algebra. R. Colpi, G. D'Este and A. Tonolo, 'Quasi-tilting modules and counter equivalences, J. Algebra 191 (1997), 461-494. R. Colpi, G. D'Este and A. Tonolo, Corrigendum, J. Algebra 206 (1998), 370-370. R. Colpi and K. R. Fuller, Cotilting modules and bimodules, Pacific J. Math. 192 (2), (2000), 275-291. E. L. Green, 0. Solberg and D. Zacharia, Minimal projective resolutions, Trans. Amer. Math. Soc. 353 (7) (2001), 2915-2939. D. Happel and C. M. Ringel, Construction of tilted algebras, Springer LMN 903 (1981), 125-144. N. Jacobson, Basic Algebra II, H. Freeman and C., San Francisco (1980). F. Mantese, Hereditary cotilting modules, J. Algebra 238 (2001), 462-478. F. Mantese, Generalizing cotilting dualities, J. Algebra 236 (2001), 630-644. C. M. Ringel, Tame algebras and integral quadratic forms, Springer LMN 1099 (1984). A. Tonolo, Generalizing Morita duality: a homological approach, J. Algebra 232 (2000), 282-298. B. Zimmermann Huisgen, Syzygies and homological dimensions over left serial rings, Methods in Module Theory 140, M. Dekker (1993), 161-174.

A Constructive Solution to the Base Change Decomposition Problem in Clorinda De Vivo Claudia Metelli Dipartimento di Matematica e Applicazioni, Universita Federico II di Napoli devivo8matna2 . dma . unina . it cmetelliSiaath.uiii pel. it Abstract. The relationship between Butler B(1J-groups (a class of torsionfree Abelian groups of finite Tank) and their tents (certain finite Z2-representations) is used to solve constructively the base change decomposition problem, aimed at realizing every base change as a succession of base changes, each of which changes at most two base types. For this purpose a finer analysis is needed of the way to attach a tent to the group; this also allows to determine all groups with a given tent.

Introduction A Butler B^-group G is a torsionfree Abelian group of rank m — 1 that is the sum of m rank one (pure) subgroups: G = (<7i)* + • —h (9m)* for suitable elements Qi € G. Throughout, we will assume G regular, that is, the relation for these elements is gi + • • • + gm = 0. For each i e / = {1,..., m} let (&)* = Rtgi with Z < Ri < Q; r{ = type G ( ffi ) the type of R^. Then 7 = (gi,... ,gm) is called a -B^-base (in short, a base) for G, and (TI, . . . , rm) a type-base for G. If we also have G = (hi)* -\ + (hm}f with hi-\ h hm = 0, let 7' = (hi,..., hm) be the new base, with (vi,..., vm) the new type-base. Then, for suitable a,ij 6 RJ and for all i e I, we have hi = djigj', the m x m Q-matrix M = (a,.,-) is called a base change for G. The i-th column of M yields the i-th element hi of the new base; it also yields the i-th type Vi of the new type-base (as explained in Section 1). thus M works also as a type-base change. 2000 Mathematics Subject Classification. 20K, Q6B, 06F. Key words and phrases. Butler B^'-group, finite Z2-representation, base change decomposition.

119

120

C. De Vivo and C. Metelli

Clearly, one may be interested in changing either the base or the type-base of G (for instance a free group has base changes, but only the identical typebase change). The second problem has been privileged in research, and it is the one we consider in this paper. Thus, in the above notation, we consider the two bases 7, 7' of G equivalent if (r\,... ,rm) = (vi,... ,vm)', and two base changes equivalent if they produce the same type-base change. We will use quasiisomorphism (= isomorphism up to finite index) instead of isomorphism as the main equivalence between groups, hence equality between groups will mean quasiequality. Our goal is a construction that realizes any type-base change for G as a succession of exchanges1, that is type-base changes each of which changes at most two types. This problem was introduced by Peter Yom in his doctoral thesis and in a subsequent paper [12]; he gave a solution, but his is an existence result that does not allow for a concrete realization of the decomposition. The difficulty of the problem comes from the fact that if

From Yom's term: 2-vertex exchanges

A Constructive Solution . . .

121

1. Notation and basic results In this section we summarize our approach to the translation of the problem from Q to Zs, consisting in the realization of the typeset of G as a tent; we take it essentially from [6], where there are complete proofs and extensive examples. The notations already introduced remain valid; we also denote by T(V, A) the lattice of all types (with oo added as the type of 0). For an element x = X^e/ r j3j °f ^ expressed in terms of the base 7 (in fact, for the column vector (TJ \ j € J) of its coefficients) define the partition of the index set / into 'equal coefficient blocks', by putting indices i' , i" in the same block if and only if ry = r;» (e.g.: part(#i + 2#2 + 2g3) = {{!}, {2, 3}, {4, . . . , m}}; part(O) = part(#i + • • • + gm) = {/}). It is easy to check that this definition is independent from the choice of the linear combination of the redundant base expressing x. Setting, for E C I, TE = A n, i€E

we have: Lemma 1.1 (e.g., [6]). The type of x is determined by part 7 (x): (1)

type G (x) = i(part 7 (z)) = t(#) = r/XCl V • • • V

Hence, associating to the i-th column of M the partition we have (vi, . . . , vm) = (t(tfi), . . . , t(
D

The map t from the lattice P(m) ( V, A) of partitions of / to the lattice T of types defined by t(^?} = 17^ V • • -\/rj\ck is an order map, preserves all A and certain V; its abstract properties are studied in [4]. Im(t) = typeset (G), the set of types of G (= set of types of all elements of G), is a sub-A-semilattice of T, and being finite is itself a lattice, with a supremum \/ that in general exceeds the supremum V of T. In fact, as can be seen in the next Proposition 1.2, typeset(G) takes its infimum from T, its supremum from P(m). The map t needn't be injective. For a € typeset(G), the minimum partition *& such that a = t(^) is denoted by ^ = part t (0-). Thus a — i(part t (cr)). Note that part 7 (5j > part t (type G (5j). Proposition 1.2 ([6]). partt : (typeset (G),V) —> (P(m), V) is a monomorphism of join-semilattices. D Minimum partitions play a fundamental role in Step 1 : they allow us to represent as a .B^-group, for every a £ typeset(G), the classical fully invariant subgroup of G

122

C. De Vivo and C. Metelli

In fact, define for E 6 / 9E = E 9i = iSS

and for a partition ^ = {Ci, . . . ,

Note that 5^ = &. Proposition 1.3 ([6], Lemmas 1, 2). For partitions 38, *£ of I we have: ii) GCtf) = (gci)* + ---- 1- (9ck)*, hence it is a regular B^ -group of rank \c&\ — 1; iii) G(^) is a pure subgroup of G. Moreover iv) G( a. hence t(part.7(o)) > a. then by definition part 7 (g) > part t (a), hence g £ G(part t (cr)); conversely since part7( part t (cr) and t is an order map, we have t(part 7 (g)) > £(part t (cr)) — a. D A type a e typeset(G) is ^-irreducible or a jump if a > \/{T 6 typeset(G) | T < a }, that is, if a covers at most one type of G. As in any finite lattice, jumps are the smallest set of V-generators of typeset(G). Due to this observation, jumps have been chosen to characterize typeset(G) in many occasions (and were called primes). In all of these, though, typeset(G) was known, so y could be used. But if one must construct a .B^-group H from a given set of its types, hence V is not known, we need a supremum that is independent of H. Fortunately (1) of Lemma 1.1 comes to rescue: each type in typeset(G) is a V (supremum in T) of types in typeset(G), thus, even if typeset(G) is not closed with respect to V, we can give the following Definitions 1.4. A type a e typeset(G) is called prime, if it is V-irreducible, i.e., if a > V{°" £ typeset(G) T < a}. We list the primes of G: TTI, . . . , 7rn, with r = {l,...,n}. A partition all whose blocks but at most one are singletons is called a pointed partition. For A C /, we denote by p^ the pointed partition

with p = {/} and p/ the minimum of P(m).

D

In Section 5 we discuss an example clarifying the jumps-primes choice. Lemma 1.5. 1) Jumps are primes. 2) A prime TT is a jump if and only if the type Vi^V < T \ r &T}, which is ^ TT by hypothesis, is in typeset(G). 3) If 7Tr is a prime, then for some subset Ar of I we have 7rr = TAT\ and part t (?r r ) is the pointed partition pAr •

123

A Constructive Solution . . .

Proof. 1) When V applies in typeset(G), it coincides with \/; therefore V-irreducible implies V-irreducible. 2) TT is not a jump if it satisfies TT — V{ 7 r r < 7 r | r e r } > \f{irr < TT | T € F}; the last item, in T, is > TT, hence cannot belong to typeset(G). 3) We have 7rr = t(part t (7r r )) = t(<#) = t({Ci, - . - ,Ck}) = T/\CI V ••• Vr A C f c ; but being V-irreducible means 7rr = 17^ = t(pi\d) f°r some 1 < i < fc. Thus tf = part t (7r r ) < @i\d; but pi\d < ^- Hence ^ = P/\Ci and -Ar = I \ G». D Remark 1.6. By their definition, the primes V-generate typeset(G). For instance, Ti = V{ 7Tr

7I> < TTi } = V{ ^r

i € Ar }.

From Lemma 1.5 we have Corollary 1.7. Let 7i> &e a prime o/typeset(G) and sei Ar = {ii,... , i k } - Then for the regular B^ -group G(nr) we have G(TIY) = (g^})* H ----- h {^{ifc.!})* + (5/\^ r }*. and, if G(irr) ^ G, then rkG(7r r ) = \Ar . In particular, G(irr) = (gi | i e A r )» is purely generated by elements of the redundant base 7. D Associate now to the type cr of G the set of indices c"| = { r e r J 7 r r < f f } . Then we have Since each type a & typeset(G) is the V (hence the V) of the primes 7rr < a, and partt preserves V (Proposition 1.2), we have Lemma 1.8. part t (cr) = part t (\/{ KT

D

}) = \/{ f>Ar

We now build the m x n incidence table T of the G(irr) (r 6 F) with respect to the base 7, which we call the "/-tent of G (see the next Example 1). (Note that up to Theorem 4.1 included, everything can be done using only jumps instead of primes; it is in Proposition 4.2 that primes become indispensable.) In the tent T, Ar is the support of the r-th column; if G(7iy) ^ G then the r-th column must have at least two zeros (m - 1 base elements are enough to generate G, which then contains the m-th as well). For instance in the tent

Example 1

gl

1

92 53 54 95

0 1

gQ

0

1 1

G(7T 2 )

G(7T 3 )

G(7T4)

G(7T 5 )

0 1

0

1

1

0

1 1

1

0 0

1

1 0

1 1

1 1

0

1

0 0

1

1

we have G(TTI) = (31)* + (o 3 )» + (c/ 4 )» + (ffs)* + (ff{26}>* = (91,93,94,95)*; AI = {1,3,4,5}; pAl ={{!}, {3}, {4}, {5}, {2,6}}; rkGfri) = |^| = 4 . For a = TTZ V 7T4 VTTS we have part t (a) = pA, V pA4 V pA5 - {{1, 3, 4}, {2, 5}, {6}}, G(
124

C. De Vivo and C. Metelli

knowledge of G we couldn't have chosen a = TTS V 714 V TTS, because we wouldn't know whether such a a was a type of G. Observation. part t (cr) can be easily computed directly from the tent T: select the columns of the 7rr < cr; string together zeros horizontally and vertically; then pull on one of the strings: the rows that come out strung together yield the blocks of the minimum partition of a (for more details see [6]). Theorem 1.9. Let elements xi, . . . , Xk ofG satisfy (k — 1)-independence and zerosum. Let TI be the incidence table of those elements with the subgroups G(TTI), . . . , G(7r n ), where TTI, . . . , Trn are the primes o/typeset(G). Then a) for all i £ /, type G (xj) can be read from the i-th row o/Ti; b) If TI = T, that is k = m and iypeG(xi) = typeG( 7i>; moreover type G (xj) is the V of the primes 7rr below it. Thus typeG(:E;) can be read from its row: it is the V of those primes whose index belongs to the support of its row. E.g., in the above example the type of g\ is TI = iri V it± V 7r$. b) Prom the type TJ we get (xi)« up to isomorphism; the subgroup K = (xi)* + • • • + (xm)* of G is then quasi-isomorphic to G (see e.g., Lemma 1.3 of [9]). Then K is of finite index in G, hence in our setting coincides with G. D

2. Step 1: from Q to Z2 Consider now a new base 7' = (hi,..., hm) of G with type-base (vi,... ,vm) as given at the beginning; call T" the 7'-tent of G. Let party be the analog of part 7 ; for F C /, set VF — AigF^' define the order map u: P(m) —> T by u(^) = v i\Ci V • • • V u/\Cfc> with part u the analog of part t . For each r s F let Br be the subset of I such that partjTiv) = pBr,

that is, G(KT) = (hi

i £ Br)*;

then Br is the support of the r-th column in the 7'-tent T'. Since the rank of G(irr) is clearly unchanged, we have \Ar\ = \Br for each r in F; that is, in T' the number of ones in each column is the same as it was before. Analogously, by Proposition 1.3, for every a in typeset(G) the number of blocks in the new minimum partition of the type a: part u (
125

A Constructive Solution . . .

full,

T' hi h2 HZ

h^ h§ hfi

G(7Tl)

G(7T2)

1

1

0 1 1 1 0

G(7T3)

1

0

0

0 0 1

0 1 1

1

1

G(7T4)

0

1 1 1 0

1

G(7T5)

1 1

0 0

1 1

Here the type of hi is v\ = wi V 7T2 V ir^ V ir^. For the type a = ir^ y 7^4 V ir^ we have part u (
T

b{\} b{2}

b{3} ^{4}

b{5} bie\ Here for instance V2 = (&{ 2 ), ^{3},

V2

1 0 1 1 1 0

V3

V5

1 1 1 1 0 1 1 1 1 0 0 0 1 0 0 1 0 1 1 1 1 1 0

0

C. De Vivo and C. Metelli

126

Note that by choosing the subspaces of the representation via an incidence table, we restrict the selected subspaces to be coordinate subspaces, i.e., generated from elements of /?. Definition 2.1. A representation (V; VT \ r e F) is a tent if the selected subspaces Vr are coordinate subspaces. An & -isomorphism2 (f> between two tents T = (V~ Vr \ r e T), T' = (V; Wr r £ F) is an automorphism of V such that
it is not difficult to verify that V(^) = (bCl , - • • , bCk and dim particular, recalling Lemma 1.5 3), we have

= k - 1. In

In the previous example, interpreting analogously the 7'-tent T' of G, we obtain new subspaces Wi, . . . , W$ of V:

T' Wl b{i] 1 b{2} 0 b{3} 1 b{4} 1 b{5} 1 b{6} 0

W2 1 0 1 0 0

1

1 0

1 0

1 1

W4 0

1 1 1 0

1

W5

1 1

0 0

1 1

Setting as before W(a) = P|{ Wr r eCTI} we have Theorem 2.2. For each subset a± o/F we have dimV(a) = dim W(cr). Proof. Since, by our hypothesis on G, the partition V{ PAr r € °"1 } has the same number of blocks as V{ Psr | r 6 a\_ }, we only need to show that V(a] (— n{^ / (pA I .) I r £ crj }) coincides with V(\/{ pAr \ T € a\ }). But this is a consequence of (8) in [7], stating that for all partitions ^, ^ of / we have V(£t§ V n V(^). D stands for 'representation'. We cannot use 'tent isomorphism' because in [4] this term had a more ore restrictive meaning. There, ^-isomorphic tents were called 'equigroupal'.

A Constructive Solution . . .

127

In fact, collecting previous results, it is easy to prove that the associations KT —» PAT —> Vr extend to lattice isomorphisms: Corollary 2.3. typeset(G) is isomorphic to the sub-M-semilattice o/P(m) generated by { £>Ar Ir G r }; and is anti-isomorphic to the tent lattice L(T). D 3. Step 2: solving the problem in Z2 We are now in condition to apply the main theorems of [3] and [8], that in our setting state: Theorem 3.1 ([3, Theorem 5.1]). If Theorem 2.2 holds, there is an &-isomorphism
TO/,™'*

~~4~*-:~~ .*-u~

— JUM/ / 1 1 DCttlll^, tile

on

natural object is the base (&{i}, • . . ,&{m}) which the partitions are built, while what changes is the family of distinguished subspaces (from the {Vr} to the {Wr}). Thus, in the 7'-tent T' of G, for the base change M we have i G Br if and only if hi G G(irr). When T' is interpreted on V, this becomes: i e 5r if and only if &{i} £ Wr: that is, for the ^-isomorphism tp, i € Br if and only if b^y G v(K-), i.e., i 6 Ar if and only if ip~l(b^) G K- This is why all our vector space isomorphisms, as the one in the previous Theorem, go backwards. The map (p can now be decomposed: Theorem 3.2 ([8, Theorem 2]). An &-isomorphism (p: T' - (V; Wr \ r e T] -» T — (V; Vr | r G F) decomposes as a product of exchanges ip = (p\ ... y>k, that is: 1) for each 1 < k' < k, is an &-isomorphism; 2) each
The proof realizes the decomposition as a finite algorithm. 4. Step 3: from Z2 to Q We first show that the automorphism ip obtained via Theorem 3.1 gives rise to a base change of G equivalent to M. Let E be an m x m Z2-matrix of

128

C. De Vivo and C. Metelli

and let A = diag(ai, . . . , a m ) be the diagonal m x m matrix of the a*; then the matrix .EA transforms the base (gi, . . . ,gm) into the m-tuple (xi, . . . ,xm), which satisfies both (m — l)-independence and zero-sum. For it to be a base, we need it to generate G up to quasi isomorphism. By Theorem 1.9 b) applied to the base 7' of G, we reach this result if we prove that typeG(:Ti) = Vi for all i £ /. By Theorem 1.9 a), to obtain this information we consider the incidence matrix of (xi,..., xm) with respect to the G(irr)Theorem 4.1. In the above situation, EA is a base change for G, equivalent to M. Proof. Since we only need to prove that type G (xi) = Vi, and by Remark 1.6 fj is a supremum of 7rr's, we only need to show that X{ € G(7rr) if and only if hi G G(7iy). Let then hi £ G(7rr); this is equivalent to b^ e Wr. Now b^j. e Wr = ip~1(VT) if and only if bEi = (p(b^) £ Vr. In turn, bEi £ Vr means by definition bEi > AT\ since bEi — part7 (g^v ) , this is equivalent to gEi e G(pAT) = G(TIV), hence Xi £ G(7r r ), this being a pure subgroup of G. The equivalence to M is now ensured by the fact that EA changes the type base (TI, . . . , r m ) into (vi, . . . , vm). D Observe that this result states that any new type-base of G can be realized via a base of G obtained from the original base "by bipartitions" (e.g., Xi = a^^). We are left to prove that the Z2-decomposition of E translates into a sequence of Q-base changes leading to the desired type base. For this, we need to build successive base changes corresponding to the factors (pi of if> given by Theorem 3.2. We have the B^-group G, its 7- tent, the attached incidence table T interpreted on V, and an ^"-isomorphism tjj: TI = (V; Wr \ r £ F) -> T = (V;Vr \ r € T). We will prove that 7\ is the tent of a B^-group quasi-isomorphic to G. Calling E = (bEl , . . . , bEm) the matrix of ip, for i G I we have in G elements 3/i = digEi ,

of type Ci = tEi ,

satisfying (m — l)-independence and zero-sum; here A = diag(o?i, . . . , d m ). Note that we do not know whether TI is a tent of G, as we did in Theorem 1.9. We need to prove that the subgroup Y = (yi)f + • • • + (ym}* of G is (quasi-)equal to G. Let F = (6pi , • • • , &F m ) be the matrix of i/1"1; as above for i e I we find in Y elements y'i = CiJ/Fi , of type zFi — CF; V d\Fi (see Lemma 1.1), satisfying (m - l)-independence and zero-sum. Proposition 4.2. zpi = TJ /or a/Z i e 7. Proof. <: CF, = A{ Cj \3 & K } = A{*^ I J € ^ } = i(A{^ | J e Ft }) <

^(E( &^ \i e -Fi }) = *(&{i>) = T«; the same nolds for C/\FJ-

>: We show first that each prime 7rr of G, TIY < TTJ, is also < C^. Since part t (Tj) < b^j, we have pAr = part 4 (7r r ) < part t (7Tj) < b^y, equivalently, i e Ar, and b{i} e Vr. Then 6Fi = V^H^i}) &, V'"1^) = ^f, thus 6^ > ps r , where we set Wr = V(pBr). Then

A Constructive Solution . . .

129

But i e Br, that is b^y £ Wr, if and only if bEi = ^(b^y) e i>(Wr] = VT, that is bEi > f>Ari which implies r^ > 7rr; thus zpi > Tiy for every 7rr < T». Since TJ = Vi^r I ""r < Ti}, we can conclude zpv > TV Note that if we only had jumps, this conclusion couldn't be reached: the V of the jumps below TJ could be smaller than TV D Theorem 4.3. Let the automorphism ^ of V transform the tent T, originating from the -y-tent of the primes of a JE^1' -group G, into a tent 7\. Then there is a base of G having Ti as its tent. Proof. Take for Y the subgroup of G constructed above with if} = 7rr, that is Og^ e Vr, or equivaientiy 6{i} € Wr; therefore TI is the 7"-tent of G, as desired. D Note that the new base is given by the columns of EA. We can now summarize Theorem 4.4 (Yom's decomposition). Every non identical type-base change M of the B^-group G can be realized by a sequence of exchanges. Proof. Start with the group G, its final base 7', and its tent T". Factorize the ^-isomorphism (p:T'—*T ensured by Theorem 3.1 into exchanges: (p = / p i . . . \ transforming T"2 into TI = T, with a base ipi that is equivalent to 7, having the same tent T. D The matrix which is the product of the Q-matrices obtained from the ipi, though, is not in general of the form EA. The decomposition is type-theoretical, not linear. 5. Jumps versus primes Theorem 4.3 dictates the rule under which a tent can replace the B^-group G for the determination of base changes, in the sense that every base change of the tent produces one on the group: the columns must be headed by the V-irreducible types (the primes) of G. The minimal requirement for a type IT to head a column of an incidence table is that G(rr) be purely generated by base elements, or in other words that the type TT be of the form TA for some A C I; primes (hence also jumps) satisfy this requirement by Lemma 1.5 3). Still, for many uses (for instance for Lemma 4.2 of [4], or for decompositions, [6]) jumps are enough (they were sometimes called primes). To clarify what happens we give an example. Let RO flj R2 RS RI

= = = = =

( 0 , 0, 0, 0, ...zeros...); (oo, 0, 0, 0,...zeros...); ( 0, oo, 0, 0 , . . . zeros...); (oo, oo, 0, 0 , . . . zeros...); (oo, oo, oo, 0 , . . . zeros . . . ) ;

set Ti = type of Ri for i = 0 , . . . , 4; observe that RI V R-2 = #3 < R^.

C. De Vivo and C. Metelli

130

Consider the B^-groups G = (go)* + (gi)* + (92}, + (53)* + (d'3)* where typeG(gi) = type G (c^) = TJ, hence its type base is (To,Ti,T2,T 3 ,T3); and where typeH(hi) = type ff (/i^) = TJ, hence its type base is (T 0 ,Ti,T 2 ,T 4 ,T4). Their typesets (not dashed) are isomorphic: T

4 = TI Y T2

— TI V T2 — TI Y T2

TO

TO

with TI, T2 the V-irreducible types in both cases. In H, r& is V-irreducible; so, while jumps are the same, primes are different in G and H. Setting TTI = TI, 7r2 = T2, TTs = T4, the tent T applies to both G and H if we consider only jumps in H; if we use primes, H has the tent TI.

TI

T

0 0 1 1 1

71"!

7T2

7T4

0

0 0

0 0 0

1 0

1

1 1 1 1 1 1

The base change i p o f V : E = (645, & 2 , &3, ^35, & 2 4), which transforms T into the tent

r

T'

7T2

1

0

1 1

0

multiplied by the diagonal matrix with entries (1,1,1, —1, -1), transforms the base of G into (#3 +53,51,02,-02 - 5s,-0i - 5s) with types (T 3 ,T 1 ,T 2 ,T 2 ,r 1 ); here type(5a) = type(5i + (-51 - 53)) = (T: A n) V (r3 A T2 A r2) = r3. Its inverse F = (&45,62,63,&25,&34) gets back ((-52-53) + (-5i-5 3 ),5i.52,5i-5i-53,52-52-53)> with the original type-base (TQ,TI, r2, r 3 ,r 3 ), as we checked above.

A Constructive Solution . . .

131

Instead, E transforms the base of H into (h± + /i4, hi, h2, —h2 — h'^, —hi — /i4) with types (T 4 ,Ti,T2,T2,Ti), but this is not a base of H, because here we have type(/i 4 ) = type(/ii - hi — ft4)) = (TI A TI) V (T4 A T2 A T2) = TS, not T4; hence the F-transform (—h 2 — h'4 — hi — h&, hi, h2, h2 — h2 — /i4, h2 — h2 — h'^) has types (TO,TI, T2, TS, TS), thus is the base of a proper subgroup of H. On the other hand, it is easy to verify that E is not a base change for Ti, since it transforms TI into the non-^-isomorphic tent '1

7T2

7T4

1

1

1

1

0

0

0 0 1

1 1 0

0 0 0

G. S^'-groups represented by a given tent The simplest way to build a B^-group from a given table is to write oo in place of 1 all over, and to interpret the rows as reduced types, with all entries after the n-th equal to 0; this is what we did in the last example. But if we want all such groups, here is how to choose the primes. Let T be an arbitrary (m, ra) table of zeros and ones, with the only condition that there is no column with exactly m — 1 ones. For r e F = {l,...,n}, call Ar the support of the r-th column. Choose a map { Ar \ r 6 F } —» T, Ar i—> 7rr satisfying the following conditions for all r,s,h € F: i) TIY > V{TT S | As D Ar}, ii) 7rr A TTS = Vl71"/!11 Ah D Ar U As }; where V 0 = /\{ TIY r e F }. It is easy to check that the map is then injective and anti-bi-ordered. As in Remark 1.6, for i 6 / set

Theorem 6.1. The B^-group G with type-base (ri,...,Tm) has (TTI, . . . , 7rn) as its primes and T as its tent] and any B^-group with tent T is obtained this way. Proof. Necessity is straightforward: i), ii) hold for primes. For sufficiency, using distributivity of T and condition ii) it is easy to show by finite induction that the sublattice of T generated by the TIY is in fact V-generated by them. Clearly TJ > 7rr if i e AT; conversely, if T^ = Vi 7 ^ i ^ e As } > ?rr, we need to show that i € Ar. By contradiction say i £ Ar; using i) and ii) we have TIY = TIY A tt = V{ TTr A TTS | i £ As } = V{ ^ \ Ah 2 Ar U As and i e As } > V{ ""ft Afc D Ar }; but this is not possible, since i £ Ar ensures that all the TT/J occur among the TTfc. Thus T is also the table of (TI, . . . ,T m ) with respect to (TTI, . . . ,7r n ). The table defines as usual the map t: P(m) —» T, t({Ci,..., C^}) = T/\CI V • • • V T/\ Cfc , with Im(t) = typeset (G) contained in the sublattice of T generated by TTI, . . . , TTU. Thus the vrr are V-generators of Im(i), hence they comprise the jumps, and are in fact primes by i). As in Lemma 1.5, a prime must be of the form TT = i(pyi) = TA,

132

C. De Vivo and C. Metelli

with TA > Vi71"' Ti"' is a prime < TT }. But TA is a supremum of TT/J by ii), hence can only satisfy ii) if it is a TT/J itself. D Here we determine all B^-groups with a given tent, by determining their primes, since the correspondence between a group and its primes is one-to-one. Although indirect, this is also the best way we know to compute all possible typebases (TI, . . . , r m ) for the tent T. A different question is to determine all type-bases for a given B W -group G (the type-base-change problem), since this essentially means describing all tents obtainable from a tent T of G by a type-base change. This can be done in an orderly fashion by first using the algorithm given in [8, Lemma 1] to list all exchanges on T (whose number is bounded by (2 m ~ 1 - 1)(™)); then applying the same algorithm on all tents thus obtained; repeating this last step 'enough times' will yield all tents ^-isomorphic to T, hence all type-bases of G. An upper bound for 'enough times' can be deduced from [8, Section 3] and is < ( m j 1 )- This is only a rough procedure, since the same type-bases will be obtained many times over; but it is the best we know to this day. References [1] D. M. Arnold, Abelian groups and representations of finite partially ordered sets, CMS books in Mathematics, Springer-Verlag, New York, 2000. [2] D. M. Arnold and C. Vinsonhaler, Finite rank Butler groups: a survey of recent results, Abelian groups (Curagao, 1991), Dekker, New York, 1993, pp. 17-41. [3] F. Barioli, C. De Vivo, and C. Metelli, On vector spaces with distinguished subspaces and redundant base, preprint. [4] C. De Vivo and C. Metelli, Finite partition lattices and Butler groups, Comm. Algebra 27 (1999), no. 4, 1571-1590. [5] , Admissible" matrices as base changes of B^1'-groups: a realizing algorithm, Abelian groups and modules (Dublin 1998) (Basel), Birkhauser, 1999, pp. 135-147. [6] , Decomposing B^-groups: an algorithm, Comm. Algebra, to appear. [7] , Zz-linear order-preserving transformations of tents, Ricerche Mat., to appear. [8] , A transvection decomposition in GL(n,2), Colloq. Math., to appear. [9] L. Fuchs and C. Metelli, On a class of Butler groups, Manuscripta Math. 71 (1991), 1-28. [10] H. P. Goeters and C. Megibben, Quasi-isomorphism and Z%-representations for a class of Butler groups, Rend. Sem. Mat. Univ. Padova 106 (2001), 21-45. [11] H. P. Goeters, W. Ullery, and C. Vinsonhaler, Numerical invariants for a class of Butler groups, Abelian group theory and related topics (Oberwolfach, 1993), Amer. Math. Soc., Providence, RI, 1994, pp. 159-172. [12] P. D. Yom, A characterization of a class of Butler groups. II, Abelian group theory and related topics (Oberwolfach, 1993), Amer. Math. Soc., Providence, RI, 1994, pp. 419-432.

On a Property of the Adele Ring of the Rationals Dikran Dikranjan Dipartimento di Matematica e Informatica, Universita di Udine, Via delle Scienze 206, 33100 Udine (Italy) dikranj aSdimi.uniud.it

Umberto Zannier Istituto Universitario di Architettura di Venezia, S. Croce, 191, 30135 Venezia (Italy) zannierSiuav.it, zannierSdimi.uniud.it Abstract. We show that the adele ring of the rationals has no automorphisms beyond the identity and discuss the application of this fact to questions related to dualities of locally compact modules over the rationals.

1. Rigid rings The computation of the automorphisms group Aut(A) of a commutative ring A must be of primary interest in ring theory. In this note we are concerned mainly with commutative unitary rings A with Aut(A)| = 1, we call them briefly rigid rings. It is well known that the field R of real numbers is rigid and for every prime p the field Qp of p-adic numbers is rigid. The adele ring A of the rationals Q is the product of the field R with the restricted direct product A of the fields Qp, i.e., A is the subring of the product Tlpep QP defined by A = J a e JJ Qp : k • a £ JJ Zp for some natural k > 0 I, *• per PGP ' 1991 Mathematics Subject Classification. Primary 22A05, 54H11; Secondary 54A25, 54A35, 54G20. Key words and phrases, ring automorphism, adele ring, dualities of locally compact groups. The first author was partially supported by Research Grant of the Italian MURST in the framework of the project "Nuove prospettive nella teoria degli anelli, dei moduli e dei gruppi abeliani" 2000.

133

134

D. Dikranjan and U. Zannier

where Zp is the ring of p-adic integers and P is the set of prime numbers. It is natural to expect that also the ring A is rigid. Since this fact does not seem to be widely known we offer a complete proof here. It splits in two parts. The first one, strongly relying on specific arithmetic properties of A, probably presents less novelty. The second one, less related to the nature of A, is isolated as a separate lemma. It works in a more general situation, permitting to establish easily rigidity of certain commutative rings. In §2 we discuss an application that motivated our interest in this property of A. Theorem 1.1. Let A be the adele ring of Q. Then the only ring monomorphism (f>: A —> A is the identity of A. Proof. First we prove that all fields R, QP (p 6 P) are rigid. Although this fact seems to be well known, we give here a complete proof for reader's convenience. To see that the field R is rigid notice first that every ring homomorphism (p: R —> R preserves the order, due to the fact that every positive real is a square. This entails that tp is continuous. Since non-zero ring homomorphisms R —> R are identical on Q, the density of Q implies that

Qp- Let t/(Z p ) = Zp \pZp. We prove first that (£) = Q p , while b witnesses that there exist no ring monomorphism Qp —> R. It remains to check that if p and q are prime numbers and there exists a ring monomorphism tp: Qp —> Q g , then p = q. Assume p ^ q. If q = 2, then p is odd. Take an integer n such that n = I (mod p) and n = — 1 (mod 8). Then n is a square in Qp, while n =
On a Property of the Adele Ring of the Rationals

Now we can apply Lemma 1.2.

135

D

Lemma 1.2. Let {-Ki}ig/ be a family of rigid fields such that (1)

there exist no ring monomorphism Ki —> Kj for i =£ j in I.

Then every subring B of A — Yli&1 Ki containing the ideal © l€/ Ki is rigid. Proof. For every i 6 / consider Ki as a subring of A in the obvious way and let ei denote the unit element of K^. Now take an arbitrary ring automorphism 7: B —> B. Fix an arbitrary i £ /. Then et- e B and Bei — Aei = K^. Since f ( e i ) is again a non-zero idempotent of B, the principal ideal ~f(Bei) = B~f(ei) of B generated by j(ei) is a unitary subring of B. Moreover, it is a field since Bei = Ki is a field. Hence the idempotent 7(6,) must coincide with some of the idempotents BJ , j e /. Then the restriction 7 \K. of 7 to Ki induces a ring isomorphism between Ki and Kj. Hence j = i by hypothesis (1) and 7 IK,— idpd as the field Ki is rigid. In particular, for every b e B one has 7(6)et- = 7(6)7(64) = 7(66;) = 6ej. Hence, (7(6) — 6)e» = 0. Since this is true for every i e /, we conclude that 7(6) — 6 = 0. This proves that 7 is the identity of B. D Remark 1.3. (a) The reader can easily check that the proof of the above lemma still works if the fields {-Ki}jg/ are replaced by indecomposable unitary rings. For example, the ring FJ eP Z/p np Z is rigid for every choice of the sequence fjip) of natural numbers. (b) Recall that an idempotent e of a commutative unitary ring A is minimal if the subring Ae is indecomposable. The proof of Lemma 1.2 is based on the following two simple facts: (t>i) every ring automorphism sends the minimal idempotents to minimal idempotents; (b2) the rings A and B have sufficiently many minimal idempotents in the sense that aei = 0 for some a 6 A and all Cj implies a = 0. In particular, the above proof allows us to show that two automorphisms 7,7': B —» B coincide whenever 7 |~®. J /c 1 = 7' t0. ^ without the assumptions that the fields Ki are rigid and (1) holds (in other words, the group homomorphism p& '• Aut(A) —> Aut((J)ie/ Ki) defined by the restriction, is a group monomorphism; note that the ring 0 icr Ki is not unitary if / is infinite). This entails Aut(A) = Aut(0 i6/ Ki) and these automorphism groups are also isomorphic to Aut(-B) whenever the subring B of A is stable under all automorphisms of A. (c) One can easily show that if a commutative unitary ring B has sufficiently many minimal idempotents (in the sense of (bg)), then B is isomorphic to a subring of a cartesian product A = Y[iel Ai of indecomposable rings Ai, such that B contains the ideal 0 i€ / Ai of A.

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2. An application to dualities over the rationals Q Here we discuss briefly a recent application of Theorem 1.1 to questions related to dualities over the rationals Q. Further details and proofs can be found in [2, 7]. Consider the field Q equipped with the discrete topology. The category £Q of all locally compact topological Q-modules has an involutive contravariant functor * : £Q —> £Q defined by the Pontryagin duality as follows: M* = Chom(M, R/Z) is the Q-module (equipped with the compact-open topology) of all continuous group homomorphisms of M in the circle group R/Z. The canonical isomorphism M —* M** is defined by the evaluation in points of M. The compact module K = Q* plays a relevant role in this since M* is isomorphic, as a topological Q-module, to Chom(M, K) (note that K belongs to £Q, while R/Q does not). Any (abstract) contravariant functor F: £Q —> £Q will be called duality of C, (see [7]; for involutive dualities, i.e., those satisfying F • F = idcQ, see [3, 2]). The compositions p, of the Pontryagin duality * with F (after appropriate presentation of F as Chom(—, K) at the level of abstract modules and identifying M** = M) can be thought of as an equivalence p,: £Q —> £Q such that for every module M (E £Q the module p.(M) has the same underlying module M (so only the topology is modified under p.). Moreover, p, has also the following properties: (a) p, preserves finite products and exact sequences, (b) fj, is the identity on discrete modules. Roughly speaking, p, is a functorially defined twist of the topology, so that F(M) is obtained from the Pontryagin dual as p,(M*). We call the duality .F continuous if p, is the identity, otherwise F is discontinuous. Therefore, F is continuous iff F = *. For every commutative ring R one can define in a similar way dualities of the category CR of all locally compact topological /^-modules ([3]) and their (dis)continuity. The Pontryagin duality *: LR —> LR is always available, but it is not clear whether this is the unique functorial duality of C.R ([!]). The failure of uniqueness is often due to the existence of discontinuous dualities. Roeder [9] proved that LI admits no discontinuous dualities (for an alternative proof due to Prodanov, see [3] or [4, §3.4]), while CR admits discontinuous dualities for fields R of cardinality > c and char R = 0 ([3]) or when R = Z[i] is the ring of Gauss integers ([!]). It was proved in [2] that every involutive duality of £Q is continuous. Here we offer a sketch of the proof of the following stronger result obtained in [8]. The proofs are similar, both use essentially the rigidity of the ring A (it follows from the proof of Theorem 1.1), as well as an appropriately strengthened version of Mackey's factorization theorem of the modules in £Q [6]. Theorem 2.1 ([2, 8]). The category £Q admits no discontinuous dualities. Sketch of the proof. It is enough to see that every //: £Q —> £Q as above must be the identity. The proof of the equality /n(R) = R is based on the fact that the field M is rigid (the line of the proof is the same as in [1], in the case of Z[i]-modules). Extending further the same idea from [1] also for the Q-module A one can show

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the equality /u(A) = A using this time the rigidity of the ring A. Now (a) yields M (A) = A. Consider now the module A € £Q and the canonical immersion of Q in A. It is known that Q —> A -* IK —> 0 apply fj, to get the exact sequence 0 —> Q —> A —* |i(K) —> 0 (using the equalities /z(Q) = Q, due to (b), and //(A) = A). The continuous surjective homomorphism /: A —> //(K) is open as A is locally compact and u-compact ([5]). Thus /u(K) is the topological quotient group of A under /, so /x(K) = K. It is known that this yields p, is the identity [8] (see [3, 2] in the case of involutive dualities). D References [1] D. Dikranjan, Uniqueness of dualities, in: A. Facchini and C. Menini eds., Abelian Groups and Modules, Proc. of the Padova Conference, Padova, Italy, June 23-July 1, 1994, Mathematics and its Applications, vol. 343, Kluwer Academic Publishers, Dordrecht-Boston-London, The Netherlands, 1995, pp. 123-133. [2] D. Dikranjan and C. Milan, Dualities of locally compact modules over the rationals, J. Algebra, 256 (2002) 433-466. [3] D. Dikranjan and A. Orsatti On an unpublished manuscript of Ivan Prodanov concerning locally compact modules and their dualities. Comm. Algebra, 17 (1989) 2739—2771. [4] D. Dikranjan, I. Prodanov, and L. Stojanov, "Topological groups (Characters, Dualities, and Minimal Group Topologies)1', Marcel Dekker, Inc., New York-Basel. 1990. [5] E. Hewitt and K. Ross, Abstract Harmonic Analysis, vol. 1, Springer-Verlag, Berlin (1963) [6] G. Mackey, A remark on locally compact Abelian groups, Bull. Amer. Math. Soc. 52 (1946), 940-944. [7] C. Milan, Continuous dualities of locally compact modules, Ninth Prague Topological Symposium, Prague 2001, Extended Abstracts, pp. 122-128. [8] C. Milan, Continuity of dualities of locally compact modules over commutative rings, in preparation. [9] D. W. Roeder, Functorial characterization of Pontryagin duality, Trans. Amer. Math. Soc. 154 (1971), 151-175. [10] A. Weil, Basic Number Theory, Classics in Mathematics. Springer-Verlag, Berlin, 1995. xviii+315 pp.

On Strong Going-between, Going-down, and Their Universalizations David E. Dobbs Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1300, U.S.A. dobbsflmath.utk.edu

Gabriel Picavet Laboratoire de Mathematiques Pures, Universite Blaise Pascal, 63177 Aubiere Cedex, France Gabriel.PicavetOmath.univ-bpclermont.fr Abstract. We consider analogies between the logically independent properties of strong going-between (SGB) and going-down (GD), as well as analogies between the universalizations of these properties. It is shown that if A is a (commutative integral) domain, then A C B satisfies SGB for each domain B containing A (resp., each simple overring B of A; resp., each valuation overring B of A) if and only if A is a going-down domain. Also, a domain A is a universally going-down domain if and only if A C B is universally SGB for each overring B of A. However, for any nonzero ring A and indeterminate X over A, the inclusion map A °-> A[X] is not universally SGB.

1. Introduction All rings considered below are commutative with identity; all ring extensions and ring homomorphisms are unital. In part, our interest here is to further work on the going-down (GD) property of ring homomorphisms. (To facilitate the discussion, we adapt the notation in [11, p. 28], by letting GD and GU denote the going-down and going-up properties, respectively, for ring homomorphisms; we also let Spec(A) denote the set of prime ideals of a ring A.) Recall from [2] and [6] that GD has been used to introduce a class of (commutative integral) domains that includes ail 2000 Mathematics Subject Classification. Primary 13B24, 13G05; Secondary 13C15, 14A05, 13B40, 13A15. Key words and phrases. Strongly going-between, going-down, universalization, integral domain, overring, going-down domain, universally going-down domain, spectral map, flat, universally catenarian.

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Priifer domains and all domains of (Krull) dimension at most 1. Indeed, according to [6, Theorem 1], a domain A, with quotient field K, is a going-down domain if and only if A ^-> B satisfies GD for all domains B that contain A (resp., for all overrings B of A; resp., for all valuation overrings B of A; resp., for all rings of the form B = A[u], u £ K). The first goal of this paper is to develop an analogous result for the strong going-between (SGB) property that was recently introduced in [13]. Following [13], we say that a ring homomorphism /: A —> B satisfies SGB in case the following condition is satisfied: if PI C P2 C P3 in Spec(A) and Qi C Q3 in Spec(jB) are such that f~l(Q\) = PI and /~ 1 (Qs) = PS, then there exists Q^ in Spec(B) such that Qi C Q2 C Q3 and /~1((52) = PI- The terminology has been chosen in order to avoid confusion with the different "going-between" (GB) property introduced by Ratliff. (Observe that SGB =>• GB. Also, for motivation, note that the class of GB-rings was introduced in [14] and studied further in papers such as [15].) Families of examples of ring homomorphisms satisfying SGB have been given in 13, Propositions 4.13 and 4.16, Corollary 4.17]. There are some evident structural connections involving the SGB, GD, and GU properties. It has been noted 13, Proposition 4.2] that SGB can be characterized in terms of GD (and also in terms of GU). For the sake of easy reference, we collect this result and some other equivalences of SGB in Proposition 2.1 (a). Our first goal is achieved in Corollary 2.3, where it is shown that the above statement of [6, Theorem 1] remains valid if "GD" is replaced throughout by "SGB." In other words, what might be called an "SGB-domain" is precisely the same as a going-down domain. In view of Corollary 2.3, it seems natural to ask if the analogy between SGB and GD can be extended. Proposition 2.4 identifies an obstacle in this regard, as it constructs examples showing that the SGB and GD properties are logically independent. Moreover, Section 2 closes with Remark 2.5 (f), which presents an example of an overring extension A C B of domains that satisfies GD but not SGB. This example depends, in part, on an example of Kaplansky [10] that resolved a conjecture of Krull [12] concerning the GB property for certain extensions of domains. Section 3 treats possible analogies between the "universally SGB" and "universally going-down" properties. Success is achieved in Theorem 3.1 (c) for overring extensions of domains: the concepts are shown to coincide in this context. This result, whose proof depends extensively on the work in [4] on universally goingdown homomorphisms, leads us to consider whether the class of universally goingdown domains (in the sense of [5]) admits an analogue of Corollary 2.3 (and [6, Theorem 1]). This search to characterize what might be termed the "universally SGB-domains" achieves partial success in Corollary 3.2. This result establishes, i.a., that a domain A is a universally going-down domain if and only if A =—> B is universally SGB for each overring B of A. This work leaves open the following question (which does not have an interesting SGB- or GD-theoretic analogue). If A is a universally going-down domain, is it possible to enlarge the class of "test domains" B appearing in the statement of Corollary 3.2? Along these lines, one could ask for characterizations of the domains A such that A <—> B is universally SGB for

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all domains B containing A. It turns out that there are none (Corollary 3.9)! In fact, our deepest result, Theorem 3.7, asserts that if A is a nonzero ring and X is an indeterminate over A, then A <—> A[X] is not universally SGB. The lion's share of its proof is relegated to Lemma 3.4, which features our most intricate argument. More concrete formulations that avoid the "universally SGB" terminology are given in Proposition 3.5, Corollary 3.6 and Remark 3.8. In addition to the notational conventions indicated above, we mention the following. If A is a ring, then dim(yl) denotes the Krull dimension of A. If A is a domain with quotient field K, then an averring of A is any ring B such that A C B C K. Any unexplained material is standard, as in [7], [8], [11]. 2. Relating SGB and GD We begin by stating some elementary but useful equivalences from [13] and a useful consequence concerning diagonal maps. Proposition 2.1. (a) (G. Picavet [13, Propositions 4.2 and 4.7]) Let f : A -> B be a ring homomorphism. Then the following conditions are equivalent: (1) / satisfies SGB; (2) The induced map AP —> BQ satisfies GUfor all Q <E Spec(B) and P := f ~ 1 ( Q ) ; (3) The induced map A/P <-> B/Q satisfies GD for all Q e Spec(B) and P :=

/-'(Q);

(4) The induced map A/P <-> B/Q satisfies SGB for all Q 6 Spec(B) and P :=

/-'(Q). (b) // /i: A —> Si, • • • , f n : A — > Bn are ring homomorphisms satisfying SGB (resp., universally SGB), then the diagonal map A —> B-\_ x • • • x Bn also satisfies SGB (resp., universally SGB). The following definitions are needed for Corollary 2.2 (b). As recalled in the Introduction, a domain A is called a going-down domain in case A C B satisfies GD for each domain B containing A. Following [3], a ring A is called a going-down ring in case A/P is a going-down domain for each P s Spec(J4). Corollary 2.2. Let f : A —•> B be a ring homomorphism. Then: (a) If B is a domain and f is injective and satisfies SGB, then f satisfies GD. (b) // A is a going-down ring, then f satisfies SGB. Proof. Assertion (b) appears as [13, Corollary 4.3(1)], but its proof is included here for the sake of completeness. Both assertions follow by applying condition (3) in the statement of Proposition 2.1 (a). In detail, for (a), take Q := 0; and, for (b), use the above definitions of going-down ring and going-down domain. D In view of Corollary 2.2, it seems natural to consider what might be termed an "SGB-domain," that is, a domain A. such that A C B satisfies SGB for each domain B containing A. As Corollary 2.3 records, this notion is equivalent to that of "going-down domain." In fact, we have the following SGB-theoretic analogue of a GD-theoretic result [6, Theorem 1],

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Corollary 2.3. Let A be a domain with quotient field K. conditions are equivalent: (1) A C A[u] satisfies SGB for each u e K; (2) A C B satisfies SGB for each valuation averring B of A; (3) A C B satisfies SGB for each domain B containing A; (4) A is a going-down domain.

Then the following

Proof. (4) =>• (3): Each going-down domain is a going-down ring [3, Remark (a), p. 4]. Hence, a proof follows by applying Corollary 2.2 (b) and the definition of a going-down domain. (3) =>• (1); (3) => (2): Trivial. (1) => (4); (2) => (4): In view of Corollary 2.2 (a), these implications follow from some of the characterizations of going-down domains [6, Theorem 1] that were recalled in the Introduction. D Despite the preceding two corollaries, we show next that neither SGB nor GD implies the other, even for ring extensions of a domain. Proposition 2.4. The SGB and GD properties are logically independent. In fact: (a) There exists an inclusion map A c—> B that satisfies SGB but does not satisfy GD. For each d > 1, it can be arranged that A is a d-dimensional quasilocal domain and that B is a d-dimensional reduced ring. (b) There exists an inclusion map i: A <-> B of domains that satisfies GD but does not satisfy SGB. It can be arranged that A is a two-dimensional domain and B = A[X], in particular that B is A-flat and hence that i is universally going-down. Proof, (a) Let (A,M) be any 1). Consider the reduced ring B := A x A/M, whose spectrum is the union of a copy of the spectrum of A and an additional prime A x 0, incomparable to all the others. The diagonal map A <—> B has the asserted properties. Indeed, GD fails to hold because A x 0 lies over the nonminimal prime M; however, SGB holds by, for instance, Proposition 2.1 (b). (b) Let y, Z, X be algebraically independent indeterminates over a field K\ put A:=K[Y,Z] and B :— A\X] = K [Y, Z, X}. It is well known that the inclusion map i: A >-»• B satisfies GD. Now, consider the ideals Qi := (YX - Z) and Q3 = (Y, Z) in B. Evidently, Q\ C Qs; and, by degree arguments, Qi n A = 0 and Qs n A = (Y, Z)A. Note that there exist prime ideals of A that are properly between 0 and (Y, Z)A, but there are no primes of B properly between Q\ and Qs (since they have height 1 and 2, respectively, in B [11, Theorem 149]). Thus, i does not satisfy SGB (indeed, i does not even satisfy GB). D Remark 2.5. (a) According to a realization result of Hochster [9, Theorem 6 (a)], Spec is invertible on the subcategory of spectral spaces and surjective spectral maps. It is possible to use this result to give an alternative proof of Proposition 2.4. In particular, one can thus produce an inclusion map i: A <—> B that satisfies SGB but does not satisfy GD for which A is a one-dimensional quasilocal domain

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and B is a zero-dimensional reduced ring with exactly two prime ideals (that is, B is isomorphic to the direct product of two fields). In contrast to the construction in Proposition 2.4 (a) when d > 2, any such example has the disadvantage that SGB is satisfied vacuously. A concrete example of the situation depicted above is given next. Take A :— 1<2z and B := A[X]/((2,X) n (IX - 1)). It suffices to observe that the prime ideals (2, X) and (IX — 1) are maximal in A[X], the first of these primes meets A at 2A, and the second meets A at 0. (b) In the spirit of part (a), one can use [9, Theorem 6 (a)] to prove the existence of an inclusion map i: A <—> B that satisfies GD but does not satisfy SGB for which A is & two-dimensional quasilocal treed domain and B is a, two-dimensional quasilocal ring with exactly two minimal prime ideals. (c) One can improve the conclusion of Proposition 2.4 (b) as follows. There exists an inclusion map of domains i: A '—> B such that B is a, free A-module of finite rank (whence B is A-&a,t and i is universally going-down) and also such that i does not satisfy GB. Indeed, it is enough to use [10, p. 96]. There, Kaplansky constructed domains A C B such that B is a free A-module of rank 2 and A C. B does not satisfy GB. Any such extension has the asserted properties, since free implies flat; flat implies universally going-down (cf. 8, Corollaire 3.9.4 (ii)]); and, as noted in the Introduction, SGB implies GB. It should be noted that the above-mentioned example of Kaplansky answered in the negative a question of Krull [12], for in this example, A is integrally closed and (being a module-finite algebra) B is integral over A. Subsequently, a Noetherian example to the same effect appeared in [15, Theorem 2.3, Proposition 2.6] (cf. also [I])(d) The examples in part (c) and Proposition 2.4 (b) showed that a flat extension of domains need not satisfy SGB. Nevertheless, if B is a flat overring of a domain A, then A C B does satisfy SGB. To see this, one need only apply condition (2) in the statement of Proposition 2.1 (a). Indeed, if Q e Spec(B) and P := f~l(Q), then the "flat overring" hypotheses ensures that the induced map Ap —> BQ is an isomorphism (by the proof of [16, Theorem 2]) and, hence, trivially satisfies GU. (e) Let (V, M) = K + M be a valuation domain, where K is a field and the maximal ideal M ^ 0. If D is a subring of K, then Spec(Z? + M) is well understood as a set (cf. [7, Exercise 12 (2), p. 202]). By using this information about the classical D + M construction, it was shown in the proof of [6, Corollary] that if DI C D2 are subrings of K, then D\ + M C £>2 + M satisfies GD if and only if DI C DZ satisfies GD. By using the same information, one can infer that SGB behaves analogously to GD in this regard. More precisely, if D\ C D% are subrings of K, then Dl + M C D2 + M satisfies SGB if and only if Dl C D2 satisfies SGB. (f) The first assertion of Proposition 2.4 (b) may be sharpened as follows. There exist a domain R and an overring T of R such that R C T satisfies GD but does not satisfy SGB. To see this, let A C B be as in one of the examples, such as that of Kaplansky [10, p. 96], that were cited in part (c). Let K denote the quotient field of B, and let (V, M) = K + M be a nontrivial valuation domain, such as the

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ring of formal power series Jf[[X]]. Then, by the facts recalled or established in part (e) above, it suffices to take R := A + M and T := B + M. 3. Relating the universalizations of SGB and GD Just as Section 2 was devoted to analogies between the SGB and GD properties, we devote the present section to analogies between "universally SGB" and "universally going-down." One such analogy is already known: a ring homomorphism A —> B is universally SGB (resp., universally going-down) if and only if the induced ring homomorphism of polynomial rings A[X\,..., Xn] —> B[Xi,... ,Xn] satisfies SGB (resp., GD) for each nonnegative integer n [13, Corollary 4.11] (resp., [4, Corollary 2.3]). We focus especially on injective homomorphisms involving domains. Theorem 3.1 (a) may be viewed as the "universal" analogue of Corollary 2.2 (a). On the other hand, Theorem 3.1 (b) shows that "universalization" of SGB-GD interplay does not always lead to valid results. Indeed, Remark 2.5 (f) provided an example of an overring extension of domains that satisfies GD but does not satisfy SGB, while Theorem 3.1 (b) shows that no overring extension of domains satisfies the analogous "universalization." For the proof of Theorem 3.2 (b), it is helpful to record the following material. Recall that a ring extension A C B is said to be mated if, for each P € Spec(A) such that PB ^ B, there exists a unique Q e Spec(B) such that Q n A = P. It follows easily from the definitions that if a ring extension is mated and satisfies GD, then it also satisfies SGB. Theorem 3.1. (a) Let f:A<—>>B be an extension of domains. If f is universally SGB, then f is universally going-down. (b) Let A be a domain and let B be an overring of A. If A <—> B is universally going-down, then A ^-> B is universally SGB. (c) Let A be a domain and let B be an overring of A. Then A ^ B is universally SGB if and only if A <—> B is universally going-down. (d) If B is a flat overring of a domain A, then A ^-> B is universally SGB. Proof, (a) By the above remarks, it suffices to show, for each nonnegative integer n, that the induced map A[Xi,..., Xn] —* B[Xi,..., Xn], which does satisfy SGB, also satisfies GD. This is an immediate consequence of Corollary 2.2 (a). (b) Note that for each nonnegative integer n, A[X±,... ,Xn] C B[X\,... ,Xn] is an overring extension of domains that is universally going-down, and our task is to show that this extension also satisfies SGB. Thus, by replacing A (resp., B) with A[Xi,..., Xn] (resp., B[Xi,...,Xn}}, it suffices to prove that ACS satisfies SGB. As A C B satisfies GD, it follows from the above comments that we need only prove that the extension A C B is mated. The rest of the proof depends on the UGD property of ring homomorphisms, as well as some of the other machinery and three results in [4]. (See [4, pp. 420-421] for the rather complicated definition of the UGD property.) According to [4, Theorem 3.17] (which was called "the principal application" of [4]), an overring extension R '—> T of domains is universally going-down if and only if the induced extension

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R[X!,...,Xn] ^ T[X!,...,Xn] satisfies UGD for each n > 0. In particular, A <-^ B satisfies UGD. Therefore, by applying [4, Corollary 3.12 (b)], we now see that A °-> B is "universally strongly mated" (see [4, p. 412] for the definition of "strongly mated"). Thus, by [4, Theorem 2.5], the extension A C B is mated, as desired. (c) Combine (a) and (b). (d) Flat implies universally going-down. Apply (b). D Next, it is convenient to recall a definition from [5]. A domain A is called a universally going-down domain if A c—> B is universally going-down for each overring B of A. Recall that Corollary 2.3 established behavior for the SGB property that paralleled some basic behavior of GD [6, Theorem 1]. Corollary 3.2 presents a partial "universalization" of Corollary 2.3 and, thus, a partial analogue of [5, Theorem 2.6]. Corollary 3.2. Let A be a domain with quotient field K. conditions are equivalent: (1) (2) (3) (4)

Then the following

A t—> A[u] is universally SGB for each u £ K; A'—*B is universally SGB for each valuation averring B of A] A <—> B is universally SGB for each averring B of A; A is a universally going-down domain.

Proof. Combine the above definition of a universally going-down domain with Theorem 3.1 (c) and the appropriate characterizations of universally going-down domains given in [5, Theorem 2.6]. D We next isolate the most important instance of Corollary 3.2. Corollary 3.3. Let A be a domain. Then A is a Priifer domain if and only if A is integrally closed and A c-> B is universally SGB for each averring B of A. Proof. A domain is a Priifer domain if and only if it is an integrally closed universally going-down domain [5, Corollary 2.3]. Apply Corollary 3.2. D The comment prior to the statement of Corollary 3.2 referred to it as a "partial" analogue of Corollary 2.3 and [5, Theorem 2.6]. The "missing part" is unavoidably so, for it is not the case that a universally going-down domain A has the property that A c—> B is universally SGB for each domain B containing A. In fact, as we establish in Corollary 3.9 below, no domain A has this property! The crux of the argument is isolated in Lemma 3.4 below. Given the above remarks, it is perhaps useful to note that some non-overring extensions of domains do satisfy the universally SGB property. In fact, if a ring homomorphism /: A —> B is radiciel (in the sense of [8, Definition 3.7.2, p. 248]) and universally going-down, then / is universally SGB. (For a proof, reason as in the final comment preceding the statement of Theorem 3.1.) In particular, if k C K is a purely inseparable field extension, then k <—> K is universally SGB.

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Our original path to Theorem 3.7 and Corollary 3.9 was via the case of Corollary 3.6 in which A is an 5-domain. We are grateful to the referee for Lemma 3.4, Proposition 3.5, the present form of Corollary 3.6, and Remark 3.8. Lemma 3.4. Let A be a domain, and let b and c be nonzero elements of A such that b is a nonzero-divisor modulo cA. Let Q be the kernel of the evaluation map A(X] ->A[b/c}. Then: (a) Let P G Spec(A[X}) such that (Q, c) C P. Then b G P n A. Thus, PnA does not consist solely of zero-divisors modulo cA, and so Pt~}A is not minimal over cA. (b) If M £ Spec(A) contains b and c, then Q C Proof, (a): Since cX - b G Q C P and c G P, we have that b G P n A. The final assertion follows since minimal prime ideals consist solely of zero-divisors [11, Theorem 84]. (b): Suppose the prime ideal M of A contains b and c, and let / = f ( X ) = anXn+an-iXn~1+- • -+a-iX+a0 G Q. We shall show inductively that each a^ G M. If n = 0, the assertion holds since one then has f G QnA = Q. Next, suppose n = 1, so that / — a\X + OQ. Since f £ Q, f(b./c) = 0, whence a\b = — CO,Q G cA. As b is a nonzero-divisor modulo cA, we have «i G cA C M. Thus, ca0 = —a^b G cbA, and so canceling c shows ao G bA C M, this completing the induction basis. For the induction step, n > 1 and anbn + can-ibn~l + • • • + cn~laib + cna0 = cnf(b/c) = cn • Q = Q. Thus anbn G cA, and since 6 is a nonzero-divisor modulo cA, we have an G cA C M. With d :— an/c e A, consider the polynomial g = g(X) = (db+an-1)Xn-1+an-2Xn-2 + ---+a1X + a0 G A[X}. Since db(b/c)n-1 = an(b/c)n and f ( b / c ) = 0, we easily verify that g(b/c) = 0. In other words, g £ Q, and so, by the induction hypothesis, all the coefficients of g lie in M. Hence, a» € M for all i except possibly i = n — I. However, since db + a n _i e M and 6 £ M, we get a n _i G M as well, thus completing the induction step and the proof. D Proposition 3.5. Let A be a domain and suppose that b and c are nonzero elements of A such that b is a nonzero-divisor modulo cA. If (b, c) ^ A, then A C A[X] does not satisfy SGB. Proof. By hypothesis, we can choose M G Spec(A) containing b and c. Let Q be the kernel of the evaluation map A[X] —> A[b/c]. By Lemma 3.4 (b), Q C MA[X] =: P. Clearly, Qn A = 0 and PnA = M. Once again, we appeal to the fact that minimal prime ideals consist solely of zero-divisors [11, Theorem 84]. Indeed, since b € M is not a zero-divisor modulo cA, this result shows that the prime M cannot be minimal over cA. Thus, there exists N G Spec(^4) with N C M and N minimal over cA. If the assertion fails, A C A[X] satisfies SGB, and so there exists q G Spec(A[X]) such that Q C q C P and q n A = N. By the choice of N, this contradicts the final assertion of Lemma 3.4 (a), to complete the proof. D Corollary 3.6. Let A be a domain but not a field, and let X and Y be algebraically independent indeterminates over A. Then A[Y] C A[Y, X] does not satisfy SGB.

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Proof. By the hypothesis on A, we can choose a nonzero nonunit b of A. In -A[Y"], we see that 6 is a nonzero-divisor modulo y.A[y] and, of course, (b,Y) 7= A[Y}. Apply Proposition 3.5. D Theorem 3.7. If A is a nonzero ring and X is an indeterminate over A, then A <—» A[X] is not universally SGB. Proof. Deny. Since A ^ 0, we can choose a maximal ideal M of A. Put k := A/M. Since A <-* A[X] is universally SGB, so is k ^ k ®A A[X] = k[X}. If T is another indeterminate, algebraically independent from X, then the canonical map k[T](T) <-> k[T](T) ®k k[X] is also universally SGB. However, it is straightforward to use standard tensor product identities in order to verify that k[T](T) ®fc M-^1 is canonically isomorphic to fc[T](r)[X]. So, R := k[T](T) is such that R <-» R[X] is universally SGB. This contradicts Corollary 3.6, since R, being a DVR, is a one-dimensional domain. The proof is complete. D Remark 3.8. For readers who prefer formulations that are more concrete than the "universally P" terminology of 8, p. 240], we now give a result that not only generalizes Corollary 3.6 but is also more specific than the statement of Theorem 3.7. Let A be a nonzero ring and let Y, Z, X be algebraically independent indeterminates over A. If dim(A) > 0, then A[Y] C A[Y,X] does not satisfy SGB. If dim(A) = 0, then A[Z,Y] C A[Z,Y,X] does not satisfy SGB. The second assertion is a consequence of the first, for if dim (A) — 0, then dim(A[Z]) = 1. We next prove the first assertion. Since dim(^4) > 0, we can choose a nonmaximal prime ideal P of A. By Corollary 3.6, the domain D :— A/P is such that D[Y] C D[Y, X] does not satisfy SGB. In view of the canonical isomorphisms D[Y] ^ A[Y}/PA[Y^ and D[Y,X] ^ A[Y,X]/PA[Y,X], the assertion now follows from a standard homomorphism theorem. We pause to note, for the sake of completeness, that Theorem 3.7 and Remark 3.8 are best possible, in that the zero ring T is trivially such that T —» B is universally SGB for the unique (unital) T-algebra B(= T = T[X}). Finally, we record a way in which the behavior of "universally SGB" is fundamentally different from that of "universally going-down." Corollary 3.9. There does not exist a domain A such that A <—> B is universally SGB for each domain B that contains A. Proof. Domains are nonzero rings. Apply Theorem 3.7 or Remark 3.8.

D

References [1] M. Brodmann, Piece-wise catenarian and going between rings, Pacific J. Math. 86 (1980) 415-419. [2] D. E. Dobbs, On going-down for simple overrings, II, Comm. Algebra 1 (1974), 439-458. [3] D. E. Dobbs, Going-down rings with zero-divisors, Houston J. Math. 23 (1997), 1-12. [4] D. E. Dobbs and M. Fontana, Universally gping-down homomorphisms of commutative rings, J. Algebra 90 (1984), 410-429. [5] D. E. Dobbs and M. Fontana, Universally going-down integral domains, Arch. Math. 42 (1984), 426-429.

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[6] D. E. Dobbs and I. J. Papick, On going-down for simple overrings, HI, Proc. Amer. Math. Soc. 54 (1976), 35-38. [7] R. Gilmer, Multiplicative Ideal Theory, Dekker, New York, 1972. [8] A. Grothendieck and J. A. Dieudonne, Elements de Geometric Algebrique, Springer-Verlag, Berlin, 1971. [9] M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142 (1969), 43-60. [10] I. Kaplansky, Adjacent prime ideals, J. Algebra 20 (1972), 94-97. [11] I. Kaplansky, Commutative Rings, rev. ed., Univ. Chicago Press, Chicago, 1974. [12] W. Krull, Beitrdge zur Arithmetik kommutativer Integritdtsbereiche. HI Zum Dimensionsbegriffe der Idealtheorie, Math. Zeit. 42 (1937), 745-766. [13] G. Picavet, Universally going-down rings, i-split rings and absolute integral closure, submitted for publication. [14] L. J. Ratliff, Jr., Going-between rings and contractions of saturated chains of prime ideals, Rocky Mountain J. Math. 7 (1977), 777-787. [15] L. J. Ratliff, Jr., Four notes on GB-rings, J. Pure Appl. Algebra 23 (1982), 197-207. [16] F. Richman, Generalized quotient rings, Proc. Amer. Math. Soc. 16 (1965), 794-799.

Factorization of Divisorial Ideals in a Generalized Krull Domain Said El Baghdadi Department of Mathematics, Faculte des Sciences et Techniques, P.O. Box 523, Beni Mellal, Morocco baghdadiSf stbm.ac.ma

Dedicated to Robert W. Gilmer

1. Introduction Dedekind domains are characterized as the class of Priifer domains in which each nonzero ideal is a finite product of prime ideals. The P VMD-analogue of this factorization property holds for Krull domains. Namely, Krull domains are the PVMDs in which each i-ideal is a t-product oft-prime ideals. In [5], the author introduced the generalized Krull domains, a class of PVMDs which are closely related to Krull domains. Priifer generalized Krull domains coincide with the class of generalized Dedekind domains introduced by N. Popescu [20]. The aim of this paper is to extend the factorization property for ideals in Krull domains to generalized Krull domains, in the same spirit of a work on generalized Dedekind domains by Gabelli and Popescu [8]. A generalized Krull domain (GK-domain for short) is a PVMD such that P ^ (P 2 )t, for each i-prime ideal P, and each nonzero principal ideal has only finitely many minimal (i)-primes (cf. [5, Theorem 3.9]). GK-domains of ^-dimension one coincide with the class of Krull domains. For more details see [5]. In Section 2 we give some preliminary results. In Section 3, we study divisorial ideals in a GK-domain and prove the promised factorization property which characterizes GK-domains. Namely, a domain R is a GK-domain if and only if it is a PVMD whose divisorial ideals are precisely those of the form / = (JP\ • • • Pn)t, where J is a t-invertible t-ideal and PI, ... ,Pn are pairwise incomparable i-prime ideals. In this paper, inclusion of sets is den'pted by the symbol C. We reserve the sign C for strict inclusion. 149

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2. Preliminary results We begin by reviewing the v- and i-operations on an arbitrary integral domain. The v- and the i-operation are examples of star operations. For more details the reader is referred to [10, Sections 32 and 34] or to [16]. Let R be an integral domain with quotient field K. For an .R-submodule E of K, set E"1 = (R : E) := {x € K xE C R}. Let E be a nonzero .R-submodule of K. Then Ev = (E~1)~1 is an .R-submodule of K called the f-closure of E. Note that if E is not a fractional ideal of R, Ev = K. Also, the i-closure Et of E is denned to be \J{FV \ F C E finitely generated .R-submodule}. The map E \—» Ev (respectively, E i—> Et) induces a star operation on R, called the ^-operation (respectively, i-operation), by restriction on the set of nonzero fractional ideals. A fractional ideal I of R is said to be divisorial or a f-ideal (respectively, a i-ideal) if / = /„ (respectively, I = It). A u-ideal is a i-ideal, and the ^-closure and the i-closure coincide on finitely generated ideals. A v-ideal I is said to be of finite type or w-finite if I = Jv for some finitely generated ideal J. An ideal I is said to be i-invertible if there is an ideal /' with (//')* = R- In this case I' — I~l. A t-invertible i-ideal is a u-ideal of finite type. A i-ideal which is prime is called a i-prime ideal (or simply, a i-prime). The set of integral i-ideals of a domain has maximal elements under inclusion, called i-maximal ideals, such ideals are prime. Each proper i-ideal is contained in a i-maximal ideal and a minimal prime of a i-ideal is a i-prime. We denote by i-Max(E) the set of all i-maximal ideals of R. If / is a nonzero fractional ideal, we have It = Hi-^t^M | M £ i-Max(.R)}; in particular, R — fll-^M M 6 i-Max(ft)}. For an extension R C T of integral domains, it is often necessary to take the i-closure in both R and T. For a fractional ideal J of R, we shall usually write Jt for the i-closure of J with respect to R and (JT)t for the i-closure of JT with respect to T. A domain R is said to be a PVMD (Priifer ^-multiplication domain) if the vfinite w-ideals form a group under the i-multiplication I * J = (IJ)t- Equivalently, R is a PVMD if and only if RM is a valuation domain for each i-maximal ideal M of R. For other results on PVMDs, see [12] and [19]. An integral domain T is said to be i-linked over a subring R (or the extension R C. T is i-linked) if for each finitely generated ideal F of R with F~l = R, we have (FT)"1 = T (cf. [4] and [2]). Any generalized ring of fractions of a domain R is i-linked over R, and a i-linked overring of a PVMD is a PVMD (cf. [17, Chapter 3]). Also, recall that an overring T of R is i-flat (or R C T is a i-flat extension) if TM = RMHR for each i-maximal ideal M of T. Flatness implies i-flatness, the converse does not hold in general (cf. [18, Remark 2.12]). PVMDs have many properties similar to those of Priifer domains. Among them we mention a result in [18, Proposition 2.10], which is the i-analogue of a well-known result proved by Richman for Priifer domains [21, Theorem 4], it states that a domain R is & PVMD if and only if every i-linked overring of R is i-flat. Let R be an integral domain. A i-multiplicative system of i-ideals of J? is a set of integral i-ideals of R closed under i-multiplication. If $ is a i-multiplicative

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system of i-ideals of R, the generalized ring of fractions of R with respect to $ is the overring #$ = \J{(R : /) / G $}. An important class of i-multiplicative systems of i-ideals is given by the ilocalizing systems (of i-ideals). We recall that a i-localizing system of i-ideals of R is a i-multiplicative system $ such that: (i-LSl) / e $ and J a i-ideal of # such that / C J => J € $; (i-LS2) / e $, J a i-ideal of P such that (J :R iR) € $, for each i & I, => J e $. If P is a nonzero i-prime ideal of .R, then $p = {/ | / i-ideal of 7? and / <2 P} is a i-localizing system of R and we have P$p = Rp. More generally, if A is a nonempty set of i-prime ideals of R, then 3>(A) = p|{$p | P e A} is a i-localizing system of PL and PL$(A) = CK-^P I -P £ A}. The notion of i-localizing system is the i-analogue of that of a localizing system and we have some interconnections between these two concepts. If F is a localizing system, then Ft = {It \ I £ F} (the i-closure of F) is a i-localizing system and Rp = Rpt- Conversely, to each i-localizing system $, we can associate a localizing system * = {/ C PL It e $}, and we have $ = ($)tIf P is a localizing system, we have F = Ft. Moreover, for each i-ideal I of PL, /$ = J$. For more details, see [5, Section 1]. Finally, a localizing system F of Pi is called v-finite if each i-ideal / e F contains a ^-finite i-ideal J such that J e F. Since in a Mori domain each i-ideal is u-finite, then each (i)-localizing system of a Mori domain is u-finite. If $ is a i-localizing system, then $ is v-finite if and only if $ is a ^-finite localizing system (cf. [5, Section 1]). We recall that a generalized Krull domain is a PVMD PL which satisfies the following property: ( G K ) For each i-linked overring T of PL, there is a unique i-localizing system $ of R such that T = PL$. A Krull domain is a GK-domain and, since in a Priifer domain the i-operation coincides with the identical star operation, then the class of Priifer GK-domains coincides with the class of generalized Dedekind domains. Several characterizations of GK-domains were given in [5], the following useful result is new. Proposition 2.1. Let R be a PVMD. The following are equivalent: (1) R is a GK-domain. (2) For each t-localizing system $ of R, there is a unique set A of pairwise incomparable t-prime ideals of R such that $ = $(A); namely, A = {Mr\R M € i-Max(Pt$)}. In this case, the map: A —> i-Max(PL$), P H-> (PR$~)t = P$ is a bijection. Proof. (1)=K2). Let A = {M n PL | M e i-Max(#$)}. Since PL is a PVMD, then PL C PL$ is i-flat. Thus by [18, Lemma 2.9], all ideals in A are i-primes. On the other hand, by [5, Proposition 2.4(ii)], M = ((M n R)R$)t for each i-maximal ideal M of PL$. It follows that A is a set of pairwise incomparable i-prime ideals of PL. Further, since (#$)M = RMHR for'each i-maximal M of P$ (by i-flatness), then PL$ = fll-^Mnfi | M € i-Max(Pi$)} = PL$(A)- Hence $ = $(A) as PL is a GK-domain.

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For the uniqueness of A, let A' be a set of pairwise incomparable i-prime ideals of R such that $ = $(A'). Let P £ A'. We claim that (PR$)t ^ R$. If (P#$)t = R*, then there exists F C P a finitely generated ideal of PL such that (FtR$)t = -R$So Ft$ = PL$ (cf. [5, Proposition 1.5(2)]). Hence Ft £ $ [5, Proposition 1.5(1)], which is impossible since Ft C P and $ = $(A'), and our claim follows. Now, let M be a t-maximal ideal of PL$ = R$ such that (PR$,)t C M. Since R is a GK-domain, then $ is a ^-finite t-localizing system [5, Proposition 3.1], hence $ is a ^-finite localizing system. Thus, by [5, Proposition 1.3(iv)], M — (M n PL)$. Hence (M n #)*(A') = (M n #)$ = (M n R)$ = M. It follows that M <~\ R C Q for some Q e A.'; but P C M D R, so P = Q = M (~) R. Hence A' C A. For the reverse inclusion, let M be a i-maximal ideal of PL$. Then, as in the previous part of the proof, (M n PO<j>(A') = M. So M n R C P for some P e A'; but A' C A, so M n R = P e A'. Hence A C A'. Therefore A' = A. (2)=>(1). Let $ and $ be two t-localizing systems of -R such that R$ = Ry. Let A = {M n R | M e t-Max(P$)} = {M n P | M e i-Max(E^)}. Hence $ = if = $(A) (see the proof of (l)=s>(2)). Therefore, P is a GK-domain. The last statement follows from the proof of (1)=>(2). D Now, let us fix some notation for- the remainder of this paper. Let / be an integral t-ideal of R. Denote by Min(/) the set of (t)-primes minimal over / and by A(J) = {M e t-Max(R) \ M 2 I}- To / we can associate the t-localizing system of R, $(/) = $(Min(J)) n $(A(I)). Then R^(I) = (~]{Rp P e Min(I}}r\{RM \ M £ A(/)}. Also, in all the paper we shall make use of the following two results on GK-domains (cf. [5]): In a GK-domain Min(J) is finite and the t-maximal ideals are t-invertible. The next two results deal with the dual of an ideal in a PVMD. Lemma 2.2. Let I be a t-ideal of a PVMD R. Then, I~l is a ring if and only if Proof. Cf. [14, Theorem 4.5].

D

Proposition 2.3. Let R be a PVMD and I a t-ideal of R such that no minimal (t) -prime of I is t-maximal. Then (R : \/J) — (\/7 : \/7) = Proof. We start by showing that (R : v7/) = #$(/)• Let M 6 A(J) and s e / \ M. Let x e (R : \/7), then sx € R; so x & RM- Hence (R : \/7) C RM. Now let P € Min(J) and TV e i-Max(P) such that \/l C P C N. We have ^RN = PRN\ so (R : vT) C (R : V!)R\N Q (RN • V^-Rw) = (RN • P^w) = Rp (since RN is a valuation domain and PPu/v is a nonmaximal prime (cf. [15, Corollary 3.6])). Hence (R : yJ) C P$(/). For the reverse inclusion, let u € -R$(/) and a e V/. We show that ua € PL by checking that ya £ PLAT, for each i-maximal ideal N. Let AT e i-Max(Pi). If \/J g AT, it is clear that ua e PJV- Assume that \/7 C TV and let P € Min(J) such that v7/ C P C TV. Write w = ^ for some 5 € P \ P and r e R. We claim that f 6 RN. Otherwise, ^ = i e P^; so s = ai 6 P^AT n R = P, which

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is impossible. Therefore, ua e RN- Hence (R : V7) = R$(i). By [1], it follows that (R: VI) = Next, we characterize the t-ideals of a GK-domain which are t- products of t-primes. Before this we need some results. Recall that, for an integral domain R, the star t-operation on R can be extended to a semistar operation of finite type on R (which is denoted also by t). More generally, for any overring T of R, the map t1' : E —> Et, where E is a nonzero T-submodule of K (the quotient field of R) , defines a semistar operation of finite type on T. When Ti = T, tT induces a star operation on T by restriction on the set of fractional ideals. For more details, see [6] and [7, Section 2]. Lemma 2.4. Let R be a GK-domain and T an averring of R such that T = AJ>(A) for some set A of pairwise incomparable t-primes. Lett denote the start-operation on R and t' the star t-operation on T. Assume that tT is a star operation on T (i.e., Tt =T). Then

It = If = f}{IRp I P e A} for each nonzero fractional ideal I of T. Proof. Since tT is a star operation of finite type on T, then tT < t' [6, Proposition 1.6(5)]. Hence It C J t /. For the reverse inclusion, since T is a PVMD, as a t-linked extension of a PVMD, then Iv = f}{ITM M e t'-Max(T)}. Also, It = [}{ItTM \ M e tT-Max(T)} (cf. [12, Proposition 4]). To conclude it suffices to show that tT-Max(T) C t'-Max(T). Let M e tT-Max(T), then M n R is a t-prime ideal of R. Hence TM 2 RMC\R are valuation domains (since R is a PVMD). It follows that M is a t'-prime of T. Now, let N 2 M be a t'-maximal ideal of T. Since tr < t' , then N is a iT-prirne ideal of T. Hence N = M. So tT-Max(T) C t'-Max(T). Hence Iv C It. Thus It = It, = f\{ITM M e i'-Max(T)}. On the other hand, we have A = {M n R | M e i'-Max(T)} [Proposition 2.1], and TM = RMnR for each M e i'-Max(T) (by t-flatness); hence It = Iv = ^\{IRP P € A}. D Let / and J be two t-ideals of a domain R. We say that / and J are t-comaximal if (I + J)t = R. It is easy to see that / and J are t-comaximal if and only if J and J are not contained in a common i-maximal ideal. In a PVMD, any two incomparable t-primes are i-comaximal [19, Proposition 4.4(1)]. Lemma 2.5. Let R be a domain and Ji, . . . , In pairwise t-comaximal t-ideals of R. Proof. It suffices to check the result for the case n = 1 since /$ and Jj = (ELii Ij)t are i-comaximal, for i = 1, . . . , n. It is clear that (I\Iz)t C I\ n /2. For the reverse inclusion, we note that (/i+/2)(/in/ 2 ) C IJ^. Hence ((h+h)(hf\h}}t C (I\Iz)tAs (/i + 1-2)1 = R, we conclude that (/i n 72)t = /i n I2 C (/i/ 2 ) t . D Now we state our main result of this section. Proposition 2.6. Let R be a GK-domain and I a t-ideal of R. Then the following are equivalent:

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(i) / = (P! .. • Pr)t, where PI, ... ,Pr are not t-maximal pairwise incomparable t-primes; ( i i ) (R :/) = ( / : /); (m) / = (J(R : J))t for some t-ideal J of R. In this case, I is a radical ideal. Proof, (i) =4> (ii) follows from Proposition 2.3 since \/T = PI n • • • fl Pn = I (cf. [Lemma 2.5]) and the minimal t-primes Pi are not t-maximal. (ii) => (i). We first show that IRP - PRP for each P e Min(J). Deny. Assume there exists P e Min(J) such that IRp C PRp, and let a 6 P such that IRP C aflp C PRP. Since fl is a GK-domain, then P = V^t f°r some finitely generated ideal H of R (cf. [5, Theorem 3.5]). Set J = # + aR. We have P = y/Jj. We claim that / C Jt. It suffices to check i-locally. Let N £ t-Max(U). If P £ TV, then J t fljv = #w 2 /-fyv since P = V^t- If P C AT, then IRN C 7flP. Hence a ^ JPtjv- So IRj\t C Jt-Rjv since PLJV is a valuation domain, and our claim is proved. Now, since I~l is a ring, then I"1 C RP [Lemma 2.2], and hence JJ-1 C P/-1 C PRP. So 1 6 (JJ~l)t C P, a contradiction. Set T = /-1 = (/ : J). By Lemma 2.2, T = R$(I}; hence T is a t-linked overring of R. Let Min(7) = {Q 1 ; ... ,-Qn} (cf. [5, Corollary 3.2]). Let t' be the star t-operation on T and tT the semistar operation on T induced by the star toperation on R. Since T = (R : I), then Tt = T. Note that / is also an ideal of T (since T = (I : I ) ) . Thus I = It = It, = ^IRQ, f]{#M M e A(/)} [Lemma 2.4]. So J = nt((3i-R<9i n B) = ^ <9<- Hence / = (QiQ2 • • • Qn)t [Lemma 2.5]. Write / = (Pi • • • PrMi • • • Ms)t, where PI, . . . , Pn are not t-maximal pairwise incomparable t-prime ideals and MI , . . . , Ms are t-maximal ideals (which are tinvertible). Set J = (Pi • • • Pr)t and H = (Ml • • • Ms)t. Then / = (JH)t. We have J = (I(R : H))t C (II-1^ = (1(1 : J)) t - /, and clearly / C J; so / = J = (Pi • • • P r ) t , as desired. (ii) & (Hi) cf. [3, Proposition 1]. The last statement follows from (i) and Lemma 2.5. D Remark 2.7. Recall that a domain R is an RTP (radical trace property) domain if I(R : /) is a radical ideal for any noninvertible ideal I of R. A generalized Dedekind domain is an RTP domain [8, Proposition 0.1]. A similar property holds for GKdomains. We can define a t-RTP domain as a domain R such that (I(R : I ) ) t is a radical ideal for any t-ideal I of R that is not t-invertible. By Proposition 2.6, a GK-domain is a t-RTP domain. 3. Factorization of divisorial ideals In this section, we investigate factorization of divisorial ideals into prime ideals in a GK-domain. We start with the following proposition. Proposition 3.1. Let R be a GK-domain. Then a t-product of finitely many tprimes is divisorial. Before proving Proposition 3.1, we need two Lemmas.

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Let / be a nonzero ideal of a domain R. The /-transform, is defined to be T(I) = \Jn>i(R '• In)- The /-transform is an overring of PL. Lemma 3.2. Let R be a GK-domain and P a prime t-ideal of R. Then (Pn)t is divisorial for each n > 1. Proof. If P is a maximal t-ideal, then (Pn)t is a t-invertible t-ideal of R, and hence a divisorial ideal. Assume that P is a nonmaximal t-prime. By [13, Proposition 2.5], it suffices to show that P"1 ^ T(P). Deny. Let S = f\{RM M e A(P)}. By Proposition 2.3, P"1 = RP H S, and by [13, Proposition 1.5], T(P) = RPo n S, where P0 = fln>i(- p ")t- Set $ = $P n $(A(P)) and * = $PO D $(A(P)). Then P$ = P"1 = T(P) = Ry. Hence $ = *, because R is GK-domain. Moreover, P ^ (P")t for each n > 2, since P ^ (P 2 ) t in a GK-domain. Hence P £ * = $, a contradiction. D Lemma 3.3. let P 6e a PT/MD. Tften (1) /// and J are two divisorial t-comaximal ideals, then (IJ)t is divisorial. (2) Let P, Q be two prime t-ideals such that P C Q, then (PQ)t — P • Proof. (1) By Lemma 2.5, (U)t = I C\ J which is divisorial as an intersection of divisorial ideals. (2) Since PL is a PVMD, it suffices to check that PRM = PQRu for each t-maximal ideal M of PL. Let M be a t-maximal ideal of Pt. If Q £ M, PRM = PQRM. If Q C M, then PRM C QRM- Let a e Q such that a £ PRM. Then PRM C O,RM\ so a~lP C PLM- We have a(a~1P)RM = PRM, hence a~lPRM C PP_M (since a £ PRM)- Thus PRM ^ aPRM Q QPRM- Hence PRM = PQRM, as desired. D Proof of Proposition 3.1. Let / — (QiQz • • • Qn}t for some prime t-ideals Qi, . . . , Qn. Since PL is a PVMD, then for any two prime t-ideals P, Q, either P and Q are comparable or t-comaximal [19, Proposition 4.4(1)]. Hence by Lemma 3.3, we can assume that / = (P1aiP2a2 • • • Pmm)t, where PI, • • • ,Pm are pairwise t-comaximal prime t-ideals. The conclusion follows by induction using Lemma 3.2 and Lemma 3.3(1). D Now we state our first main result of this section. Theorem 3.4. Let R be a GK-domain and let I be a nonzero proper ideal of R. Then I is divisorial if and only if I — (JPi • • • Pr)t, where J is a t-invertible integral t-ideal of R and PI, . . . , Pr (r > 1) are pairwise t-comaximal prime t-ideals of R. Proof. Let / be a divisorial ideal of R. If / is t-invertible, then / = (JM)t, where M D / is a t-maximal ideal and J = (IM~l)t is an integral t-invertible t-ideal. Next we assume that / is not t-invertible. Set H — (I(R : I))t- We claim that H~l = (H : H) = (I : I). The equality H~l = (H : H) follows from Proposition 2.6. Clearly (/ : /) C (H : H). For the1 reverse inclusion, let x e (H : H). Then xll~l C (II~l)t C P; so xl C (/~ 1 )~ 1 = /„. Hence xl C I since / is divisorial. Therefore H~l = (H : H) = ( / : / ) .

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Thus, by Proposition 2.6, H = (Pi • • • Pr)t, where PI, . . . , Pr are not t-maximal pairwise incomparable t-primes. Set T = /?$(#). By Lemma 2.2, we have T = H~l = (H \ H) = (I : I). Note that H and I are also ideals of T. Let t' be the star t-operation on T and t the star t-operation on R. By Lemma 2.4, t' = tT (since Tt = (R : H)t = (R : H} = T). Hence H = Ht = Ht> = H^(H) [Lemma 2.4]. Thus H = r\i PiRpi r\{RM M e A(/f)} = P14(H) n . . . n P r $ (ff) . Since R is a GK-domain, then Pi$(ff),... ,Pr<s>(H) are t'-maximal ideals of T [Proposition 2.1], and hence t'-invertible t'-ideals of T (since T is a GK-domain (cf. [5, Corollary 3.2]). Hence H = (Pns,(H) • • • P r $(#))t' is a t'-invertible t'-ideal of T. Write H = (JiT)t' = (J\T}t and / = (J2T)t, = (J2T)t, where Ji and J2 are two finitely generated ideals of R. Since I C H, we can assume that J2 C Ji. Set •/ — (Ji(R '• J\))t, which is an integral t-invertible t-ideal of R. We claim that / = (JH)t. Indeed, we have (JH)t = ((J2(R : J i ) ) t ( J i T ) t ) t = ((J2T)t(J\(R : Ji))t)t = (J2T)t = /. Conversely, assume / = (JP\ • • • Pr}t as in the theorem. Set H = (Pi • • • Pr)tBy Proposition 3.1, H is a divisorial ideal of R. Thus (JH)V = ( ( J J - l ] ( J H ) v ) t C (J(J-lJH}v)t = (JHv)t = (JH\. Hence / is a divisorial ideal. D Remark 3.5. 1) In Theorem 3.4, when / is not proper (7 = R), we have the factorization I = (M~lM}t, where M is an arbitrary i-maximal ideal of R. In this case, we can take J = M~l, but J cannot be chosen integral. 2) If / is not t-invertible in Theorem 3.4, the i-prime ideals PI, . . . , Pr in the factorization of J are uniquely determined by /, we have (I(R '• I})t = (Pi • • • Pr)tHowever, J is not uniquely determined by /, see [8, Section 3] for the case of generalized Dedekind domains. Our next goal is to characterize GK-domains in terms of the factorization property of divisorial ideals given in Theorem 3.4. We start by extending to PVMDs some results proved in [11] for Priifer domains. Lemma 3.6. Let R be a PVMD and P a proper t-prime ideal of R. Then Rp 2 P|{f?M M e A(P)} if and only if there exists a finitely generated ideal J C P such that each t-maximal ideal of R containing J contains P. Proof. Let S = (~}{RM M e A(P)}. (=>). Let x e S\RP and set / = (R ;R x). Since R is a PVMD, then I = Jv for some finitely generated ideal J of R. Clearly, / g M for each M € A(P), and hence J 2 M for each M e A(P). On the other hand, since x <£ Rp, then x"1 € PRp, as R,p is a valuation domain. Hence / = (R :# x) C P, and so J C P. The ideal J satisfies the desired property. (<=)• Let J C P be a finitely generated ideal of R such that each t-maximal ideal of R containing J contains P. Let T = T(J), the J-transform. We have T = Rp, where F — {J" | n > 0}. Since J is finitely generated, then P is a u-finite multiplicative system of ideals of R (cf. [5, Section 1]). Hence (JT)t = T (cf. [5, Proposition 1.3(ii)]). So (PT)t = T. We claim that T £ RP. Otherwise, the extension T C RP is i-linked [18, Lemma 2.7]. Since (PT)t = T, then (PRP)t =

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RP, which is impossible since (PRp)t = PRp. This prove the claim. On the other hand, since J g M for each M t A(P), then T C 5. Hence S £ RP. D Lemma 3.7. Let R be a PVMD. Then the following are equivalent: (1) For each t-maximal ideal M, RM 2 fl{#M' I M' <E A(M)}; (2) For each t-maximal ideal M and for each t-prime ideal Q 7^ M, RM 2 (3) R is uniquely representable as an intersection of a family {Va}a of essential valuation overrings of R such that Va <2 V@ fora ^ /?; namely, R = {~}{RM M e t-Max(P)}. Proof. (l)=£-(2). Let M be a t-maximal ideal of R. By Lemma 3.6, there exists J a finitely generated ideal of R such that M is the only t-maximal ideal which contains J. Let x € M \ Q and set J' = ( J, x). So, we can assume that J £ Q. To conclude, we proceed as in the proof of (<=) of Lemma 3.6. (2)=>(3). Let R = C\aRpa be a representation of R as an intersection of essential valuation overrings of R. We show that t-Max(JJ) = {Pa}a- Since there are no containment relations among the Pa's, then, for each a, there exists a tmaximal ideal Ma D PQ with Ma ^ M^ for each J3 ^ a (since PQ and P@ are t-comaximal) . Suppose that there exists a such that Pa C Ma. We have PM,, 2 R =• (~}p Rpg = Rpa Pli-^M I M & A(MQ)}, contrary to the assumption. Hence Pa = Ma, for each a. Thus {PQ}a C t-Msx(R). For the reverse inclusion, let M e t-Max(P) with M ^ MQ for each a. We have RM D P = f|a ^AfQ 2 fl{-RM' ] M' e A(M)}, a contradiction. Hence t-Max(P) = {PQ}Q. (3)=>(1) is clear. D Lemma 3.8. Let Rtbe a PVMD. Then the property: (*) For eacft t-prime ideal P of R, RP 2 fK-^M M £ A(P)} is stable under t-linked extension. Proof. Let P' be a t-prime ideal of T such that TP> 3 fli^M' | M1 e A(P')}. Set P = P' n -R and M = M' n R for M' e A(P'). Since R C T is t-linked, then it is t-flat. So P and M are t-prime ideals of R and, Tp/ = .Rp and TM> = RM [18]. Hence #P 2 ni-^M M = M' n R,M' € A(P')}. Again, by t-flatness, P' = (PT)i and M' = (MT)t [5, Proposition 2.4(ii)], so P and M are t-comaximal, thus there exists a t-maximal ideal of R which contains M but not P. Hence RF 5 (~}{RM | M e A(P)}, a contradiction. D Lemma 3.9. Let R be a PVMD such that for any proper t-prime P of R, RP 2 fK-^M M e A(P)}. Then each nonzero principal ideal of R has only finitely m.any minimal (t) -prime ideals. Proof. Let / = (x) be a nonzero principal ideal of R and {PJi6s the set of minimal (t)-prime ideals over /. Notice that there are no containment relations among the PI'S, and since in a PVMD any two incomparable t-primes are t-comaximal, then {Pi}tgs is a collection of pairwise t-comaximal t-primes. Set T = Plies^-fV Let i € E and set Mj = PiRPinT. Then TM{ — RPi is a valuation domain (in particular,

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M, is a it-prime ideal of T). Thus, as R C T is i-linked and T = fl ie2 TMi, then by Lemma 3.8 and Lemma 3.7(3), {Mj}jeS is tne set of i-maximal ideals of T. Choose i e E. By hypothesis ^Pi 2 fli-^M | M € A(Pj)}. Hence there exists Ji C PJ a finitely generated ideal such that if Jj C M for some i-maximal ideal M of .R, then Pi C. M [Lemma 3.6]. We claim that Jj £ P,, for each j 7^ i. Otherwise, Ji C Pj C M for some j ^= i and some i-maximal ideal M. So PJ C M, which is impossible since PJ and PJ are i-comaximal. Hence MJ is the only i-maximal ideal of T which contains (JjT) t (since M,- n P = PJ). Let Jj = ( a i , . . . , a n ). Let r an integer such that rvMi(o-k) > VM^X) for every k = 1 , . . . , n, and set Jj = ((x, a j , . . . , arn)T}t. Remark that JjT1^ = xTj^ and MJ is the only i-maximal ideal which contains Jj. Now, since Jj is a i-invertible i-ideal of T (as T is a PVMD), then xT = (JjJ^t for some i-ideal /? of T. We claim that If g MJ and I[ C n^ M,, Indeed, we have zTMi = (Ui)tTMi = (IiTMJpMt)t = xJ^TMi. So TM, = /jTMi. Hence I( <£ MJ. On the other hand, let j ^ i. Since xT = (lityt C MJ and Jj £ MJ, then J^ C M,-. Hence f|,¥i MJ g Mj, for each z6E. Now, to show that S is finite, we proceed as in [9, Lemma 8]. If S is infinite, there exists a well-ordering < on £ under which it has no largest element. For i e S, let Ni = fl^j MJ and let TV =• !J.eE ty. Then N is a proper i-ideal of T as a union of a chain of proper i-ideals of T. The i-ideal N is not contained in any maximal i-ideal of T. If TV C MJ for some i £ S, then Ni C MJ, and hence rW» MJ C TVj C MJ, which is impossible. Hence E is finite. D We are now in the position to state the promised characterization of GKdomains. Let 'D(R) denote the set of divisorial ideals of a domain PL. Theorem 3.10. Let R be a PVMD. Then R is a GK-domain if and only if Z>(PL) = {(JPi • • • P r ) t , J is a t-invertible t-ideal and PI, . . . ,Pr (r > 1) pairwise t-comaximal prime t-ideals}. Proof. Sufficiency follows from Theorem 3.4 and Remark 3.5. To establish necessity, assume that the u-ideals of PL can be represented in the given form. First we show that P 7^ (P 2 )t for each i-prime ideal P of R. We adapt the proof of [8, Theorem 3.3]. Let P be a i-prime ideal of PL. By assumption P is divisorial. If P is i-maximal, then P is i-invertible [13, Proposition 2.1], and hence P ^ (P2)t- Next we assume that P is not i-maximal. Let 0 ^ p e P. Then p(P : P) = (JPi • • • P r ) t , where J is a i-invertible i-ideal and PI, . . . , Pr (r > 1) are i-primes that are not i-maximal. By Proposition 2.3, we have P^1 = (P : P) = (P: • • • Pr : P: • • • Pr) = (Pi • • • Pr)"1. So Pv = (Pi • • • Pr)v, that is P = (Pi • • • P r ) t . Hence PRP = (P: • • • Pr)tRp = pJ~l(P : P)Rp = pJ~lRp is a principal ideal (since Rp is a valuation domain and J is a i-invertible i-ideal). Therefore P ^ (P 2 ) t . Next, we show that each nonzero principal ideal of R has only finitely many minimal (i)-prime ideals, by using Lemma 3.9. Let P be a i-prime ideal of R (and hence P is divisorial). Set S = (~]{RM M 6 A(P)}. If P is i-maximal, then P is i-invertible [13, Proposition 2.1]. Let P = Jv for some finitely generated ideal J.

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So, by Lemma 3.6, Rp 2 S. Now assume that P is a i-prime that is not ^-maximal. Since (Pn)t is divisorial for each n > 1 and P ^ (P 2 ) t , then by [13, Theorem 2.7], p-i ^ T(P), where T(P) is the P-transform. That is T(P) g P"1. On the other hand, we have P~l = RP n S [Proposition 2.3] and T(P) C S [13, Proposition 1.5]. Hence S <2 Rp. The conclusion now follows from Lemma 3.9 and [5, Theorem 3.9]. D Note that for Priifer domains, Theorem 3.4 recovers [8, Proposition 3.2] and Theorem 3.10 recovers [8, Theorem 3.3]. As a consequence of the previous results, we get a new characterization of Krull domains. Corollary 3.11. Let R be an integral domain. Then the following statements are equivalent: (i) R is a Krull domain; (n) R is a GK-domain and each divisorial ideal of R is t-invertible. Proof. (i)=>(ii) is clear. (ii)=>-(i). By Theorem 3.10, any i-prime ideal is divisoriai. Hence PL is a domain in which each i-prime ideal is t-invertible. Such a domain is a Krull domain [17, Theorem 6.5]. D Remark 3.12. The technical results Lemmas 3.6 and 3.7 are related to a notion ^-analogue of the ^-property. For more details, this concept is a project of a paper by the authors: S. Gabelli, E. G. Houston, and T. G. Lucas. Also, Professor Gabelli informed me that in this paper they have obtained results similar to the above-mentioned lemmas. References [1] D. F. Anderson, When the dual of an ideal is a ring II, Houston J. Math. 9 (1983), 325-332. [2] D. D. Anderson, E. G. Houston, and M. Zafrullah, t-linked extensions, the t-class group, and Nagata's Theorem, J. Pure Appl. Algebra 86 (1993), 109-124. [3] V. Barucci, Strongly divisorial ideals and complete integral closure of an integral domain, J. Algebra 99 (1986), 132-142. [4] D. E. Dobbs, E. G. Houston, T. G. Lucas, and M. Zafrullah, t-Linked overrings and Priifer •D-multiplication domains. Comm. Algebra 17 (1989), 2835-2852. [5] S. El Baghdad!, On a class of Priifer ^-multiplication domains, Comm. Algebra 30 (2002), 3723-3742. [6] M. Fontana and J. Huckaba, Localizing systems and semistar operations, in "Non Noetherian Commutative Ring Theory" (S. Chapman and S. Glaz, Eds.), Kluwer Academic Publishers 2000, Chapter 8, 169-197. [7] M. Fontana and K. A. Loper, Kronecker function rings: a general approach, in "Ideal Theoretic Methods in Commutative Algebra" (D. D. Anderson, I. J. Papick, Eds.), M. Dekker Lecture Notes Pure Appl. Math. 220 (2001), 189-206. [8] S. Gabelli and N. Popescu, Invertible and divisorial ideals of generalized Dedekind domains. J. Pure Appl. Algebra 135 (1999), 237-251. [9] R. Gilmer, Overrings of Priifer domains, J.',Algebra 4 (1966), 331-340. [10] R. Gilmer, Multiplicative ideal theory, Marcel Dekker, New York, 1972. [11] R. Gilmer and W. Heinzer, Overrings of Priifer domains II, J. Algebra 7 (1967), 281-302. [12] M. Griffin, Some results on u-multiplication rings, Canad. J. Math. 19 (1967), 710-721.

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[13] E. G. Houston, On divisorial prime ideals in Priifer ^-multiplication domains. J. Pure Appl. Algebra 42 (1986), 55-62. [14] E. Houston, S. Kabbaj, T. Lucas, and A. Mimouni, When is the dual of an ideal a ring?, J. Algebra 225 (2000), 429-450. [15] J. Huckaba and I. Papick, When the dual of an ideal is a ring, Manuscripta Math. 37 (1982), 67-85. [16] P. Jaffard, Les Systemes d'Ideaux, Dunod, Paris, 1960. [17] B. G. Kang, *-Operations on integral domains, Ph.D. Thesis, The University of Iowa, 1987. [18] D. J. Kwak and Y. S. Park, On t-flat overrings, Ch. J. Math. 23(1) (1995) 17-24. [19] J. Mott and M. Zafrullah, On Priifer ii-multiplication domains, Manuscripta Math. 35 (1981), 1-26. [20] N. Popescu, On a class of Priifer domains. Rev. Roumaiiie Math. Pures Appl. 29 (1984), 777-786. [21] F. Richman, Generalized quotient rings. Proc. Amer. Math. Soc. 16 (1965), 794-799.

Divisorial Multiplication Rings Jose Escoriza Department of Algebra and Mathematical Analysis, University of Almeria, 04120, Almen'a, Spain jescorizSual.es Bias Torrecillas Department of Algebra and Mathematical Analysis, University of Almeria, 04120, Almeria, Spain btorreciSual.es Abstract. Completely integrally closed domains, in particular Krull domains, are characterized in terms of being divisorial multiplication rings with respect to a Gabriel topology. We prove for a family of rings including generalized Dedekind domains that a ring R is a divisorial multiplication ring if and only if there is a family of height one prime ideals such that the localization in each prime is a multiplication ring. A new version of Mott's Theorem allows us to find a large number of examples of this class of rings. We give a structure theorem for relative Noetherian torsionfree divisorial multiplication rings.

1. Introduction A commutative ring R is said to be a multiplication ring if for any ideals A C B there exists another ideal C such that A = BC. Commutative multiplication rings have interest in multiplicative ideal theory (see [8, 10, 11]). In the Noetherian case, they are finite direct sums of Dedekind domains and special primary rings and the general case includes von Neumann regular rings. If we consider the ^-multiplication in an integral domain R, i.e., A*B = (AB)V, where Av = (R : (R : A)), for fractional ideals A,B of R, then it is natural to introduce v-multiplication rings as those rings satisfying that for every pair of ideals A = Av C B = Bv of R there exists a fractional ideal C = Cv such that A = B * C. This notion is stronger than the notion of •u-domain given in [8, 2000 Mathematics Subject Classification. Primary 13F05; Secondary 13D30. Key words and phrases, multiplication ring, Gabriel topology, generalized Dedekind domain, Krull domain, half-centered ring. The two authors were supported by grant PB98-1005 from DGES and PAI FQM 0211.

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p. 418] and it is also different from the concept of Priifer ^-multiplication ring (see [8, p. 429, Exercise 5] and Theorem 3.7). These rings can be considered as multiplication rings relative to a Gabriel topology. We can think of Krull domains as a particular class of divisorial multiplication rings by taking as multiplication the modified one. Associated to a Krull domain, there is a canonical Gabriel topology over R (see [17, p. 205]) determined by the set of height one prime ideals. This set is not affine but geometrically stable. The geometrically stable subsets of the prime spectrum of a ring R correspond exactly to Gabriel topologies over R. We investigate rings satisfying the following multiplication property: given .^-closed ideals A = Cl^(A) CB = Cl^(B), there is another .F-closed ideal C < R such that A — Cljr(BC) where dp denotes the closure with respect to a Gabriel topology f'. Many results will be established more generally for divisorial multiplication modules. The paper is organized as follows: basic results and definitions are exposed as a starting point for the subsequent investigation in Section 2. In Section 3, some properties and motivating examples of divisorial multiplication rings are exposed. We characterize divisorial multiplication domains with respect to a Gabriel topology f. Krull domains turn out to be exactly Noetherian divisorial multiplication rings for the canonical Gabriel topology. In the last section we deal with Gabriel topologies T having enough .F-critical ideals. We have determined all Gabriel topologies over R such that R is a divisorial multiplication ring in the Noetherian case. Divisorial multiplication rings are characterized locally and new relative versions of Nakayama's Lemma and Mott's Theorem are shown. As a consequence, we can find many examples of divisorial multiplication rings. Finally, we determine the structure of Noetherian torsionfree divisorial multiplication ring relative to a Gabriel topology. 2. Preliminaries Throughout this note, every ring R will be a commutative ring with an identity element and T will represent a Gabriel topology over R or its associated Gabriel filter. If TV is a submodule of M, the closure of N in M is the set Cl™ (N) = {x e M : 31 G F A Ix C N}. If TV = Cl^(N), the submodule N is called ^-closed. The submodule Cl-p (0) is denoted also by t(M) and it is called J-'-torsion of M. A module M is called ^-torsion if M — t(M) and it is called T'-torsionfree if t(M) = 0. The set of all ^-closed submodules of M is denoted by Cyr(M). Several authors have used the lattice of jF-closed submodules to characterize different kind of rings such as relative distributive or Noetherian rings and obtain information about the structure of certain kind of modules (see [6, 13, 17]). The formula AB = Cl^-(AB) for any couple of ^-closed ideals A,B defines a multiplication in Cf(R). We have taken into account not only the structure of complete modular lattice of Cy^(R) but its multiplicative properties as well. Recall that an .R-module M is said 'to be a multiplication module relative to f (an T'-multiplication module or a divisorial multiplication module) if for every ^-closed submodule N, there exists an ideal A < R such that N = Cl^ (AM)

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(see [4]). When F = {R} (trivial Gabriel topology), we obtain the concept of multiplication module. In the sequel, we will assume that every Gabriel topology is different from the trivial one. An ideal A of R is a divisorial multiplication ideal if it is a divisorial multiplication .R-module, i.e., for every ideal B = Clf(B) C A there exists an ideal C such that B = Cl^(AC). Notice that if A is .^-closed, then the above condition is equivalent to B = Clp(AC) because Clp(I) = Clp(I) f~l A for every ideal /. A ring is a divisorial multiplication ring with respect to a Gabriel topology F if every ^"-closed ideal is a divisorial multiplication ideal. Given a Gabriel topology F over R and a multiplicative subset S of R, we denote by FS the Gabriel topology corresponding to the filter FS = {Is '• I £ F}, where 7g = {| : a G /, sG S}. Unless otherwise stated, if A and B are .R-modules we write (.As : BS) for {x e RS • xB$ C AS} and if N is an .R-module, the Gabriel topology considered over N$ is FS • In case S — R - P with P a prime ideal of R, we write Rp and Fp as usual. Some helpful properties of this kind of Gabriel topologies appear in [5, Lemma 2.1]. If P is a set of prime ideals of R closed under generalization (if P G P and Q G Spec(#) satisfies Q C P then Q & P), then Fp denotes the Gabriel topology whose filter consists of {/ < R: I <£ P VP 6 P}. Recall that if F is a Gabriel topology over R, F is called semicentered if for every ideal / ^ F there exists a prim'e ideal P £ F such that I C P. A. ring over which every Gabriel topology is semicentered is said to be a half-centered ring or, more briefly, a HC ring. It is well-known that seminoetherian or semiartinian rings and discrete valuation domains of arbitrary rank are HC rings (see [6, 14] for further information). It is said that F has enough F-critical ideals if for every ideal / ^ F there is an ideal P (£. F and I C. P being maximal with respect to this condition. An integral domain R is said to be a generalized Dedekind domain if it is a Priifer domain over, which every Gabriel topology is finitely generated, i.e., there is a filter basis consisting of finitely generated ideals (see [6]). Therefore, every Gabriel topology over a generalized Dedekind domain has enough J-"-critical ideals.

3. Divisorial multiplication domains Now R stands for an integral domain. Let F be a Gabriel topology over R. Let K be the field of quotients of R. Recall that a fractional ideal A of K is an .R-module satisfying that there exists a non zero element d € R such that dA C R. The set of all fractional ideals of R is denoted by Frac(.R) and frac(J?) stands for the set of all finite generated fractional ideals. An ideal A of R is F-invertible if there exists a fractional ideal B such that CA.-p (AB) = R. This condition is equivalent to Clf(AB] = R if the domain is .F-closed in its quotient field. In the absolute case, multiplication domains are just Dedekind domains. In the relative case, we have the following result. Theorem 3.1. Let T be a Gabriel topology over a domain R which is F-closed in its quotient field. Then, R is a divisorial multiplication domain with respect to P if and only if every F-closed ideal of R is F-invertible.

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Proof. Suppose that R is a domain whose ^"-closed ideals are ^"-invertible. Let A be an .F-closed ideal of R. By hypothesis, A is ^"-invertible. By [5, Proposition 2.5], A is a divisorial multiplication ideal and therefore R is an divisorial multiplication ring with respect to f. Now, suppose that R is a divisorial multiplication domain with respect to J- '. Let 0 7^ A an .F-closed ideal of R. Take 0 ^ a € A. Since R is a domain, there exists a fractional ideal C of R such that CRa = R. By hypothesis, Cl^(Ra) = Cl^(AB) for some B < R. It holds R = Cl^(CRa) = Cl*(CAB) = Cl*(CAB) and hence, A is ^-invertible. D Remark 3.2. Every Krull domain is a divisorial multiplication domain with respect to the canonical torsion theory (see [4, Example 3.4]). Recall that an .R-module M is called J- -cyclic if M — Cl1^ (Rm) for some m e M. Now we are interested in knowing when a divisorial multiplication module with respect to T is ^-cyclic. Let R be an integral domain and let K be its field of fractions. We denote by E(M) the injective hull of an .R-module M. The classical Gabriel topology on R will be denoted by .F(u) and it is the Gabriel filter associated to the torsion theory JJL defined in the following way: if M is an .R-module, its torsion submodule is H(M) = {m e M : 3d e R - {0} A dm - 0}. Theorem 3.3. Let M be a non zero R-module. Then, the following statements are equivalent: 1. M is a divisorial multiplication module with respect to J-(fi). 2. M is F(n)-cyclic. 3. There exists a monomorphism of R-modules from M into K. Proof. We prove the equivalence between (1) and (2). Every J-"-cyclic module is a divisorial multiplication module by [4, Corollary 3.9]. Now, suppose that M is a divisorial multiplication module with respect to Firstly, we assume that n(M) — 0. Let 0 ^ m € M. By hypothesis, there exists A < R such that Cl™ (Rm) = Cl^(AM). Moreover, A ^ 0 because if not, m e fJi(M), a contradiction. Given n + Rm e M/Rm, there exists a non zero element a e A such that an £ Cl^(Rm). Therefore, there exists / e F(v>) such that Ian C Rm. It follows that there exists 0 ^ i £ I satisfying that ian e Rm. Thus ia(n + Rm) e M/Rm. Consequently, M/Rm is /i-torsion and hence, M = Cl^ (Rm). Consider the general case. Then, M/p.(M) is jU-torsion free. By the above case, there exists m € M such that M///(M) = C^ /M(M) (m + ^(M)). As a consequence, for every x e M — /i(M), there exists a non zero element, a e R, such that az + ^(-^0 = m + /Lt(M). It is deduced that M/Rm is /^-torsion and hence, M is /^-cyclic. Prove that (2) is equivalent to (3). Suppose that M = Cl1^ (Rm) for some TO e M. Then, M/Rm is //-torsion and' .Rm is essential in M. Let 0 ^ x e M. Then, 0 ^ rx for some Q ^ r & R and jRm n Rr ^ 0. Since Rm = R, it holds E(M) ^ E(Rm) ^ E(R) ^ K. It is deduced that (2) => (3).

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Suppose that
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—> K is a monomorphism of ^-modules. Since M 7^ 0, a G v(^) n •& ^ is clear that K/Ra is ^-torsion. Since is /^-torsion. Let m & M such that a =
As every ideal of any integral domain satisfies condition (3) of Theorem 3.3, the following result is straightforward. Corollary 3.4. Every integral domain is a divisorial multiplication ring with respect to Recall that a star- operation over a domain R with field of quotients K is a mapping * from the set Prac(-R) of all fractional ideals of R to itself satisfying the following properties: 1. (Ra)* = Ra for every Q^a&K. 2. (la)* = I*a for every 0 7^ a & K and for every / e Frac(E). 3. ICJ^PCJ* for every /, J e Frac(.R). 4. / C /* and (I*)* = I* for every / e Frac(tf). A *-ideal is a fractional ideal / satisfying that 1 = 1*. A domain J? is a *multiplication domain if for every *-ideal A there exists a fractional ideal B such that (AB)* = R. If J-" is a Gabriel topology over the domain R such that R is .F-closed in its field of quotients K, then the mapping * defined by I* = Clp(I) is a star-operation in R. In this situation, we have the following result. Proposition 3.5. Let R be an integral domain and T a Gabriel topology such that R is f -closed in its field of quotients. Then, R is a divisorial multiplication ring with respect to JF if and only if R is a * -multiplication ring. Proof. Suppose that R is a divisorial multiplication ring. Let A < R be a *-ideal. By hypothesis and Theorem 3.1, there exists B £ Frac(.R) such that R = Cl^(AB). Thus (AB)* = R and B* is a *-ideal satisfying that (AB*)* = Clp(AB*) = R. The result can be extended easily for fractional *-ideals. Conversely, we assume that R is a ^-multiplication ring. Let A be an .F-closed ideal of R. By hypothesis, there exists B = B* such that (AB)* = R = Cl^(AB). Therefore, A is .F-invertible. By Theorem 3.1, A is a divisorial multiplication ideal and therefore R is a divisorial multiplication ring with respect to f. D On the other hand, if * is a star-operation on R, then J^* — {I < R : I* = R} defines a Gabriel topology on R, Proposition 3.6. Let R be an integral domain and * a star operation on R. If R is a divisorial multiplication ring with respect to f* , then it is a * -multiplication ring. Proof. Let A = A* < R. Take y € Ct?.(A). Then, there exists / e f* such that ly C A. Since I* — R, (Iy)* C A* = A. By the properties of star-operations, I*y C A and hence, Ry C A. It follows that y e A and A is JT*-closed. By Theorem

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3.1, there exists B e Frac(.R) such that Clf.(AB) = R. Therefore, AB & T and consequently, (AB*)* = R. It is deduced that A is *-invertible. u Given A e Frac(R), we denote by A'1 = (R : A) = {k e K : kA C R}. The map from Frac(R) to itself which maps A to Av = (R : (R : A)) is called v-operation and it is a star-operation (see [8, Theorem 34.2]). Therefore, R = Rv. The fractional ideal A is called a w-ideal if A = Av. It is said that R is a v-multiplication ring if it is a ^-multiplication ring for the ^-operation. Let f\ be the Gabriel topology corresponding to the torsion theory cogenerated by the injective hull of K/R, E(K/R) (see [17]). Then, an .R-module is J^-torsion when HomR(M,E(K/R)) = 0. Given a subring R of R', an element a e J?' is almost integral over R if there exists a finitely generated submodule of the .R-module R' containing all powers of a. The set R0 of elements of R' which are almost integral over R is called the complete integral closure of R in R'. If R = RQ, it is said that R is completely integrally closed in R.'. If R is completely integrally closed in its rings of quotients, it is said that R is completely integrally closed. A fractional ideal A of J? is called divisorial ;( -n. A — — I'D D .. \\\ 11 ^t, .. (^-it j-ijj.

Theorem 3.7. For an integral domain' R the following sentences are equivalent: 1. R is a v-multiplication ring. 2. R is completely integrally closed. 3. R is a divisorial multiplication ring with respect to T\. Proof. The first step is to prove that T\-multiplication domains are just u-multiplication domains. Firstly, we will prove that J-\ = {A < R : Av = R}. Suppose that A < R verifies Av = R. Let J < R and A C J. Let /: J/A -» K/R be a homomorphism of .R-modules. By injectivity, we extend it to a morphism /': R/A -> E(K/R) such that Im/ = /'(J/A). Then, Aim/' = 0 and hence, Im/' n (K/R) C (R : A)/R. Thus Im/' n (K/R) = R(lmf n (K/R)) =

Av(lmfr\(K/R))

C (A^fi: A)+R)/R = 0. Because K/R is essential in its injective hull, Im /' = 0 and therefore, / — 0. This proves that HomR(J/A, (K/R)) = 0. It follows that A £ J^i. On the other hand, if A e ^"i, it is clear that .R = Cl^ (A) and consequently, Av — R. Now, notice that A £ Frac(J?) is ^i-invertible if and only if there is a fractional ideal B such that Cl^(AB) = R, but it is equivalent to the existence of B e Frac(.R) satisfying (AB)V = R, that is, A is n-invertible. By [2, Proposition 3.2], for every f-invertible ideal A, it holds Av = Cl%^ (A). Now, it suffices to apply Proposition 3.5. By [2, Theorem 3.1 (f)], R is completely integrally closed if and only if every fractional ideal is w-invertible. It suffices to prove that i>-ideals are v-invertible. By the first step, this is equivalent to proving that every fractional ideal in R is JVinvertible. D

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An J?-module M is J- -finitely generated if there is a finitely generated submodule TV < M such that M = Cl^(N). A ring is J^-Noetherian if every ideal is ^-finitely generated. From [10, Theorem 8.12], the following result is straightforward. Corollary 3.8. fi-Noetherian ^Pi-multiplication domains are exactly Krull domains. Corollary 3.9. If the integral domain R is completely integrally closed, then its localizations in the height one primes are multiplication rings. Proof. By [7, Corollary 3.11], Rp is a discrete valuation domain for every height one prime ideal (divisorial prime ideal) of R and therefore Rp is a multiplication ring. D If we consider the t-operation over an integral domain R, i.e., It = L){FV : F C I,F € frac(#)}, then {/ < R : It = R} is a filter for a Gabriel topology Ft. By using Theorem 3.1, Proposition 3.5 and Proposition 3.6, we deduce immediately the following characterization. Proposition 3.10. Krull domains are exactly divisorial multiplication domains with respect to Ft-

4. Main results Recall that a submodule AT of M is called F -maximal if it is maximal between the ^-closed submodules of M. The set of all jF-maximal submodules of M is denoted by Maxjr(M). The following results shows that relative multiplication modules share properties with relative projective and finitely generated modules. Proposition 4.1. If M ^ t(M) is a divisorial multiplication module with respect to F, then every f -closed submodule is contained in an T'-maximal submodule. In this case, N = Cl^ (N) is F -maximal if and only if N = Cl^(PM) for some Pe Proof. By hypothesis we can choose m € M - t(M). The ^-closed ideal (t(M) : Rm) is not zero because of the choice of m. Thus there exists P e Maxf(R) containing (t(M) : Rm). Since M is a divisorial multiplication module, there is an ideal A such that Cl%(Rm) = Cl^(AM). If M = Cl^(PM), Cl^ (Rm) = Cl^(ACl^(PM)) = Cl^(APM) = Cl^(P(AM)) = Cl^(Pm). It follows that there is / G f such that Im C Pm. Since / belongs to the Gabriel filter, / g P and therefore there exists c <E I D (JR - P) such that cm = 0. Hence c € (t(M) : Rm), a contradiction. Thus M ^ Cl™ (PM). Let L be an .F-closed submodule of M such that Cl^(Pm) C £ c Af. Since L = CZ^((L : M)M) = Cl% (Cl%(L : M)M), it is clear that Clp(L : M) ^ R. As a consequence, there exists Q & Maxf(R) such that Cl*(L : M) C Q. Since PM C Cl^(PM) C L, P C (L : M) and therefore P C Cl%(L : M) C Q = Cl*(Q). As P,Q e Max^(fi), it follows that P = Cl%(L : M) = Q. As a result, L = Cl%((L : M)M) = Cl^(PM) and Cl^(PM) is maximal between the .F-closed submodules. D

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By the above result, it makes sense to define the relative Jacobson radical of a multiplication ^-module M as being

Jf(M] = r\{N = Cl™(N) < M : N e Max^(M)}. The definition in the general case of relative Jacobson radical and the first relative version of Nakayama's Lemma can be found in [15, 16]. The next result is a new relative version of Nakayama's Lemma for multiplication modules. Corollary 4.2. If M is a divisorial multiplication module with respect to F and N is an f'-closed submodule of M such that M — Cl^/l (N + J f ( M ) ) , then M — N. Proof. Suppose that M ^ Cl^(N + J>(M)). Prom Proposition 4.1, there exists K € Max^(M) such that Cl^(N) C K. Since Cl¥(J?(M)) C K, it follows that Cl%(N + Jr(M}) = Cl%(Cl%(N) + Cl^(J^(M)} C Cl^(K) = K, & contradiction. D We need some previous results to characterize locally divisorial multiplication rings. An ./-"-closed .R-submodule N of M is said to be an T-meet-principal submodule if for every ^-closed ideal A of R and every .F-closed submodule L of M, Clf(ANr\L) = Clf([Ar\(L : N)}N}. By using the usual properties of the closure we obtain the following results. Proposition 4.3. The submodule N = Cl^l (N) < M is an F-multiplication submodule if and only if it is J- -meet-principal. Corollary 4.4. //1, J are divisorial multiplication F-closed ideals, then the ideal Clyr(IJ) is a divisorial multiplication ideal with respect to T. We set r)(N) = 52xeN(Clp (Rx) : N) for any submodule N of M. Recall that an ideal A of R is called ^"-critical if it is maximal in the set of .F-closed ideals. This set will be denoted by Max^(J?). The following results can be proved similarly to the absolute case. Lemma 4.5. // TV is an ^F-closed divisorial multiplication submodule of M, then for every submodule L = Cl™(L) C TV, L = Cl¥(r)(N)L). In the next result we consider the Gabriel topology T over R and Tp over Rp. Lemma 4.6. Given A e Cf(R) and N e CV(Af), it holds 1. r)(A)r)(N) C rj(AN). 2. r](N)s C ri(Ns) for every multiplicatively closed subset S of R. Henceforth, we need to work with Gabriel topologies F having enough .F-critical ideals. Henceforth f is such a Gabriel topology over the ring R. Proposition 4.7. Assume that P e Max^(JZ) and N £ Cf(M) is a divisorial multiplication module with respect to f'. If rj(N) C P, then NP = 0 and if r/(N) <2 P, then Np is cyclic as an Rp-module.

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Proof. Suppose that r,(N) C P and take x 6 TV. By Lemma 4.5, Cl^(Rx) = Clf(Rr/(N)x). Therefore, there exists J e F such that Jz C rj(N). Thus there exists c £ J n (.R — P) such that ex = 0. It follows that Np = 0. Since TV is a divisorial multiplication module and P is an JF-closed prime, TVp is a multiplication .Rp-module (see [5, Theorem 4.18]). But every multiplication module over a local ring is cyclic (see [1, Theorem 1]). D Proposition 4.8. Every J-'-closed R-submodule TV of M is a divisorial multiplication submodule if and only if for every P £ Maxjr (R) such that 77 (TV) C P, it holds NP = 0. Proof. Necessity is a consequence from Proposition 4.7. To prove the sufficiency, we check that TV is jF-meet-principal. Let A € C?(R) and L 6 Cf(M). Since J7 is semicentered, we have [ C l f ( N ) ] P = NP = Cl*pp(NP). Therefore we shall prove that ([A n (L : TV) TV] TV)P = (AN D L)P for every P e M&xjr(R). Let P 6 Maxyr(R). First, suppose that rj(N) C P. To check the equality, we localize in the jF-closed critical ideals. Then, {(A H (L : TV))TV)jp = (AP n (L : N)P')NP = 0. Therefore, Cl^((A n (L : N))N) = t(M). On the other side, (AN n L)P = ApNp n Lp = 0. It follows that Ciy (AN n L} = t(M). Now, we suppose that ??(TV) £ P- By definition of 77 (TV), there exists x e TV such that (Cl^(Rx) : N) <£ P. Then there exists t' £ R - P satisfying t'TV C Cl^(Rx), that is, there is J e T with i' JTV C Rx. But some element of J does not belong to P because otherwise P G J- . Thus there exists t & R — P such that iTV C flx. Now, we can proceed as in the absolute case. Given y = a^xi + • • • anxn € ATV n L with dj e A, Xj € TV (1 < j < n), we have tXj € tN C T?x. Consequently, for every j, tx,,- = r^x for some r, e -R. Hence ty = a\(tx\) + • • • + an(txn) = ai(rix) H ----- h an(rnx) = (a^ri H ----- h a n r n )x. Then, (oir-i H ----- h anrn)tN C (airi + • • • + anrn)Rx = Rty C L. Therefore, (a^i + • • • anrn)t £ An (L : N). Hence (airi H ----- h anrn)/l £ [A n (L : TV)] P . Then, 6

(ft\~

tb

ft\~

air-i + • • • + anrn x

(L : TV) P )TVp.

We have (ATV n L)P C [A n (L : 7V)]PTVp = [(A n (L : N))N]P. Since the other inclusion is obvious, this gives the equality.

D

Proposition 4.9. An R-module M is a divisorial multiplication module with respect to T if and only if for every P e Maxyr(R) with MP ^ 0, MP is a cyclic Rp-module and (L : M)P = (Lp : MP)'for every L e Cf(M). Proof. Let P e Max:F(J?) and suppose that the left part of the equivalence is true. By Proposition 4.8, if MP ^ 0 then rj(M) <£ P, i.e., there is x e M such

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that (Rx : M) g P. Thus there exists c e R - P such that cM C Rx. Hence MP = (Rx}p. Let L be an ^-closed submodule of M. By using [5, Lemma 2.1], (LP : Mp) = (LP : (Rx)P) = (L : Rx)P D (L : M)P. On the other hand, (L : M)P — (L : cM)P because if f e (L : cM)P then acM C L and therefore f = ff € (£ : M)p and the other inclusion is always true. Thus (L : M)p = (L : cM)P D (L : Rx)P = (LP : (Rx)P) = (LP : MP). Now, we assume the right side of the equivalence is true. We shall prove that M is .F-meet principal. Let A £ C^(R), L 6 O(M) and P 6 M&xf(R). First, we suppose that MP = 0. Then (AM n L)P = APMP n LP = 0 = [(A n (L : M))M] P . Now, we assume that Mp ^ 0. By hypothesis, MP is cyclic as an .Rp-module. Since every cyclic module is a multiplication module, MP is meet-principal. Thus (AM n L)P = APMp r\LP = \AP n (LP : MP)}MP. By using the second part of the hypothesis, [A n (L : M)M]P = [AP n (L : M}P}MP = [AP n (LP : MP}}MP.

D

We can define a divisorial almost multiplication ring with respect to a Gabriel topology f as a ring R such that RP is a multiplication ring for every P 6 Spec-p(R). When R has enough /"-critical ideals, RP is a multiplication ring for every P e Spec:F(R) if and only if RQ is a multiplication ring for every Q 6 Max^-(E). The proof is completely similar to the absolute case (see [10, pp. 216-225]). Compare next result to [13, Corollary 7.4]. Corollary 4.10. Let T be a Gabriel topology over R having enough ^--critical ideals and let M be an R-module. Then 1. if M is a divisorial multiplication R-module, then MP is a multiplication Rp-module for every P £ Maxyr(R). 2. if moreover M is ^-finitely generated, the converse is true. 3. if R is F-Noetherian, R is a divisorial multiplication ring with respect to T if and only if R is a divisorial almost multiplication ring with respect to T. Proof. (1). Suppose that M is a divisorial multiplication module with respect to Jand P is an ^"-critical ideal of R. If Mp = 0, then Mp is clearly a multiplication /Zp-module. If not, Proposition 4.9 gives that Mp is cyclic as an .Rp-module and every cyclic module is a multiplication module. (2). Now, we assume that Mp is a multiplication .Rp-module for every P e Maxjr(JZ). Since Mp is a multiplication module over a local ring, it is cyclic. As M is ^-finitely generated, (L : M}p = (LP : MP) for any L < M. By Proposition 4.9, M is a divisorial multiplication -R-module. By (2), (3) is trivial. D An integral domain with ascending chain condition over divisorial ideals is said to be a Mori domain. As a consequence of Corollary 4.10 (3), Krull domains are Mori domains whose rings localized in height one primes are multiplication rings.

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Local multiplication rings are Noetherian, namely, they are discrete valuation domains of rank 1 or special primary rings. However a local divisorial multiplication ring with respect to T is not necessarily jF-Noetherian as the following example shows. Consider a valuation domain R having an Archimedean value group G and v being the corresponding valuation. Then we can consider G C JR. An ideal of G is a subset / C G+ = {g 6 G : g > 0} such that if x € I and y > x then y £ I. Since G — I has a supremum r in R, ideals in G can be of the form Ir = {x 6 G : x > r} (when r £ G) or Ir = {x € G : x > r} for some r 6 R (when r £ G). Therefore, ideals of R are of the form AT = {x 6 R : v(x) > r} or AT = {x e G : v(x) > r}. The only prime ideal of R is the maximal ideal of R, A0, which is clearly idempotent. We can consider the Gabriel topology f having filter equal to {AQ, R}. The ring R is not /"-Noetherian because F is not semicentered. Nevertheless, R is a divisorial multiplication ring with respect to T. In fact, the ^"-closed ideals are those of the form AT because Cl-p(Ar) = Ar. Given Ar C As, then r - s > 0. Since TT — Ts + / r _ s , we have ~A~T = CI^(ASAT-S}. We denote by T-p the Gabriel topology induced by the set of prime ideals P and PI stands for the set of the height one primes. Theorem 4.11. Let R be a HC ring. If R is a divisorial multiplication ring with respect to a Gabriel topology T then there exists P C PI such that Rp is a multiplication ring for every P £ P. If moreover R is fp-Noetherian then the converse is true. Proof. If R is a HC ring, then every Gabriel topology over R is of the form jF-p with P C Spec(R). By Corollary 4.10 (1), if R is a divisorial multiplication ring with respect to T-p then Rp is a multiplication ring for every P & P. Since every nonzero prime ideal of a local multiplication ring is maximal, the height of PRp, and therefore the height of P, is less or equal to one. Thus that P C P^. For the second part, it suffices to consider Corollary 4.10 (3). D Notice that Noetherian divisorial multiplication rings generalize strictly Krull domains. For example, consider the ring R = K[x, y]/(x2 —y3) which is not a Krull domain but it is a Noetherian domain. Every localized ring is a discrete valuation domain except the localized one in zero. Thus R is a divisorial multiplication ring with respect to the Gabriel topology induced by the pointed spectrum, i.e., the one induced by the powers of the maximal ideal (x, y)/(x2 — y 3 ). The construction can be generalized by taking the coordinate ring of any curve with a finite number of singularities. Thus we have achieved a way to obtain new examples of divisorial multiplication rings. Examples. The Gabriel topology induced by the set Pn of all primes of height less or equal to n is denoted by Tn. If R is a divisorial multiplication ring with respect to Tn (not necessarily ^-Noetherian), by Corollary 4.10 (1), Rp is a multiplication ring for every P 6 Pn. By the proof of Theorem 4.11, P has to belong to PI and therefore there is no divisorial multiplication ring with respect to J-n for n > 2. In a generalized Dedekind domain R, every Gabriel topology F is of finite type and therefore T is semicentered and R is .F-Noetherian, i.e., the hypothesis of

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Theorem 4.11 is satisfied. By [6, Theorem 2..2], Rp is a discrete valuation domain of rank 1 for every P 6 P when T C 'Pi. Thus every generalized Dedekind domain is a divisorial multiplication ring with respect to F-p for every P C PI . Assume that R is a discrete valuation domain of rank n. Then, the set of primes is the chain 0 = PO C PI C • • • C Pn. The ring R is HC and by [3, Theorem 4.2] and every non-trivial Gabriel topology over R is of the form J-pi for some i e { l , 2 , . . . , n } . By Corollary 4.10 (1), if i > 2, R is not a divisorial multiplication ring. For i — 1, we obtain a discrete valuation domain of rank 1 which is a multiplication ring. By [4, Proposition 4.14], R is a divisorial multiplication ring with respect to J-pl. Semiartinian rings are characterized as HC rings having Krull dimension equal to 0 and a commutative ring R is perfect if and only if it is semiartinian and Spec(-R) is finite. Suppose that R is perfect. If J? is a divisorial multiplication ring with respect to J-', then every Rp is a multiplication ring for every P e Specyr(R) by Corollary 4.10. Conversely, if every RP is a multiplication ring for every P e Spec^-R), then every Rp is either a special primary ring or a Dedekind domain and therefore it is Noetherian. Since Spec(.R) is finite, [17, Corollary 13.4.6] gives that R is J-p-Noetherian. It follows that if R is a perfect ring, then being a divisorial multiplication ring is equivalent to being a divisorial almost multiplication ring. Now. our aim is to characterize divisorial multiplication ^"-critical ideals. Proposition 4.12. // N is an T'-closed f-finitely generated f'-locally cyclic submodule of M, then the ideal rj(N) belongs to the Gabriel filter. Proof. Let P £ Maxj:(R). By hypothesis, NP is cyclic. It follows that there is x G N such that Np = (Rx)p. Since JV is ^-"-finitely generated, [5, Lemma 2.1] gives RP = ((Rx)P :'/VP) = (Rx : N)P = (Cl^ (Rx) : N)P. Then, ri(N)P = {^(Cl*((Rx) : N ) ] P = (^(Cl*(Rx : N ) ] P = RP. x€N

x£N

As a result, Cl*(n(N)) = R. Proposition 4.13. Let N 6 Cf(M).

D Then

1. // Clf(rj(N)) = R, then N is T'-locally cyclic. 2. If rj(N) = R, then N is F-finitely generated. Proof. Since r,(N) = Y.^Cl^(Rx) : N) and Cl*(r](N)) = R, it follows that there exists x e N such that (Cl™(Rx) : N) g P for every P e M&xf(R). Thus there is t £ R - P such that tN C Rx. As a result, NP = (Rx)P. Hence N is j^-locally cyclic. Now we suppose that rj(N) = R. Then, there exist xi, • • • ,xn 6 N such that (Cl¥(Rxi) : TV) + - - • + (Cl^(Rxn) : N} = R. By multiplying by N, N = (Cl¥(Rxi) : N)N + • • • + (Cl^(Rxn) : N)N C Cl^(Rxl + • • • Rxn) C N. Therefore, N is jr-finitely generated.

D

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Theorem 4.14. Given P £ M.ax.^(R), P is a divisorial multiplication ideal if and only if either P is J--finitely generated and ^-locally principal or P is F-locally principal and P c r/(P) C Cl^(r/(P)} = R or RP is a field. Proof. Notice that by the maximality of P, there are two possibilities, namely P = Cl^(n(P)) or CZ£fa(P)) = R. If P = a£(7j(P)), then r?(P) = P and by Proposition 4.8, P is a divisorial multiplication ideal if and only if Pp = 0, that is, RP is a field. Suppose that P is a divisorial multiplication ideal with respect to J-. If R = Clf(r)(P)), by Proposition 4.13, P is always ^-locally principal. If t](P) = R, the same proposition gives that P is jF-finitely generated. Otherwise, r/(P) ^ R but Cl^(n(P)} = R, that is, P C 7?(P) C Cl%(q(P)) = R. Now we suppose that the right side of the equivalence is true. If P is Ffinitely generated, then being ^-locally principal is equivalent to being a divisorial multiplication ideal (see [4, Theorems 3.7 and 4.18]). If P C 7?(P) C C/£(r?(P)) = R, then there is no Q £ Max^(.R) such that r/(P) C Q and by Proposition 4.8, P is a divisorial multiplication ideal with respect to F. D The next result is a relative version of Mott's Theorem (see [12]), which establishes that if every prime ideal of a ring P is a multiplication ideal, then R is & multiplication ring. We write Spec^P) for the set of all ^"-closed prime ideals of R. Theorem 4.15. Let R be an J--Noetherian ring. Then R is a divisorial multiplication ring with respect to J- if and only if every f-closed prime is a divisorial multiplication ideal. Proof. Suppose that' every jF-closed prime is a divisorial multiplication ideal. Let P e Specjr(P) and let QP e Spec(PP). Then Q e Spec(P). Take an ideal A < Q. By hypothesis Cl*(A) = Cl*(BCl%(Q)} = Cl*(BQ) for some B < R. By localizing, Ap = BpQp. It follows that every prime ideal of Rp is a multiplication ideal. By Mott's Theorem (in its original version), Rp is a multiplication ring. By Corollary 4.10, P is a divisorial multiplication ring. The other side of the equivalence is trivial. D This result provides us with a large number of new examples of divisorial multiplication rings. We can think of any Noetherian ring R which is not a multiplication ring and any family of prime ideals being multiplication ones. Then the corresponding semicentered torsion theory T makes R into a divisorial multiplication ring with respect to J-. For example, R = Z[x] is not a multiplication ring because it is an integral domain which is not a Dedekind domain. We can consider the semicentered torsion F having Rx and 0 as ^-closed prime ideals. By Theorem 4.15, R is a divisorial multiplication ring with respect to T. It is well-known that if I? is a Noetherian UFD (unique factorization domain), then every height one prime is principal.1 Since every principal ideal is a multiplication ideal, Theorem 4.11 and Theorem 4.15 give that R is a divisorial multiplication ring with respect to T-p if and only if P CP1.

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Next result is a structure theorem for Noetherian torsionfree divisorial ring with respect to a Gabriel topology T. It is a generalization of the one given by Gilmer and Mott about the structure of Noetherian multiplication rings (see [9, Corollary 3]). Theorem 4.16. If R is a divisorial multiplication ring relative to T which is fNoetherian, then R/t(R) is isomorphic to a finite direct sum of divisorial multiplication domains and rings whose lattice of f-closed ideals consists of one J--critical ideal and a finite number of closures of its powers. Proof. First, we shall prove that if A i , . . . , A n € Cf(R) and the closure of any pair of them is R, then Cl-p(Ai n • • • fl An) = d^(A\ • • • An) by induction over n. The case n = 1 is trivial. Suppose n = 2 and let Ai,A 2 £ C?(R) with Cl*(Ai + A 2 ) = R. The inclusion AiA 2 C A\ D A2 is always true. Then C7^(AiA 2 ) = Cl*(Ai n A 2 ) = AI n A2. We have (Ai + A2)x C Ai(Al n A 2 ) + A 2 (Ai n A2) C A! A2 + A 2 Ai = AiA 2 for every x € AI n A2. Since AI + A2 belongs to the Gabriel filter by hypothesis, x 6 Clf(AiA2) and the equality holds. By [5, Theorem 4.13], the lattice Cp(R) is distributive and hence CZ£(Ai + (Az n • • • n An)) = CZ£((Ai + A 2 ) n • • • n (Ai + A n )) = R. Then Cl^(A1 n - • - n A n ) = C?f ( A i ( A 2 n - • - n A n ) ) = Cl^(AlCl^(A2n- • - n A n ) ) = Clf(Ai • • • An), where the first equality follows from the case n = 2 and the last one from the hypothesis of induction. Since Cf(R] is Noetherian, every A e Cy^(R) can be written as a reduced finite meet of primary ideals belonging to Cr(R) (see [13, Theorem 3.3]). Suppose that A = AI n • • • n An with Ai Pj-primary ideals. It is well-known that Pt 6 Cf(R). Since Pi and Pj are coprime for i / j, Clp(Ai + A,-) = R. By the first step, A = C?£(Ai n • • • n A™) = Cl^(Al • • • An). Now, we shall prove by induction over n that if P € SpeCjr(-R), then there are no ^-closed ideals properly contained between P, Clp(P2),..., Cl^(Pn) for every positive integer n. By the same argument used in the proof of Theorem 4.11, every P S Spec-F(R) has height less or equal to 1 and therefore either P e Max.yr(R) or P C M for some M & Maxjr(fl). In the last case, P = Cl^(CM) for some C e Cf(R). Since P is prime, C C P and therefore P = Cl~(PM). Suppose n = 2 and A e Cf(R) with Cl^(P2) C A C P. Since R is a divisorial multiplication ring, A = Cl%(BP) for some B e C^-(fi). If B C P, then A C Ci^(P2) and we have A = Cl%(P2). Otherwise, either Cl*(B + P) = R 01 Cl*(B + P) = M for some M e Max J r( J R). In any case P = Cl%(P(B + P)) = d*(PB + P 2 ) = A. Assume n > 2 and take A e Cyr(R) such that Cl^(Pn] C A C P. By hypothesis A = Clf(BP). Since we can take B = (A : P), Cl^(Pn~l) C B. If B C P, then the hypothesis of induction gives us that B = Cl^(Pk] for some 1 < k < n — 1 and hence A = Clp(Pk+1). Otherwise, by repeating the above reasoning, P = A. As a result, every jF-closed ideal of R is the closure of a finite product of prime ideals because Aj is an ^"-closed ideal between P and the closure of some power of P.

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Suppose t(R) = C7J?(Pi • • • P* Af*1 • • • Mrr} where Pi is a ^"-closed prime not belonging to Max^(R) for each 1 < i < k and M,- e Maxjir(E) for each 1 < j < r, all ideals being pairwise coprime. We define (p: R/t(R) —> R/Pi x - • • x .R/Pfc x R/Cl%(M{1} x • • • x R/Cl%(M?r) which applies the element a £ .R in the product of the classes of a in each ring of the codomain of if. The morphism if is well-defined because iix+Clf(P\ • • • Mrr) = y+ Clr(Pi • • • Mr>-), then x-y € Cl*(Pi • • • M^} = Pin- • -r\Clf(M^} and, therefore <^(x + Clp(Pi • • • Mrr}) = (p(y + Clf(Pi • • -Mrr)). The map tp is injective: if Clp(Pi • • • M*r) and, x + Cl^(Pi • • • M*T] =y + Cl^(Pi • • • Mrr}. By the Chinese Remainder Theorem, ip is surjective and as a result, it is an isomorphism. The ring R/Pi is a divisorial multiplication domain with the induced topology and (M Sj ) satisfies the required conditions for each 1 < j < r. D References [1] D. D. Anderson, Multiplication Ideals, Multiplication Rings, and the Ring R(X), Can. J. Math. Vol. XXVIII (4) (1976) 760-768. [2] D. D. Anderson, J. Mott, M. Zafrullah, 'Some Quotient Based Statements in Multiplicative Ideal Theory, Boll. Un. Mat. It. (7) 3-B (1989) 455-476. [3] W. Brandal, E. Barbut, Torsion theories over commutative rings, J. Algebra 101 (1986) 136-150. [4] J. Escoriza, B. Torrecillas, Multiplication modules relative to torsion theories, Comm. Algebra 23 (11) (1995) 4315-4331. [5] J. Escoriza, B. Torrecillas, Relative Multiplication and Distributive Modules, Comment. Math. Univ. Carolinae 38 (2) (1997) 205-221. [6] M. Fontana, N. Popescu, Sur une classe d'anneaux qui generalisent les anneaux de Dedekind, J. Algebra 173 (1995) 67-96. [7] R. M. Fossum, The divisor class group of a Krull domain, Springer, Berlin, 1987. [8] R. W. Gilmer, Multiplicative ideal theory, Marcel Dekker, New York, 1972. [9] R. W. Gilmer, J. L. Mott, Multiplication rings as rings in which ideals with prime radical are primary, Trans. Amer. Math. Soc. 114 (1965) 40-52. [10] M. D. Larsen, P. J. McCarthy., Multiplicative theory of ideals, Pure and Applied Mathematics 43, Academic Press, New York, 1971. [11] S. Mori, Axiomatische Begriindung des Multiplikationringes, J. Sci. Hiroshima Univ. 3 (1932), 43-59. [12] J. L. Mott, Equivalent conditions for a ring to be a multiplication ring, Can. J. Math. 16 (1964) 429-434. [13] C. Nastasescu, La structure des modules par rapport a une topologie additive, Tohoku Math. J., The second series 26 (2) (1974) 173-201. [14] J. V. Pakala, T. S. Shores, On compactly packed rings, Pacific J. Math. 97 (1) 1981 197-201. [15] T. Porter, A relative Jacobson radical with applications, Radical Theory. North-Holland. Amsterdam, 1985 pp. 405-408. [16] T. Porter, The kernels of completion maps and a relative form of Nakayama's Lemma, J. Algebra 85 (1983) 166-178. [17] B. Stenstrom, Rings of Quotients, Springer, Berlin, 1975.

Global Deformations of Lie Algebras Alice Fialowski Department of Applied Analysis, Eotvos Lorand University, Pazmany Peter setany 1, H-1117 Budapest, Hungary fialowskScs.elte.hu Abstract. By considering non-trivial global deformations of the Witt (and the Virasoro) algebra given by geometric constructions it is shown that, despite their infinitesimal and formal rigidity, they are globally not rigid. This shows the need of a clear indication of the type of deformations considered. The families appearing are constructed as families of algebras of Krichever-Novikov type.

1. Introduction This talk is based on a joint work with Martin Schlichenmayer (see [6]). Deformations of mathematical structures are important in most part of mathematics and its applications. Considering deformations of a given object, a natural question arises: Can we equip the set of nonequivalent deformations with the structure of a topological or even geometric space. In other words, does there exists a moduli space for these structures. If so, then for a fixed object the deformations of this object should reflect the local structure of the moduli space at the point corresponding to this object. In this respect, a good example where the picture is completely clear is the classification of complex analytic structures on a fixed topological manifold. Also in algebraic geometry one has well-developed results in this direction. Recall for instance that for the moduli space M.g of smooth projective curves of genus g over C (or equivalentiy, compact Riemann surfaces of genus g) the tangent space T\C}Mg can be naturally identified with H1(CI, TC), where TC is the sheaf of holornorphic 1991 Mathematics Subject Classification. Primary: 17B66; Secondary: 17B56, 17B65, 17B68, 14D15, 14H52. Key words and phrases. Deformations of algebras, rigidity, Witt algebra, Virasoro algebra, Krichever-Novikov algebras, elliptic curves, Lie algebra cohomology. This research was partially supported by the grants OTKA T030823, T29525.

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vector fields over C. This extends to higher dimension. In particular, it turns out that for compact complex manifolds M, the condition H1(M, TM} implies that M is rigid, [9, Thm. 4.4]. Rigidity means that any differentiable family TT: M —> B C R, 0 G B which contains M as the special member MQ := 7r -1 (0) is trivial in a neighbourhood of 0, i.e., for t small enough M% '.= TT~1(t) = M. (see also [11] for definitions, results, and further references). Such results lead to the impression that the vanishing of the relevant cohomology spaces will imply rigidity with respect to deformations also in the case of other structures. In this talk I will consider an infinite dimensional Lie algebra example. The results will show that the theory of deformations of infinite-dimensional Lie algebras is far from being satisfactory. What I will show here, can also be done in the case of associative algebras. Consider the complexification of the Lie algebra of polynomial vector fields on the circle with generators ln := exp(in
neZ

where y> is the angular parameter. The.bracket operation in this Lie algebra is

[ln,lm] = (m-n)ln+m. We call it the Witt algebra and denote it by W. Equivalently, the Witt algebra can be described as the Lie algebra of meromorphic vector fields on the Riemann sphere P1(C) which are holomorphic outside the points 0 and oo. In this presentation ln = z n+1 ^, where z is the quasi-global coordinate on P 1 (C). The Lie algebra W is infinite dimensional and graded with the standard grading deg ln = n. By taking formal vector fields with the projective limit topology we get the completed topological Witt algebra VV. In this talk I will consider its everywhere dense subalgebra W. It is well-known that W has a unique nontrivial one-dimensional central extension, the Virasoro algebra V. It is generated by ln (n e Z) and the central element c, and its bracket operation is defined by (1.1)

[ln,lm] = (m-n)ln+m + ^(m 3 - m) J n ,_ m c,

[Z n ,c] = 0.

The cohomology "responsible" for deformations is H 2 (W, W). It is known that H 2 (W, W) = {0} (see [4]). Hence, guided by the experience in the theory of deformations of complex manifolds, one might think that W is rigid in the sense that all families containing W as a special element will be trivial. But this is not the case as we will show. Certain natural families of Krichever-Novikov type algebras of geometric origin (see Section 3 for their definition) will appear which contain W as special element. But none of the other elements will be isomorphic to W. In fact, from H 2 (W, W) = {0} it follows that the Witt algebra is infinitesimally and formally rigid. But this condition does not imply that there are no non-trivial global deformation families. The main point to learn is that it is necessary to distinguish clearly the formal and the global deformation situation. The formal rigidity of the

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Witt algebra indeed follows from H 2 (W, W) = {0}, but no statement like that about global deformations. There is a lot of confusion in the literature in the notion of a deformation. Several different (inequivalent) approaches exist. One of our aims here is to clarify the difference between deformations of geometric origin and so called formal deformations. Formal deformation theory has the advantage of using cohomology. It is also complete in the sense that under some natural cohomology assumptions there exists a versal formal deformation which induces all other deformations. Formal deformations are deformations with a complete local algebra as base. A deformation with a commutative (non-local) algebra base gives a much richer picture of deformation families, depending on the augmentation of the base algebra. If we identify the base of deformation—which is a commutative algebra of functions—with a smooth manifold, an augmentation corresponds to choosing a point on the manifold. So choosing different points should in general lead to different deformation situations. As we will see in the case of the infinite dimensional Witt algebra, there is no tight relation between global and formal deformations. I am going to introduce and recall the necessary properties of the KricheverNovikov type vector field algebras. Trjey are generalizations of the Witt algebra (in its presentation as vector fields on P 1 (C)) to higher genus smooth projective curves. We construct global deformations of the Witt algebra by considering certain families of algebras for the genus one case (i.e., the elliptic curve case) and let the elliptic curve degenerate to a singular cubic. The two points, where poles are allowed, are the zero element of the elliptic curve (with respect to its additive structure) and a 2-torsion point. In this way we obtain families parameterized over the affine line with the peculiar behaviour that every family is a global deformation of the Witt algebra, i.e., W is a special member, whereas all other members are mutually isomorphic but not isomorphic to W, see Theorem 3.4. Globally these families are non-trivial, but infinitesimally and formally they are trivial. The construction can be extended to the centrally extended algebras, yielding global deformations of the Virasoro algebra. 2. Global deformations and formal deformations 2.1. Intuitively Let us start with the intuitive definition. Let £ be a Lie algebra with Lie bracket //o over a field K. A deformation of £ is a one-parameter family £t of Lie algebras with the bracket (2.1)

p,t =/z 0 + i
where the
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£ where ^ are linear maps over K, i.e., elements of C1 (£,£), such that

(2.2)

^(z, y) = ^\nt(^(x

The Jacobi identity for the algebra £4 implies that the 2-cochain <j>i is indeed a cocycle, i.e., it fulfills d^i = 0 with respect to the Lie algebra cochain complex of £ with values in £ (see [7] for the definitions). If (pi vanishes identically, the first nonvanishing
# - 0 i = diVi-

Hence, every equivalence class of deformations defines uniquely an element of H2 (£,£). This class is called the differential of the deformation. The differential of a family which is equivalent to a trivial family will be the zero cohomology class. 2.2. Global deformations Consider now a deformation £ t not as a family of Lie algebras, but as a Lie algebra over the algebra K[[t]]. The natural generalization is to allow more parameters, or to take in general a commutative algebra A over K with identity as base of a deformation. In the following we will assume that A is a commutative algebra over the field K of characteristic zero which admits an augmentation e : A —> K. This says that e is a K-algebra homomorphism, e.g., e(lx) = 1- The ideal m£ := Kere is a maximal ideal of A. Vice versa, given a maximal ideal m of A with A/m = IK, then the natural factorization map defines an augmentation. In case that A is a finitely generated K-algebra over an algebraically closed field K then A/m = K is true for every maximal ideal m. Hence in this case every such A admits at least one augmentation and all maximal ideals are coming from augmentations. Let us consider a Lie algebra £ over the field K, e a fixed augmentation of A, and m = Ker (. the associated maximal ideal. Definition 2.1. A global deformation A of £ with the base (A,m) or simply with the base A, is a Lie A-algebra structure on the tensor product A ®K £ with bracket [. , .}\ such that (2.4)

is a Lie algebra hornomorphisrn (see [5]). Specifically, it means that for all a, 6 t A and x,y 6 £,

(1) [a x, b ® y]x = (ab ® id)[l <8> x, I y}\, (2) [ . , . ]\ is skew-symmetric and satisfies the Jacobi identity, (3) e ® id([l x, 1 ® j/] A ) = l®[x,y}. By Condition (1) to describe a deformation it is enough to give the elements , ~L<S>y]\ for all x, y E £. By condition (3) follows that for them the Lie product

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has the form i

with tti € m, Zj € £. A deformation is called trivial if A ®K £ carries the trivially extended Lie structure, i.e., (2.5) reads as [1 x, 1 y]\ — I <8> [x, y}. Two deformations of a Lie algebra £ with the same base A are called equivalent if there exists a Lie algebra isomorphism between the two copies of A <8> £ with the two Lie algebra structures, compatible with e ® id. We say that a deformation is local if .A is a local K-algebra with unique maximal ideal 771.4. By assumption TTLA = Kere and A/rriA = K. In case that in addition = 0, the deformation is called infinitesimal. Example. If A = K[i], then this is the same as an algebraic 1-parameter deformation of £. In this case we sometimes use simply the expression "deformation over the affine line." This can be extended to the case where A is the algebra of regular functions on an affine variety X. In this way we obtain algebraic deformations over an affine variety. These deformations are non-local, and will be the objects of our study. Let A' be another commutative algebra over K with a fixed augmentation e': A' —> K, and let : A —> A' be an algebra homomorphism with (!) = 1 and e' o <j) — e. If a deformation A of £ with base (A, Ker e = m) is given, then the push-out A' = (j>f\ is the deformation of £ with base (A', Ker e' = m'), which is the Lie algebra structure [a( ®A (ai ® li),a'2 ®A (0-2 ® ^)]A' := a(a'2 ®A [a\ ® /i,a 2 <S>l2\\, (a(,a2 € A',ai,a2 e A,k,l2 6 £) on A' ® £ = (A' ®AA)® £, = A' ®A (A ® £). Here A1 is regarded as an A-module with the structure aa' = a'(a). Remark. For non-local algebras there exist more than one maximal ideal, and hence in general many different augmentations e. Let £ be a K-vector space and assume that there exists a Lie A-algebra structure [ . , .}A on A ®K £. Given an augmentation e: A —> K with associated maximal ideal me = Kere, one obtains a Lie K-algebra structure ££ = ( £ , [ . , . ] e ) on the vector space £ by (2.6)

e ® id([l ® x, 1 ® j/] A ) = 1 ® [i, j/] £ .

Comparing this with Definition 2.1 we see that by construction the Lie A-algebra A K £ will be a global deformation of the Lie K-algebra £ e . On the level of structure constants the described construction corresponds simply to the effect of "reducing the structure constants of the algebra modulo me." In other words, for x, y, z G £ basis elements, let the Lie A-a\gebi& structure be given by (2.7)

[l®z,l®yU

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Then £e is defined via (2.8)

[x, y}e := Y,(C^,y

mod m

^z'

In general, the algebras C€ will be different for different e. The Lie A-algebra will be a deformation of different Lie K-algebras U. Example. For a deformation of the Lie algebra £ = £Q over the affine line, the Lie structure £Q in the fiber over the point a 6 K is given by considering the augmentation corresponding to the maximal ideal ma = (t — a). This explains the picture in the geometric interpretation of the deformation. 2.3. Formal deformations Let A be a complete local algebra over K, so A — lim n _ >00 (A/m n ), where m is the maximal ideal of A and we assume that A/m = K. Definition 2.2. A formal deformation of £ with base A is a Lie algebra structure on the completed tensor product A ® £ = limn^00((A/mn) ® £) such that

e § id : A® £ —> K ® £ = £ is a Lie algebra homomorphism. Example. If A — K[[t]], then a formal deformation of £ with base A is the same as a formal 1-parameter deformation of £ (see [8]). There is an analogous definition for equivalence of deformations parameterized by a complete local algebra. 2.4. Infinitesimal and versal formal deformations i In the following let the base of the deformation be a local K-algebra (A, m) with A/m = K. In addition we assume that dim(m fe /m fc+1 ) < oo for all k, Proposition 2.3 ([5]). With the assumption dimH 2 (£;£) < oo, there exists a universal infinitesimal deformation rjc of the Lie algebra £ with base B = K ® H2(L\C,}', where the second summand is the dual o/H 2 (£;£) equipped with zero multiplication, i.e., (ai,hi) • (a 2 ,/i 2 ) = (aia 2 ,ai/i 2 + a:2/ii). This means that for any infinitesimal deformation A of the Lie algebra £ with finite dimensional (local) algebra base A there exists a unique homomorphism : K ® .H"2(£; £)' —> A such that A is equivalent to the push-out *?/£. Although in general it is impossible to construct a universal formal deformation, there is a so-called versal element. Definition 2.4. A formal deformation r/ of £ parameterized by a complete local algebra -B is called versal if for any deformation A, parameterized by a complete local algebra (A,m^), there is a morphis'rn /: B —» A such that 1) The push-out ffr) is equivalent to A. 2) If A satisfies m^ 2 = 0, then / is unique (see [2, 5]).

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Remark. A versal formal deformation is sometimes called miniversal. Theorem 2.5 ([3], [5, Thm. 4.6]). Let the space H 2 (£;£) be finite dimensional. (a) There exists a versal formal deformation of £. (b) The base of the versal formal deformation is formally embedded into H 2 (£;£), i.e., it can be described in H 2 (£;£) by a finite system of formal equations. A Lie algebra £ is called (infinitesimally, formally, or globally) rigid if every (infinitesimal, formal, global) family is equivalent to a trivial one. Assume H 2 (£,£) < oo in the following. By Proposition 2.3 the elements of # 2 (£;£) correspond bijectively to the equivalence classes of infinitesimal deformations, as equivalent deformations up to order 1 differ from each other only in coboundary. Together with Theorem 2.5, Part (b), follows Corollary 2.6. (a) £ is infinitesimally rigid if and only i/H 2 (£,£) = {0}. (b) # 2 (£;£) = {0} implies that £ is formally rigid. Let us stress the fact, that H2(£, £) = {0} does not imply that every global deformation will be equivalent to a trivial one. Hence, £ is in this case not necessarily globally rigid. In this talk we will see plenty of nontrivial global deformations of the Witt algebra W. Hence, the Witt algebra is not globally rigid. The Lie algebras considered here are infinite dimensional. Such Lie algebras possess a topology with respect to which all algebraic operations are continuous. In this situation, in a cochain complex it is natural to distinguish the sub-complex formed by the continuous cohomology of the Lie algebra (see [1]). It is known (see [4]) that the Witt and the Virasoro algebra are formally rigid. 3. Krichever-Novikov algebras 3.1.

The algebras -with their almost-grading

Algebras of Krichever-Novikov types are generalizations of the Virasoro algebra and all its related algebras. We only recall the definitions and facts needed here. Let M be a compact Riemann surface of genus g, or in terms of algebraic geometry, a smooth projective curve over C. Let N, K e N with N > 2 and 1 < K < N be numbers. Fix

I = (Pi,...,PK),

and O = (Qi,... ,Qjv-x)

disjoint ordered tuples of distinct points ("marked points", "punctures") on the curve. In particular, we assume P» / Qj for every pair ( i , j ) . The points in I are called the in-points, the points in O the out-points. Sometimes we consider I and O simply as sets and set A = IU O as a set. Denote by £ the Lie algebra consisting of those meromorphic sections of the holomorphic tangent line bundle which are holomorphic outside of A, equipped with the Lie bracket [ . , . ] of vector field's. Its local form is /O T \ (3.1)

r

xl f \ £ f \ ~ \ [e,/]| = \[e(*)-,/(z)-] :

I

/

\

J /

\

r/

\

/

\ I

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To avoid cumbersome notation we will use the same symbol for the section and its representing function. For the Riemann sphere (g = 0) with quasi-global coordinate z, I = {0} and 0 = {oo}, the introduced vector field algebra is the Witt algebra. For infinite dimensional algebras and modules and their representation theory a graded structure is usually of importance to obtain structure results. The Witt algebra is a graded Lie algebra. In our more general context the algebras will almost never be graded. But it was observed by Krichever and Novikov in the two-point case that a weaker concept, an almost-graded structure, will be enough to develop an interesting theory. Definition 3.1. Let A be an (associative or Lie) algebra admitting a direct decomposition as vector space A = ® n gz-4n- The algebra A is called an almost-graded algebra if (1) dim.4n < oo and (2) there are constants R and S with n+m+S

(3.2)

An-Am

C

0

Ah,

Vn,meZ.

h=n+m+R

The elements of An are called homogeneous elements of degree n. For the 2-point situation for M a higher genus Riemann surface and / = {P}, 0 — {Q} with P, Q 6 M, Krichever and Novikov [10] introduced an almost-graded structure of the vector field algebras L by exhibiting a special basis and defining their elements to be the homogeneous elements. 3.2. The family of elliptic curves

We consider the genus one case, i.e., the case of one-dimensional complex tori or equivalently the elliptic curve case. We have degenerations in mind. Hence it is more convenient to use the purely algebraic picture. Recall that the elliptic curves can be given in the projective plane by (3.3)

Y2Z = 4X3-g2XZ2-g3Z3,

52,
with A :=
The condition A ^ 0 assures that the curve will be nonsingular. Instead of (3.3) we can use the description Y2Z =

(3.4)

4(X-eiZ)(X-e2Z)(X-e3Z)

with (3.5)

ei + 6 2 + 63 = 0, and A = 16(ei - e 2 ) 2 (ei - e 3 ) 2 (e 2 - e3)V 0.

These presentations are related via (3.6)

g2 = -4(e1e2 + e1e3 + e 2 e3),

g3 = 4(eie 2 e 3 ).

We set

(3.7)

5:={(ei,e2,e 3 ) 6 C3 \el + e2 + e3=0, 2

et ^ e0 for t ^ j}.

In the product B x P we consider the 'family of elliptic curves £ over B defined via (3.4). The family can be extended to (3.8)

5 := {ei.ea.ea) 6 C3

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The fibers above B \ B are singular cubic curves. Resolving the one linear relation in B via 63 = —(e\ + 62) we obtain a family over C 2 . Consider the complex lines in C2 (3.9)

Ds:={(e1,e2)eC2\e2 = s-e1},

s e C,

Dx := {(0,e 2 ) e C2}.

Set also (3.10) for the punctured line. Now (3.11)

B^C

Note that above D^ we have e\ = 62 ^ 63, above D^,2 we have 62 = 63 ^ ei, and above D12 we have e\ = 63 ^ 62- In all these cases we obtain the nodal cubic. The nodal cubic EN can be given as (3.12)

Y2Z =

where e denotes the value of the coinciding &i = &j (— 2e is then necessarily the remaining one). The singular point is the point (e : 0 : 1). It is a node. It is up to isomorphy the only singular cubic which is stable in the sense of Mumford-Deligne. Above the unique common intersection point (0, 0) of all Ds there is the cuspidal cubic EC (3.13)

Y2Z = 4X3.

The singular point is (0 : 0 : 1). The curve is not stable in the sense of MumfordDeligne. In both cases the complex projective line is the desingularisation, In all cases (non-singular or singular) the point oo = (0 : 1 : 0) lies on the curves. It is the only intersection with the line at infinity, and is a non-singular point. In passing to an affine chart in the following we will loose nothing. For the curves above the points in D* we calculate 62 = sei and 63 = — (l + s)ei (resp. 63 = —62 if s = oo). Due to the homogeneity, the modular parameter j for the curves above D* will be constant along the line. In particular, the curves in the family lying above D* will be isomorphic. 3.3. The family of vector field algebras We have to introduce the points where poles are allowed. For our purpose it is enough to consider two marked points. One marking we will always put to oo = ( 0 : 1 : 0 ) and the other one to the point with the affine coordinate (ei,0). This marking defines two section of the family £ over B = C2. With respect to the group structure on the elliptic curves given by oo as the neutral element (the first marking) the second marking chooses a two-torsion point. All other choices of two-torsion points will yield isomorphic situations. For this situation (and for a three-point situation) a basis of the KricheverNovikov type vector field algebras was given in [12].

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Theorem 3.2. For any elliptic curve £( ei , e2 ) over (ei, e 2 ) € C2\(.D*U.D11/2UD:12) the Lie algebra £( e i > e 2 ) of vector fields on E^^ has a basis {Vn, n G Z} suc/i that the Lie algebra structure is given as (3.14) (m-n}Vn+m, n,m odd, (m - n)(Vn+m + Se^n+m-2 , m even, [Vn,Vm} = - e 2 )(ei - e3) Vn+m-A , (m - n)Vn+m + (m-nn odd, m even. +(m — n - 2)(ei — By defining deg(Vn) := n, we obtain an almost-grading. The algebras of Theorem 3.2 defined with the structure (3.14) make sense also for the points (e\, e 2 ) 6 Di\jD_i/%\jD-2- Altogether this defines a two-dimensional family of Lie algebras parameterized over C2. In particular, note that we obtain for (ei, 62) = 0 the Witt algebra. We consider now the family of algebras obtained by taking as base variety the line Ds (for any s). First consider s ^ oo. We calculate (e\ — e 2 )(ei — 63) = ef (1 — s)(2 + s) and can rewrite for these curves (3.14) as

n,m odd,

(m - n)Vn+m, (m - n) (Vn+m (3.15) [Vn,Vm] =

n,meven, (m - n)Vn+m + (m-n- l)3eiV n+m _ 2 +(m - n - 2)e 2 (l - s)(2 + s)V r n+m _ 4 ,

n odd, m even.

For .Doo we have 63 = — e2 and e\ = 0 and obtain (m - n)Vn+m, (m - n) (K +m - e^K l+m _ 4 ) , (m - n)Vn+m - (m - n - 2)e2V n+m _ 4 ,

n, m odd, n, m even, n odd, m even.

If we take V£ — (^/^i)~nVn (for s 7^ oo) as generators we obtain for e\ ^ 0 always the algebra with ei = 1 in our structure equations. For s = oo a rescaling with (^/^)~nVn will do the same (for e2 ^ 0). Hence we see that in all cases the algebras will be isomorphic above every point in Ds as long as we are not above (0,0). Proposition 3.3. For (ei,e 2 ) the Witt algebra.

(0,0) the. algebras

are not isomorphic to

Proof. Assume that we have an Lie isomorphism $ : W = £(°>°) —> £( e i'«2). Denote the generators of the Witt algebra by {/„', n 6 Z}. In particular, we have [/Q, ln] = nln for every n. We assign to every ln numbers m(n) < M(n) such that $(in) = SfcJmin) a fc( n ) v fc

witil Q

m(n)( n ))Q ; M(n)(") 7^ 0. From the relation in the Witt

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algebra we obtain M(0)

M(n)

M(n)

i=m(n)

l=m(n)

We can choose n in such a way that the structure constants in the expression of [Vk, Vi] at the boundary terms will not vanish. Using the almost-graded structure we obtain M(0)+M(n) = M(n) which implies M (0) = 0, and m(0) + m(n)-4 = m(n) or m(0)+m(n)-2 = m(n) (for s = 1 or s - -2) which implies 2 < m(0) < M(0) = 0 which is a contradiction. D It is necessary to stress the fact, that in our approach the elements of the algebras are only finite linear combinations of the basis elements Vn . In particular, we obtain a family of algebras over the base Ds, which is always the affine line. In this family the algebra over the point (0, 0) is the Witt algebra and the isomorphy type above all other points will be the same but different from the special element, the Witt algebra. This is a phenomenon also appearing in algebraic geometry. There it is related to non-stable singular curves (which is for genus one only the cuspidal cubic). Note that it is necessary to consider the twodimensional family introduced above to "see the full behaviour" of the cuspidal cubic ECLet us collect the facts. Theorem 3.4. For every s & C U {00} the families of Lie algebras defined via the structure equations (3.15) for s ^ oo and (3.16) for s — oo define global deformations Wt of the Witt algebra W over the affine line
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[6] Fialowski, A., and Schlichenmayer, M.: Global deformations of the Witt algebra of KricheverNovikov type. Comm. Contemp. Math, (to appear). [7] Fuchs, D.: Cohomology of infinite-dimensional Lie algebras, Consultants Bureau, N.Y., London, 1986. [8] Gerstenhaber, M: On the deformation of rings and algebras I, II, III Ann. Math. 79, 59-10 (1964), 84, 1-19 (1966), 88, 1-34 (1968). [9] Kodaira, K.: Complex manifolds and deformation of complex structures, Springer, New-York, Berlin, Heidelberg, Tokyo, 1986. [10] Krichever I. M., and Novikov S. P.: Algebras of Virasoro type, Riemann surfaces and structures of the theory of solitons, Funktional Anal. i. Prilozhen. 21, 46-63 (1987); Virasoro type algebras, Riemann surfaces and strings in Minkowski space. Funktional Anal. i. Prilozhen. 21, 47-61 (1987); Algebras of Virasoro type, energy-momentum tensors and decompositions of operators on Riemann surfaces, Funktional Anal. i. Prilozhen. 23, 46—63 (1989). [11] Palamodov, V. P.: Deformations of complex structures. In: Gindikin, S. G., Khenkin, G. M. (eds.) Several complex variables IV. Encyclopaedia of Math. Sciences, Vol. 10, pp. 105-194, Springer, New-York, Berlin, Heidelberg, Tokyo, 1990. [12] Schlichenmaier, M.: Degenerations of generalized Krichever-Novikov algebras on tori, Jour. Math. Phys. 34, 3809-3824 (1993).

Maximal Prime Divisors in Arithmetical Rings Laszlo Puchs Department of Mathematics, Tulane University, New Orleans, Louisiana 70118 fuchsOtulane.edu

William Heinzer Department of Mathematics, Purdue University, West Lafayette, Indiana 47907 heinzerSmath.purdue.edu

Bruce Olberding Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003-8001 olberdinSenrary. nmsu. edu Abstract. We investigate for an ideal A of an arithmetical ring R the relationship between the set Max(A) of maximal prime divisors of A and the set XA of maximal members of the set of Krull associated primes of A. We show that the arithmetical rings R such that XA C Max(A) for every regular ideal A are precisely those satisfying the "strong" separation property. For a Priifer domain R, we prove that every branched prime ideal of* height greater than one is the radical of a finitely generated ideal if and only if End(^4)M = End(AM) for every nonzero ideal A and maximal ideal M of R. We use this to prove that if in addition R is a QR-domain, then every maximal prime divisor of an ideal A of R is a Krull associated prime of A (i.e., XA = Max(A)) if and only if each branched prime ideal of R of height greater than one is the radical of a finitely generated ideal.

Introduction

Let R be a commutative ring with identity. Associated to a proper ideal A of R is the set S(A) = {x e R : xy 6 A for some y e R \ A} of elements of R that are non-prime to A. The complement R \ S(A) of elements prime to A is closed under multiplication. Krull proves [14, page 732] that ideals maximal with respect to 1991 Mathematics Subject Classification. Primary 13A15; Secondary 13F05. Key words and phrases. Primal ideal, associated prime, arithmetical ring, Priifer domain.

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not meeting a multiplicatively closed set are prime ideals and defines the maximal prime divisors of A as the prime ideals P of R, that contain A and are maximal with respect to the property P C S(A). Let Max(.A) denote the set of maximal prime divisors of A. For P <E Max(A), Krull defines the ideal A(p) = {x e R : xy £ A for some y e -R \ P} to be the principal component of A with respect to P and establishes the decomposition of every proper ideal A of R as the intersection of its principal components A — DpeMaxM) ^-(.P) [14, Satz 2]. However, as we discuss in [4], a drawback to this decomposition is that we do not know what kind of ideals the ^.(p) are. For instance, there sometimes exist elements of P that are prime to the principal component A(p); indeed, the ideal A( P ) is not in general a primal ideal, where an ideal B is said to be primal if the set S(B) of elements non-prime to B is an ideal. If B is primal, then S(B) is a prime ideal called the adjoint prime of B. To obtain in a commutative ring without finiteness conditions a decomposition of the ideal A that involves components closely tied to A with structure we know, in [4] we define the set XA of maximal Krull associated primes of A and obtain using the set XA a canonical primal decomposition of A. If R is a Noetherian ring and Ass(A) = {Pi, P2, • • • , Pn} is the set of associated primes of a proper ideal A of R, then ff(A) = (JILi Pi- ^ follows that the maximal prime divisors of A are exactly the prime ideals of R that are maximal members of Ass(.A). In this sense, the maximal prime divisors of an ideal of a Noetherian ring are well-understood. However if R is not Noetherian, the set Max(A) of maximal prime divisors of A is generally less transparent, and the primes in Max(A) need not be "associated." There are several inequivalent notions of an associated prime of an ideal of a general commutative ring, but from our point of view, it is the Krull associated primes that are most natural. We review this and related notions in Section 1. Motivated by the Noetherian case, we are thus interested in the question of when the set of maximal prime divisors of an ideal A coincides with the set of maximal members of the set of Krull associated primes of A. We shall examine this question for the class of arithmetical rings, those rings R such that for every maximal ideal M of R, RM is a valuation ring, that is, a ring for which the set of ideals is linearly ordered by inclusion. Of particular interest is the class of Priifer domains, namely the arithmetical integral domains. Our purpose in the present paper is to investigate for ideals A of an arithmetical ring or Priifer domain the interrelationship between the sets Max(A) and XA- The connection between these two sets of prime ideals is of special interest in the context of arithmetical rings. Indeed, if A is a proper ideal of an arithmetical ring R, then Max(A) = XA if and only if every prime ideal containing A but not prime to A is a Krull associated prime (Corollary 1.3). Also, Krull mentions [15, p. 16] that he does not know whether the principal components ^4(p) of A, P £ Max(A), are always P-primal ideals. (A primal ideal B is Q-primal if Q = S(B).) Examples show that the answer is in the negative' (see [17],[5] and [4, Example 3.8]); we investigate in Section 3 arithmetical rings for which the answer to this question is in the affirmative for all regular ideals. (An ideal A of R is regular if it contains a

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regular element, i.e., an element that is not a zero-divisor.) This is also of special interest, because an ideal of an arithmetical ring is primal exactly if it is irreducible (Theorem 1.8 of [4]). In Section 3 we examine conditions on an arithmetical ring R in order that each maximal prime divisor of a regular ideal A of R be a Krull associated prime of A. It follows from Theorem 2.3 that a necessary condition for this to hold is that R satisfy the strong separation property: for regular prime ideals P C Q and regular element r 6 P, there exists s € Q such that P C (r, s)R C Q. We conclude that Priifer domains such as the ring of entire functions and the ring Int(Z) of integer-valued polynomials contain ideals A having the property that there exists a maximal prime divisor of A that is not a Krull associated prime of A. For a regular ideal A of an arithmetical ring, we consider in Theorem 3.7 conditions in order that the set of maximal prime divisors of A is precisely the set of prime ideals that are maximal among the Krull associated primes of A. We deduce a complete characterization of the QR-domains with this property. (By contrast, ideals of Noetherian rings always exhibit equality between these two sets of prime ideals.) The results that lay the groundwork for this theorem touch on several interesting technical aspects of arithmetical rings and Priifer domains. In particular, there is a connection between our problem and that of when Eud(X) localizes, that is, when Eud(X)M = End(XM) for a submodule X of the quotient field of R and maximal ideals M of R. Acknowledgement

We thank Gabriel Picavet and David Rush for helpful conversations on the topics of this paper and [4], and for showing us the connections of our work to the literature on Krull associated primes. 1. Krull associated primes In this section we briefly review the notion of a Krull associated prime of a proper ideal of a ring. We became interested in the Krull associated primes of an ideal because of their connection to the primal isolated components of the ideal. For a more complete treatment of these primes and their relation to issues involving the primal decompositions of an ideal, see [4]. Let A be a proper ideal of the ring R. Following [12], we define a prime ideal P of R to be a Krull associated prime of the ideal A if for every element x & P, there exists y <£ R. such that x £ (A : y] C P. A prime ideal P of R Is called a weak-Bourbaki associated prime of A if it is a minimal prime divisor of (A : x) for some x G R \ A. Following [11, page 279], we call P a Zariski-Samuel associated prime of A if P = ^/(A : x) for some x € R \ A. If -R is a Noetherian ring, then all three of these notions of an associated prime coincide, but for non-Noetherian rings these notions are in general distinct'(see [11], for example). It is clear that a Zariski-Samuel associated prime is a weak-Bourbaki associated prime. Moreover, it is noted in Lemma 2.1 of [4] that P is a Krull associated prime

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of A if and only if P is a set-theoretic union of weak-Bourbaki primes of A. It is this characterization of Krull associated primes that we shall use in what follows. We denote by Ass(A) the set of Krull associated primes of A. Notice that if A is a proper ideal of R, then Ass(A) is nonempty. By contrast, Nakano [19] gives an example of a Priifer domain such that no finitely generated ideal has a Zariski-Samuel associated prime. The set Ass(A) behaves well with respect to localization: Lemma 1.1 (Lemma 2.4(ii) of [4]). Let A be an ideal of a ring R and P be a prime ideal of R containing A. Then P € Ass(A) if and only if PM £ ASS(AM) for some (or equivalently every) maximal ideal M of R containing P. It is not hard to see that if A is a proper ideal of a ring R, then every member of Ass(A) is contained in a maximal member of Ass(A). We define XA to be the maximal members of Ass(A), that is, XA consists of the maximal Krull associated primes of A. There exist examples of rings with ideals A such that XA is infinite. (See Example 2.6 of [4] for example, or use Lemma 1.5 below.) These examples are necessarily non-Noetherian. In Lemma 2.3 of [4], we observe that S(A) — UpeA1 P- •"•n particular, we have UPGA-A P = UgeMax(A) Q- Despite this-close connection between XA and Max(.A), one cannot conclude in general that XA C Max(A) or Max(A) C XA- If Max(A) has only one member, then it is easy to see Max(A) = XA- However, in Example 3.8 of [4] we construct a ring R with Noetherian prime spectrum and an ideal A of R such that Max(A) has only two elements, but neither maximal prime divisor of A is in XA- We also construct in Example 2.9 of [4] a 2-dimensional Priifer domain R for which Spec(R) is Noetherian and for which there exist an ideal A and a finitely generated maximal ideal M such that M is a maximal prime divisor of A and yet M <£ Ass(A). When R is arithmetical, the set Ass(A) has some striking properties. In the next sections we will need the following lemmas. Lemma 1.2 (Proposition 2.7 of [4]). Let R be an arithmetical ring and let A be a proper ideal of R. If P & Ass(-A), then Q G Ass(A) for every prime ideal Q of R such that ACQCP. Corollary 1.3. Let A be a proper ideal of an arithmetical ring R. Then XA = Max(.A) if and only if every prime ideal of R that contains A and is not prime to A is a Krull associated prime of A. Proof. Assume that XA = Max.(A). If P contains A and is not prime to A, then P is contained in some Q G XA. Hence by Lemma 1.2, Q G Ass(A). The converse is clear. D Let A be a nonzero proper ideal of a Priifer domain R and let Q be a prime ideal of -R with A C Q. Since RQ is a valuation domain, End(A-Rg) = Rp for some P e SpecE with P e g . Define £A = {P £ SpecR : Rp = End(ARg) for some prime ideal Q containing A}.

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Lemma 1.4 (Proposition 2.8 of [4]). Let A be a nonzero proper ideal of a Prufer domain R. Then Ass(A) = £A- In particular, if P e XA, then there exists a maximal ideal M of R such that End(yW) — RpLemma 1.5. Let A be an ideal of an arithmetical ring R. If AM is a nonzero finitely generated ideal of RM for each maximal ideal M of R, then Ass(A) is precisely the set of prime ideals of R containing A. Proof. First assume that R is a valuation ring and A is a nonzero finitely generated proper ideal of R. Then A = yR is a principal ideal of R. Let P be the maximal ideal of R. If A = P, then clearly P £ Ass (A), so suppose there exists x e P \ A. Since R is a valuation ring, A C Rx and A = (A : x)xR. If (A : x) = A, then A = xA and y = xya for some a £ A. But this means y(l — xa) = 0. Since 1 — xa is a unit of R, this implies y = 0, a contradiction to our assumption that A f 0. Therefore each x G P is non-prime to A, so A is a primal ideal with S(A) = P and P e Ass(A). In the general case where R is an arithmetical ring, we conclude from Lemma 2.1 and the preceding argument that every maximal ideal P of R that contains A is in Ass(A). Hence by Lemma 1.2, every prime ideal of R containing A is in Ass(A). D

2. The case XA C Max(A) In this section we characterize when XA £ Max(A) for every regular ideal A of an arithmetical ring. To do this, we first recall the notion of separated prime ideals, but we reformulate this definition to include rings with zero-divisors in such a way that our definition agrees with that of the separation property for domains (see pp. 91-92 of [3]). We define a ring R to have the separation property if for each pair of distinct comparable regular prime ideals P C Q of R, there exists a finitely generated ideal A such that P C A C Q. The ring R has the strong separation property if for each pair of comparable regular prime ideals P C Q of R and regular element r e P, there exists s & Q such that P C (r, s)R C Q. Clearly every one-dimensional domain has the strong separation property. By the Altitude Theorem of Krull [18, page 26] or [13, page 110], a Noetherian domain of dimension greater than two has the separation property but not the strong separation property. By contrast, Prufer domains need not possess either separation property. The Prufer domains that have the separation property have been well-studied (see, for example, pp. 91-92 of [3] and Lemma 2.7 of [20]). Note that a Bezout domain R has the separation property if and only if R has the strong separation property. More generally, a Prufer domain with the QR-property (so in particular a Prufer domain with torsion Picard group) has the separation property if and only if it has the strong separation property. In the following lemma, we collect some technical characterizations of the separation property for Prufer domains that will be needed in the next section. Lemma 2.1. The following statements are equivalent for a Prufer domain R. (i) R has the separation property. (ii) For each nonzero prime ideal P of R, P is a maximal prime ideal o/End(P).

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(iii) For each nonzero prime ideal P of R, End(P_M) = End(P)M for every maximal ideal M of R. (iv) For each nonzero nonmaximal prime ideal P of R, if {Mi} is the collection of maximal ideals of R not containing P, then f\ -^Mi 2 -Rp(v) For each nonzero ideal A of R, if M is a maximal ideal with ACM and P is a prime ideal such that End(A)M = Rp, then no element of P is prime to A. Proof, (i) & (ii) Lemma 4.2.38 of 3]. (i) <=> (iii) Lemma 2.7 of [20]. (iii) => (iv) Apply Theorem 3.2.6 of [3] and Lemma 2.7 of [20]. (iv) =3- (v) Let A be a nonzero ideal of R and M a maximal ideal with A C M. Let P be a prime ideal such that End(A)M = Rp- It follows from Lemma 1.4 that N

where N ranges over the maximal ideals of R that do not contain A. Let Q' be the unique member of XA that is contained in M. By Lemma 1.4, End(j4jvf) = RQ< • If Q1 is a maximal ideal of R, then Q' = M and End (.AM) = RM, so Rp C RM implies that M = P = Q' 6 Ass(A), aod the claim is clear. It remains to consider the case where Q' is a non-maximal prime ideal of R. Now RP = End(A) M = ( P| RQ)RM N

Since Rp is a valuation domain, it must be equal to at least one of the two components of this intersection. However, by (iv), |~|w RN <2 RQ> since no N contains Q'. Thus f\NRN £ (C\Q£XA RQ}RM, since Q' 6 XA and RQ>RM = RQ-. Hence it must be that Rp '= (C\Q^xA ^-Q)^-M- In particular, [~]Q^XA ^Q — ^P> s° every element r € P is contained in some Q £ XA. Consequently, no element of P is prime to A. (v) => (iii) Suppose P is a non-maximal prime ideal of PL, and let M be a maximal ideal of R containing P. Let Q be a prime ideal of R such that End(P)^ = RQ. (Since End(P)M is a valuation domain such a prime Q must exist). Then by (v), the elements of Q are not prime to P. Consequently, P = Q, and (iii) follows. D Using the strong separation property, we can characterize when XA C Max(.A) for every regular ideal A of an arithmetical ring. In fact, the strong separation property is always sufficient, regardless of whether R is arithmetical, to guarantee that XA C Max(A): Lemma 2.2. If A is a regular ideal of a ring R having the strong separation property, then every P £ XA is a maximal prime divisor of A, i.e., XA C Max(A). Proof. Let P e XA. There exists Q £ Max(A) such that P C Q. We show P = Q. If XA consists only of P, then A is P-p'rimal and P = Q. Suppose XA \ {P} is nonempty, and let {PJ = XA \ {P}. Since Q € Max(A), Q C P u (\Jl Pi). Assume P is properly contained in Q. Since R has the strong separation property and A is

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a regular ideal, there exists x £ Q \ P such that P C A + (x) C Q. It follows that x is in one of the Pi, so P C A + (x) C P,. This contradicts the assumption that P £ XA is maximal among the Krull associated primes of A. Thus P = Q. D Theorem 2.3. Lei PL be an arithmetical ring. The following statements are equivalent. (i) R has the strong separation property. (ii) For each proper regular ideal B of R, XB C Max(£?). Proof. By Lemma 2.2, (i) implies (ii). To prove that (ii) implies (i), suppose P C Q are regular prime ideals of R and that a is a regular element with a & P. Define B = aP and note that the maximal ideals of R that contain B are precisely those that contain aR. We first show that XB is the set union of {P} and the set of maximal ideals of R that contain B but not P. Since a is a regular element, we have P = B : a, so P s Ass(-B). We claim that P £ XB- Suppose L £ XB and P C I / . Let M be a maximal ideal containing L. By Lemma 1.7 of [4], (BRM)(PR*,) is a P^-primal ideal of RM since BRu is a finitely generated ideal of RM- We claim BR^ — (BRM)(PRM)- To verify that this is the case, we need only show that BRj^ : y — BR^ for all y & RM \ PM- To this end, observe first that PRM — yPRM for all y £ RM \ PM • This is because RM is a chained ring, so if y £ RM \ PM, then PPiM C yRjw, and if p £ PRM, then p = yu for some u € PM- Since y ^ PRM, it follows that u £ PP^M and P £ 2/Pw- Thus PPtM = yPRM- Hence, with y £ RM \ PRM, we have that BRM '• y — yBRM '• y — BRM, since y 0 PRM implies that PPiM C yPtM and hence that y is a regular element of RM- Therefore BRM = (BRM)pRM, and we conclude that BRM is a PP^-primal ideal. However, by Lemma 1.1 LRu £ A.SS(BRM), so this forces LRM C PRM- Hence L C P, and we may conclude that P = L & XBNow suppose TV is a maximal ideal of PL that contains B but not P. Then BN = aRw ^ 0, so N is in XB by Lemma 1.5 and Lemma 1.1. This proves that the set XB contains P and the maximal ideals of PL that contain B but not P. The reverse inclusion follows from the fact that any prime ideal of PL that is a member of XB must contain B. Let {Ni} be the set of members of XB distinct from P. As noted, each TV; is a maximal ideal of PL, and {Ni} is precisely the set of maximal ideals of R that contain aR but not P. Since P 6 XB and, by (ii), XB C Max(P>), we have Q 2 S(B) — P U (Uj Ni), for otherwise P, as a proper subset of Q, would not be in Max(^4). In particular, there exists x e Q such that x is not contained in P nor in any of the N^. We have (a, x)R, C Q, and to complete the proof we show that P C (a,x)R. It is enough by Theorem II.3.1 of [1, page 88] to show that PM C (A,x)RM for all maximal ideals M of P that contain (A,x). Observe that if M is such a maximal ideal, then since by design x is not an element of any maximal ideal of R that contains Ra but not P, it must be that M contains P. Now since PLM is valuation ring, the ideals PM and (a,x)RM are comparable, so if PM 2 (O-,X)RM, then it must be that (CL,X)RM C PM. But then x £ PM and since P C M, it is the case that P(M) = P, and we have x £ P, a contradiction

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to the choice of x. Thus we conclude that P C (a, x)R C Q, and R has the strong separation property. D It follows from Theorem 2.3 that if R is an arithmetical ring that does not have the strong separation property, then there exist a proper regular ideal B of R and P 6 XB such that P g Max(5). By definition of Max(B), this means there exists Q E Max(B) such that P is properly contained in Q and by definition of XB, this means Q (I Ass(B). Therefore the strong separation property is a necessary condition on an arithmetical ring in order that maximal prime divisors (of regular ideals) always be Krull primes. Discussion 2.4. Some important examples of Priifer domains do not have the separation property, and so by Theorem 2.3, these Priifer domains have ideals B such that XB <£ Max(B). Thus there exist maximal prime divisors of B that are not in Ass(B). For example, neither the ring of entire functions nor the ring of integer-valued polynomials has the separation property. Indeed, both these rings are completely integrally closed integral domains of Krull dimension greater than one. If P is a nonzero nonmaximal prime ideal of such a Priifer domain R, then End(P) = R and P is not maximal in End(P). Hence by Lemma 2.1 (ii), such domains do not have the separation property. On the other hand, every Priifer domain having Noetherian maximal spectrum has the separation property (apply Theorem 4.2.39 of [3] and [22]).

3. The case XA = Max(,4) Our main objective in this section is to investigate conditions on an arithmetical ring PC in order that'A^ = Max(j4) for each regular ideal A of PL. We recall that an integral domain P is a QR-domain if every overring of PC is a localization of PL with respect to some multiplicatively closed subset of PL. It is well-known that QR-domains are necessarily Priifer, and that a Priifer domain with torsion Picard group (e.g., a Bezout domain) is a QR-domain. More generally, a Priifer domain PL is a QR-domain if and only if the radical of every finitely generated ideal of PL is the radical of a principal ideal of PL [21]. There exist QR-domains having nontorsion Picard group [9] . Thus the condition on a Priifer domain PL that the radical of every finitely generated ideal is the radical of a principal ideal does not imply PL has torsion Picard group. However, it is clear that a QR-dornain that has the separation property also has the strong separation property. Lemma 3.1. Let A be an ideal of a Priifer domain R.. Suppose M is a maximal ideal of R that contains A, and P is a prime ideal such that End(A)nf = Rp. If P e Ass(A), then End(A) M = Proof. Since P € Ass(A), -A(p) is a primal ideal with adjoint prime P, and it follows that Ap is a Pp-primal ideal. 'By Lemma 1.4, End(Ap) = Rp. Thus End(-Ap) = End(A)M, so AEnd(Ap) = AEnd(A)M implies AP = AM. Consequently, End(A M ) = End(Ap) = RP = End(A)M. D

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Lemma 3.2.. Let R be a Priifer domain with field of fractions F, let X be an Rsubmodule of F, and let M be a maximal ideal of R. Then End(X)M — Rp for some P e Spec-R with PCM. If P is the union of prime ideals Pi, where each Pi is the radical of a finitely generated ideal, then End(X)M — Proof. Since RM C End(X)M and RM is a valuation domain, End(X)M = -Rp for some prime ideal P C M. We have End(X}M C Eud(XM). If End(X) M = F, then clearly End(X)M = End(XM), so we assume Eud(X)M =£ F and thus P ^ (0). Let S = End(X). Then 5" is a Priifer overring of R and PS is a prime ideal of S. Moreover PS is the union of the prime ideals PiS of S and each PiS is the radical of a finitely generated ideal of 5". Thus our assumptions on R are inherited by S and we may assume that End(X) = R. We claim that Xp ^ F. By assumption there exists a nonzero Q G Spec/? with Q C P such that Q is the radical of a finitely generated ideal / of R. Since Xp C XQ, it suffices to show XQ ^ F. Since / is an invertible ideal of R and Eiid(X) = R, it is easy to see that I is the intersection of finitely many ideals of R of the form X : q, where q £ 7"1. Thus at least one of the ideals, say X : q, q € F, is contained in Q. Consequently, q £ XQ, and it follows that XQ ^ F. Since Rp is a valuation domain, there exists a nonzero element r of R such that rXp C RP. Without loss of generality, we may assume r = 1. Define A — tXpHR, where t ^ 0 is contained in some Pi. Observe that A is an ideal of R contained in Pj. Also P is a union of the prime ideals Pj of R that contain A, and each P; is the radical of a finitely generated ideal of R. To show P e Ass(A), it suffices to show each Pi with A C Pi C P is in Ass(J4). As above, Pi is the radical of an ideal of R of the form tX : q for some q & F. Thus (Pi)pi is the radical of Api : q, where q € Rpi and this residual is taken over the ring Ptp^ so by Lemma 1.1 Pj € As's(A). Therefore P e Ass(A), so by Lemma 3.1, End(AM) = End(A)^- Since AM is isomorphic to XM, End(^M) = End(Ajw) = -Rp- Therefore Endpf M ) = A prime ideal P is branched if there exists a P-primary ideal different from P. If P is a nonzero prime ideal of a Priifer domain PL, then P is branched if and only if P is not the union of the prime ideals properly contained in P, and in this case if P fails to be branched, then the valuation domain Rp is infinite dimensional and there is no maximal element among the prime ideals properly contained in P. Lemma 3.3. Let R be a Priifer domain having the separation property. If there exists a finitely generated ideal J of R having infinitely many minimal primes, then there exists a submodule X of the field of fractions F of R and a maximal ideal M containing J such that End(X)M Proof. Let PI, PS, PS, ... be countably many distinct minimal primes of J. Define S — Qi Rpi . Then S is a Priifer overring of R and the minimal primes of JS are of the form QS, where Q e Spec R is a minimal prime of J. In particular, PiS is a minimal prime of JS for each i > 1. Since R has the separation property, each Pj5 is a maximal ideal of S. This is because, if A'" is a maximal ideal of S containing PjS for some j, then since Pi is a Priifer domain, N must be of the form PS for some

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prime ideal P of R containing Pj. If P ^ Pj, then it follows that (~\iRpi C -Rp, and since Rp is a valuation domain and Rpi <2 -Rp, we have r\i^tjRpi C Rp. However, since the set {Pi} consists of comaximal prime ideals of R not containing P, this contradicts the assumption that R has the separation property (Lemma 2.1 (iv)). Since J is a nonzero finitely generated ideal, each minimal prime of J is branched. Since JS has infinitely many minimal primes, Theorem 1.6 of [7] implies there exists a minimal prime QS of JS that is not the radical of a finitely generated ideal. If Q = Pj for some j, then by [6, Theorem 2], we have S = C\ijtj RP*Therefore, by relabeling if necessary, we may assume that Q <£ {-Pi}iSiDefine A — JRQ n 5. Then A is QS-primary. In particular, QS is the unique minimal prime of A and A 2 PiS for each i > 1. Let Qi be the prime ideal of R just below Q (such a prime Q\ exists because Q is branched). Since R has the separation property, there is a finitely generated ideal B of R such that Qi C B C Q. Local verification shows that Qi C Bn C Q for each positive integer n. For each i > 1, let i 1 T. c. Ai \\ (P. ~i ~— •**^ i I^I . . . I^I •*P.i II ^ -1 A -* ~ \y -

(By Prime Avoidance (see for example.[2, Lemma 3.3]) such an element z$ exists.) Define J, = Bi+l + XiR. Then Q1 C J{ and J, g (Pi U • • • U Pi). Define X = £»>! J~1S. Let N = QS and for each i > 0, define Ni — PiS. As previously demonstrated, N and the ideals ATj, i > 0, are maximal ideals of 5. We show first that End(X;v) ^ EndpiTV For each i > 1, , i-l

^W; — / , Jk fc=l

$Ni,

since by design, J^S^k — Sf^k for all k > i. So as a finitely generated fractional ideal of S^, X^i is isomorphic to S^. Thus End(J^ArJ = S^ for each i. We have that for each i, S^ = -Rpt; hence End(X) C p|End(^JVJ = S. i

Therefore, End(^) = S. In particular, End(X)N = SN. Thus to prove the claim that End(X)jv 7^ End(-X'jv), it suffices to show that NEiid(XN) = End(^Ar). To this end we verify that XN = NXN. Since N = SQ, we have S^v = RQ, so for each i > 1, Al+lSx C A1S^- In particular, Zt+iSjv C ZjS'jv since z, e A1 \ Al+1 and z,+i ^ A*. It follow that (Jl+i)S^ C Since Ji is a finitely generated ideal of S1, it follows that (Ji+i)SV Q Consequently, J"1 C N(J~+\)SN for each i > 1. Therefore X C A^^AT, as claimed. For 5-submodules y and W of P,'Jet [VF : Y] = {q e F : qY C V7}. Since = NXfj and End(A^jv) = SN, we have [SN : XN] = [A^N : NXN]

= [A^w : XN].

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Moreover, [Sjv : XN] = fli Ji$N and QiSN C p| JiSN C p| A'SW = QiSNi

i

The latter equality follows from the fact that since SN is a valuation domain and ASx is a principal ideal of SN, the intersection f^ A1S^ is a prime ideal. Since Qi is the largest prime ideal of R properly contained in Q, A C Q and A £ Qi> we have f|i 4'Sjv =QiSN. It follows that [NN : XN} = [SW : -X"w] = (SQi)N- Since PLJV is a valuation domain, the maximal ideal iVjv of SN is m-canonical (see [10]), that is, XN = [NN : [NN : XN}} = (NN : (SQM

Since SQi is a nonmaximal prime ideal of S, [JVjv : (SQi)pf] = SSQ, [3, Theorem 4.1.21]. Thus XN — SSQI, and we have End(JCjv) = SSQ^ = RQ^- In particular, End(-Xjv) ^ RQ. Now XN — XSQ = XQ, and if M is a maximal ideal of R containing Q, then SSQ = SM, so XM = XQ. Thus while EndpOM = SM = RQ. Therefore End(X M ) ^ End(X) M -

d

Theorem 3.4. Let R be a Prufer domain with field of fractions F. The following statements are equivalent. (i) For each finitely generated ideal A of R, every weak-Bourbaki associated prime of A is a Zariski- Samuel associated prime of A. (ii) Every finitely generated ideal of R has only finitely many minimal prime ideals. (iii) Every principal ideal of R has only finitely many minimal prime ideals. (iv) Every branched prime ideal of R is the radical of a finitely generated ideal. (v) For each R-submodule X of F, End(XM) = End(X}M, for every maximal ideal M of R. Proof, (i) => (ii) Let A be a finitely generated ideal of R. Every minimal prime P of A is a weak-Bourbaki prime of A and therefore by (i) a Zariski-Samuel prime of A and hence of the form \/(A : x). Since R is Prufer, (A : x) is finitely generated, see, for example, [6, Lemma 2]. Thus each minimal prime of R/A is the radical of a finitely generated ideal of R/A. By Theorem 1.6 in [7], R/A has finitely many minimal primes. This proves (ii). (ii) => (iii) is obvious. (iii) =>• (iv) Let P be a branched prime of R. Then there exists a prime ideal Q of R such Q C P and there are no other prime ideals between Q and P. Let x £ P \ Q. Since the prime ideals contained in P are linearly ordered, P is a minimal prime ideal of xR. By (iii), xR has only finitely many minimal prime ideals. Since R is Prufer, if B is the intersection of the other minimal primes of xR, then B + P = R. Hence there exists y £ P such that B + yR = R. It follows that P is the radical of (x,y)R. This proves (iv).

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(iv) => (v) Since every nonzero prime ideal of R is the union of branched prime ideals, this follows from Lemma 3.2. (v) => (i) Let A be a finitely generated ideal of R and let P be a weak-Bourbaki associated prime of A. Then P is a minimal prime of (A : x) for some x £ R. Since R is Priifer, (A : x) is finitely generated. Lemma 3.3 implies that (A : x) has only finitely many minimal primes. Thus by Lemma 5.10 of [4] each minimal prime of (A : x) is a Zariski- Samuel associated prime of (A : x), and therefore a ZariskiSamuel associated prime of A. This proves (i). D There is an interesting connection between the conditions of Theorem 3.4 and "trace properties" of Priifer domains. T. Lucas shows in [16, Theorem 23] that a Priifer domain R satisfies (iv) of Theorem 3.4 if and only if R has the radical trace property, namely, for every ideal A of R, A A~l is either R or a radical ideal of R. Corollary 3.5. The following statements are equivalent for a Priifer domain R. (i) Every branched prime ideal of R of height greater than one is the radical of a finitely generated ideal of R. (ii) For each nonzero ideal A of R, End(AM) = End(A)M for all maximal ideals M ofR. (iii) For each ideal A of R, Ass(A) is precisely the set of prime ideals P such that AC P and PEnd(A) ± End(A). Proof, (i) => (ii) Suppose (i) holds, and let A be a nonzero ideal of R. Let M be a maximal ideal of R. If M has height one then RM is a one-dimensional valuation domain, so End(^M) = RM- Consequently, End(A)M = End(^M). If M has height greater than 1, then M is the union of branched prime ideals of height greater than 1, so by (i) and Lemma 3.2, End(Ajvr) = End(A)M- This proves (ii). (ii) => (i) Let P be a branched prime ideal of R such that there is a nonzero prime ideal Q of R with Q c P. From (ii) and Lemma 2.1, it follows that R has the separation property. Thus Q is a maximal ideal of End(Q) and End(Q)/Q is the quotient field of R/Q. From (ii) it follows that if X is an .R-submodule of End(Q) with QCX, then for all maximal ideals M of R containing Q. By Theorem 3.4, the branched prime ideal P/Q of R/Q is the radical of a finitely generated ideal. Hence P is the radical of an ideal I + Q, where / is a finitely generated ideal of R. Since R has the separation property, there is a finitely generated ideal J such that Q c J C P. Thus P is the radical of the finitely generated ideal I + J. (ii) =>• (iii) Let A be an ideal of R, and suppose P € Ass(A). Then there exists Q e X 'A such that P C Q. By Lemma 1.4, RQ = End(AM) for some maximal ideal M of R. Thus by (ii), RQ = End(A)M, and we have QEnd(A) ^ End(A). It follows that PEnd(A) ^ End(A). Now suppose P is a prime ideal coritaining A such that PEnd(A) ^ End(A). Then End(A)M ^ PEnd(A)M for some maximal ideal M of R, and it follows that End(A)M C RP. By (ii), End(A M ) C RP. If End(A M ) = RQ for some prime ideal

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Q, then by Lemma 1.2 and 1.4, it follows that Q e Ass(A). Since A C P C Q, Lemma 1.2 implies P £ Ass (A). This proves (iii). (iii) =s> (ii) Let A be a proper ideal of R and M be a maximal ideal of R containing A. If End(A)M = Rp for some prime ideal P of R, then PEnd(A) ^ End(A), so by (iii), P £ Ass(A). By Lemma 3.1, End(A) M = End(AM). This proves (ii). D Remark 3.6. Theorem 3.4 shows that conditions for good behavior with respect to localization for endomorphism rings of submodules of the fraction field involve all the primes of R. On the other hand, Corollary 3.5 shows that the corresponding property for ideals involves only conditions on the primes of height greater than one. This phenomenon is illustrated in the article Goeters-Olberding [8], where the statement of Corollary 3.5 that End(A) commutes with localizations is of central importance. Our proof of Lemma 3.3 is a refinement of the argument given in Theorem 3.7 of [8]. Theorem 3.7. Consider the following statements for an arithmetical ring R. (i) XA = Max(A) for each proper regular ideal A of R. (ii) Every branched prime ideal of R, that properly contains a regular prime ideal is the radical of a finitely generated ideal. (iii) Every branched prime ideal of R that properly contains a regular prime ideal is the radical of a principal ideal. Then (iii) =>• (i) =>• ( i i ) . Moreover, if R is a QR-domain, then all three statements are equivalent. Proof, (i) =>• (ii) We consider first the case where R is a domain. Then to prove (ii), it suffices by Corollary 3.5 to show that for every ideal A of R and maximal ideal M containing A, End(A)A/ = End(A.M)- Let A be a proper nonzero ideal of R and let M be a maximal ideal such that A C M. Since RM is a valuation domain, there exists a prime ideal P of R such that End(A)M = Rp. We claim first that the elements of P are not prime to A. Let {Ma} denote the set of maximal ideals of R that contain A, and let {Np} denote the set of maximal ideals of R that do not contain A. For each a, there exists a prime ideal Pa such that End(AM a ) = RpaMoreover, by Lemmas 1.2 and 1.4, we have XA C {Pa} C Ass(A). Since End(A) — (n Q End(A M J) n (n /3 End(A N J), we have

RP = End(A) M = (p|-RpJ#M n a

(f}RN0)RM. /3

Since R has the separation property, Lemma 2.3(iv) implies that Pi/j^s 2 RPThus because Rp is a valuation domain, it must be that (~]a Rpa C RP, and it follows that P C (Ja Pa. Since S(A) = U a P a , the elements of P are not prime to A, as claimed. Therefore there exists Q e Max(A) such that P C Q. By (i), Q € XA, so by Lemma 1.2, P e Ass(A) since A C P C Q. Thus by Lemma 3.1, End(A)M = End(Aj\4-). This proves that (i) implies (ii) in the case R is a, domain. Now consider the general case where R is not necessarily a domain. Observe that since statement (i) holds for R, statement (i) holds for R/P for all regular

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prime ideals P of R. This follows from the observation that if A is an ideal of R that contains P then for all x € R, (A/P:R/Px + P) = ( ( A : x ) In particular, Ass(A/P) = {Q/P : Q e Ass(A),P C Q}. Also, Max(A/P) = {Q/P : Q e Max(A),P C Q}. Thus statement (ii) holds for R/P for all regular prime ideals P of R. Let Q be a branched prime ideal of R containing at least one regular prime ideal. Then there exists a regular prime ideal P of R such that P C Q. By Theorem 2.3, there exists a finitely generated ideal B of R such that P C B C Q. Moreover, since (ii) holds for R/P, there exists a finitely generated ideal C of R such that Q is the radical of C + P. It follows that Q is the radical of the finitely generated ideal C + B. (iii) => (i) Let A be a proper regular ideal of PC. For each P € XA there exists M e Max(A) such that PCM and then P = M if and only if M e Ass(A). Thus it suffices to prove that M 6 Max(A) implies M € Ass(A). This is clear if M/A has height one as prime ideal of R/A. Assume that ht(M/A) > 2. Then M is the union of a chain of branched prime ideals Pj of R, where A C Pi and ht(Pj/A) > 2. Since A is a regular ideal, so is each PI. Thus by (iii), each Pj is the radical of a principal ideal of. R. Since Pl C M, the elements in Pi are non-prime to A. Since Pi is the radical of a principal ideal and S(A) — UQ^XA *?> ^» ^s contained in some member of XA. By Lemma 1.2, each Pi e Ass(A). Therefore \JiPi = M € Ass(A). Finally, we conclude that statements (i),(ii) and (iii) are equivalent for a QRdomain PC since in a QR-domain every prime ideal that is the radical of a finitely generated ideal is the radical of a principal ideal. D Corollary 3.8. If R is a Priifer domain having the property that every maximal prime divisor of an ideal A of R is a Krull associated prime of A, then every branched prime ideal of R of height greater than one is the radical of a finitely generated ideal. Proof. The corollary is a consequence of Theorem 3.7 and Corollary 1.3.

D

Example 3.9 illustrates the fact that it is possible for a 2-dimensional Priifer domain to satisfy the hypotheses of Theorem 2.3, but not Theorem 3.7. Example 3.9. There exists a Priifer domain R that has the strong separation property and yet contains an ideal A such that XA is a proper subset of Max( A) . Moreover, R does not satisfy any of the conditions ( i ) , (ii) or (iii) of Theorem 3.7. Let D be an almost Dedekind domain (that is, DM is a DVR for all maximal ideals M of D) that is not Dedekind. (See [3, page 281] for an example of such a domain.) Let P denote the field of fractions of D, let x be an indeterminate over F, and let P = xF[x](x). Define R = D + P. Then PC is a Priifer domain and P is the unique nonzero nonmaximal prime ideal of R. Moreover, if M is a maximal ideal of PC, then M contains P and if r € M \ P, then P C Rr C M, so R has the strong separation property. In particular, by Theorem 2.3, every ideal A of PC has the property that XA C Max(A). Since D is almost Dedekind, but not Dedekind, there exists a maximal ideal Q of D that is not the radical of a finitely generated

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ideal. Consequently, the (height 2) maximal ideal Q + P of R is not the radical of a finitely generated ideal of R. By Theorem 3.7, there exists an ideal A of R such that XA is a proper subset of Max(A). In Example 2.9 of [4], we construct a 2-dimensional Prufer domain with Noetherian spectrum having an ideal A for which there exists a maximal ideal M such that M is in Max(^4) but M is not a Krull associated prime of A. Since this domain R has Noetherian spectrum, R satisfies statement (ii) of Theorem 3.7. Hence condition (ii) of Theorem 3.7 does not imply condition (i), even when R is a 2-dimensional Priifer domain. This leaves the question of whether in general (i) implies (iii) in Theorem 3.7: Question 3.10. Does there exist an arithmetical ring R such that R satisfies statement (i) but not statement (iii) of Theorem 3.7? References [1] N. Bourbaki, Commutative Algebra, Chapters 1-7, Springer-Verlag, 1989. [2] D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Springer-Verlag, 1994. [3] M. Fontana, J. Huckaba and I. Papick, Prufer domains, Marcel Dekker, 1998. [4] L. Fuchs, W. Heinzer and B. Olberding, Commutative ideal theory without finiteness conditions: primal ideals, to appear. [5] R. Gilmer, A counterexample to two conjectures in ideal theory, Amer. Math. Monthly, 74 (1967), 195-197. [6] R. Gilmer and W. Heinzer, Overrings of Prufer domains II, J. Algebra 7 (1967), 281-302. [7] R. Gilmer and W. Heinzer, Primary ideals with finitely generated radical in a commutative ring, Manuscripta Math. 78 (1993), 201-221. [8] H. P. Goeters and B. Olberding, Extension of ideal-theoretic properties of a domain to its quotient field, J. Algebra 237 (2001), 14-31. [9] W. Heinzer, Quotient overrings of integral domains, Mathematika 17 (1970), 139-148. [10] W. Heinzer, J. Huckaba and I. Papick, m-Canonical ideals in integral domains, Comm. Algebra 26 (1998), 3021-3043. [11] W. Heinzer and J. Ohm, Locally Noetherian commutative rings, Trans. Amer. Math. Soc., 158 (1971), 273-284. [12] J. Iroz and D. Rush, Associated prime ideals in non-Noetherian rings, Can. J. Math. 36 (1984), 344-360. [13] I. Kaplansky, Commutative Rings, University of Chicago Press, 1974. [14] W. Krull, Idealtheorie in Ringen ohne Endlichkeitsbedingung, Math. Ann. 101 (1929), 729744. [15] W. Krull, Idealtheorie, Ergebnisse der Math., Berlin, 1935. [16] T. Lucas, The radical trace property and primary ideals, J. Algebra 184 (1996), 1093-1112. [17] M. Nagata, Some remarks on prime divisors, Memoirs College of Science, Kyoto University, Series A, 33 (1960) 297-299. [18] M. Nagata, Local Rings, Interscience, 1962. [19] N. Nakano, Idealtheorie in einem speziellen unendlichen algebraischen Zahlkorper, J. Sci. Hiroshima Univ. Ser. A 16 (1953), 425-439. [20] B. Olberding, Globalizing local properties of Prufer domains, J. Algebra 205 (1998), 480-504. [21] R. Pendleton, A characterization of Q-domains, Bull. Amer. Math. Soc. 72 (1966), 499-500. [22] D. Rush and L. Wallace, Noetherian maximal spectrum and coprimely packed localizations of polynomial rings, Houston J. Math. 28 '(2002) 437-448.

On Strongly Flat Modules over Matlis Domains L. Fuchs Department of Mathematics, Tulane University, New Orleans, Louisiana 70118, U.S.A. iuchsffltulane.edu

L. Salce Dipartimento di Matematica Pura e Applicata, Via Belzoni 7, 35131 Padova, Italy salceflmath.unipd.it

J. Trlifaj Katedra Algebry, MFF UK, Sokolovska 83, 186 75 Praha 8, Czech Republic trlifajSkar.lin.mff.cuni.cz

Let R be an integral domain and Q its field of quotients. An ^-module M is called strongly flat if Ext]j(Q, C) = 0 implies Ext)j(M, C)=0 for any C e Mod/2. The first characterization of strongly flat modules was given by Trlifaj [T2], and they were thoroughly investigated by Bazzoni and Salce [BS3]. The strongly flat modules seem to behave over Matlis domains R particularly nicely; the reason for that may be sought in the fact that they are all of projective dimension I , and—as has been demonstrated in [FS]—such modules often display features resembling the Dedekind domain case, so they are more tractable. Another reason goes back to an old result of Matlis [M] which—in modern terminology— asserts that Matlis domains can be characterized by the property that strongly flat modules form a resolving class. Flat modules always form a resolving class, and the coincidence of the two classes characterizes an important subclass of the Matlis domains: the almost perfect ones, cf. [BS2-3]. In the Matlis domain case, we can characterize the strongly flat /J-modules as torsion-free modules M that satisfy p. d. R (M # K) < 1, where K stands for the factor module Q/R', see (1.2). This characterization, along with a theorem by Lee [L] on divisible modules of projective dimension 1 over Matlis domains, leads to a more structural description: they admit continuous well-ordered ascending chains L. Salce was supported by MURST (PRIN 2000), and J. Trlifaj was supported by grant SAB20010092 of Secretaria de Estado de Educacion y Universidades MECD at CRM IEC Barcelona, and by MSM 113200007. A large portion of this paper was completed while L. Fuchs was visiting Universita di Padova, with the support of INDAM..

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whose factors are strongly flat modules of ranks of bounded cardinality, a bound being the minimal cardinality of generating sets of Q; cf. (2.1). By making use of these results, we can prove that over Matlis valuation domains the strongly flat modules are exactly the extensions of free modules by divisible torsion-free modules; see (3.3). This result has been established over arbitrary valuation domains in [BS3] only for modules of rank up to the continuum. More can be said in the general case of Matlis domains provided we are willing to restrict our considerations to the constructible universe. We show that in this case there is a universal test module to check the strong flatness of an arbitrary module of projective dimension < 1; cf. (5.4). This result motivates the search for a universal test module for projective dimension < 1. We are able to find such a test module, under the assumption V = L, for Matlis domains of global dimension 2; cf. (5.5). Combining the two test modules, we obtain, in case of V = L, a universal test module for all strongly flat modules over Matlis domains of global dimension 2; see (5.6). The situation with strongly flat modules over Matlis domains strongly resembles the theory of Baer modules. For Baer modules one can establish the existence of a continuous well-ordered ascending chain with factors of bounded number of generators (No suffices as a bound, see Eklof-Fuchs-Shelah [EPS]), but a strong set-theoretical hypothesis (e.g., V = L) is needed to ascertain the existence of a universal test module; see, e.g., [FS, XVI.8]. Over certain domains R, the existence of a universal test module for strong flatness or for projective dimension < 1 can be verified without any additional hypothesis, or in a different way. For instance, if R is an almost perfect domain (i.e., all proper factor rings are perfect), then all flat modules are strongly flat (see [BS2] and [BS4]), and there exists a universal test module for flatness. On the other hand, if R is an IC-domain (see [BS1]), then—under the hypothesis V = L—K is a universal test module for projective dimension < 1. The relevant result (6.3) in the present paper shows that, assuming the Uniformization Principle, there is no universal test module for strong flatness over a Matlis domain whenever it is not almost perfect. The conclusion is that, over a Matlis domain of global dimension 2 that is not almost perfect, the existence of a universal test module for strong flatness is independent of ZFC + GCH; cf. (6.4). 1. Preliminaries In what follows R denotes a commutative domain with 1. and Q ^ R its field of quotients. We shall use the notation K = Q/R. R is a Matlis domain if the projective dimension p.d. R Q = i. The .R-completion of a module M will be denoted by M, and genM will denote the minimal cardinality for generating sets of M. An .R-module C is weakly cotorsion or Matlis cotorsion if ExtJj(Q, C) = 0. An .R-module M is strongly flat if Ext)j(M, C) — Q holds for all weakly cotorsion .R-modules C. Evidently, strongly flat modules are flat, and hence torsion-free.

On Strongly Flat Modules over Matlis Domains

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Lemma 1.1 (Trlifaj [T2], Bazzoni-Salce [BS]). For a reduced torsion-free module M over a Matlis domain R the following are equivalent: (i) M is strongly flat; (ii) M is a summand of an R-module N fitting in an exact sequence Q-+F^N->D^Q, where F is a free and D is a torsion-free divisible R-module; (iii) the R-completion M of M is a summand of the completion of a free R-module; (iv) K M is a summand of a direct sum of copies of K . The strongly flat modules M over a Matlis domain R can easily be characterized in terms of the tensor product K ®R M. This characterization is crucial in the sequel. Corollary 1.2. Let R be a Matlis domain. A torsion-free module M is strongly flat if and only if p.d.R(K®RM) <1. Proof. This follows from the fact that (l.l)(iv) holds if and only if p. d.R(K®RM] < 1; see [PS, VII.2.4]. D Observe that p.d. N < max{p. d. F, p. d. D} = 1 for Matlis domains, consequently, Corollary 1.3. A domain R is a Matlis domain if and only if strongly flat Rmodules have projective dimension < 1. D Recall that a submodule N of & module M of projective dimension k is said to be a tight submodule if p. d. M/N < k; then p.d. N < k likewise. We now have the simple fact: Lemma 1.4. If R is a Matlis domain, then tight submodules of strongly flat Rmodules are again strongly flat. Proof. Consider the exact sequence 0 —> N —> M —» M/N —> 0, where M is strongly flat, so p.d. M < 1 by (1.3). For any weakly cotorsion .R-module C we have the induced exact sequence Ext^(M,C) -> Ext^TV.C) -»

Ext2R(M/N,C).

The two ends vanish, since M is strongly fiat and p.d. M/N < 1, respectively, so the middle term is necessarily 0, proving the assertion. D The following corollary is immediate. Corollary 1.5. Pure submodules of countably generated strongly flat modules over a Matlis domain are again strongly flat. Proof. If TV is a pure submodule in the 'countably generated strongly flat module M, then M/N is countably generated and flat, so it has projective dimension < 1 (see, e.g., [FS, VI.9.8]). An appeal to (1.4) completes the proof. D

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2. Chains with strongly flat factors of bounded size Assume from now on that R is a Matlis domain with gen Q = /j,, an infinite (regular) cardinal. This cardinal seems to be a most relevant invariant of R. (If (i = HO, then this is actually a stronger hypothesis than R being a Matlis domain, because gen Q < HO implies p. d. Q < 1.) Furthermore, suppose that M is a reduced strongly flat .R-module (reduced means that it has no divisible submodule j^ 0). We form the exact sequence where D stands for the injective hull of M; it is the direct sum of copies of Q. Let denote the canonical map D —* T. We will need the following useful device. By an H(K)-family f of submodules of a module M we mean a collection of submodules of M such that HI. 0 , M e F; H2. T is closed under sums of submodules; H3. if A £ T and X is a subset of M of cardinality < K, then there is a B e F such that A U X C B and genB/A < K. Select an £T(/i)-family V of summands of D. Owing to (1.2), p. d. T < 1, so by a theorem of Lee (see [L], or [FS, VII.2.9]) T is a direct sum of countably generated submodules. Therefore, we can select an (H(Ho)-, and hence an) H(/j,)-faxm\y T of summands of T. We now extract, by transfinite induction, from these Jf(/Lt)-families continuous well-ordered ascending chains 0 = TO < Tj < • • • < Tff < TCT+1 < • - • in T> and T, respectively, with cr < r for a suitable ordinal T such that (ii) Da is a summand of D, and <j>(Da] = Ta is a summand of T for all a < T; (iii) genD CT+1 /D CT < fj, and genT^+i/T^ < /z for all a + 1 < T. The critical condition is 0(-Do-) = Ta. Suppose that we have constructed chains of Da and Ta for all a up to some ordinal p such that all the conditions are satisfied. If p is a limit ordinal, then we have to set Dp = {jtr

(n<w)

with links in V, and a chain

Tp_! = T° < T1 < • • • < Tn < . . .

(n < w)

with links in T, with at most ^-generated factors, such that < T1 < 4>(Dl) < • • • < Tn < 4>(Dn) <... n

n

(n
Letting Dp = \Jn^D and Tp = \Jn(Dp) = Tp will hold. It is easy to arrange the selections such that (i) will be satisfied.

On Strongly Flat Modules over Matlis Domains

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Next define Ma = M D Da (a < T). Then we get a continuous well-ordered ascending chain

(2)

0 = MO < MI < • • • < Ma < Ma+l < •••

of submodules in M such that (iv) for every a < T, 0 —> Ma —> Da —» Ta —> 0 is an exact sequence. After all this preparation we can state: Theorem 2.1. Let R be a Matlis domain satisfying genQ < \i. Every reduced strongly flat R-module M admits a continuous well-ordered ascending chain (2) of submodules Ma such that all the factors Ma+\IMa are strongly flat modules of rank at most /u. Proof. In view of the preceding argument, M has a continuous well-ordered ascending chain (2) of submodules satisfying (iv). Then Ma+i/Ma = [Da + (M n (< Dn+i/Dcr) is torsion-free and ® K = (Ma+l ® K}/(Ma ®K}^ Ta+1/Ta

holds for every a < T, where we have used the isomorphism Ma ®K = Ta which is obtained by tensoring the exact sequence in (iv) above with K (recall that Da®K = 0 = Torf (Da, K) and Torf (Ta, K)^Ta). The last factor module is isomorphic to a summand of T, so it certainly has projective dimension < 1. The factor modules MO-+I/MO. are evidently of rank at most JJL. It remains to appeal to (1.2) to complete the proof. D The following immediate consequence shows that actually we have a large supply of strongly flat submodules contained in any strongly flat module over a Matlis domain. Corollary 2.2. Assuming the hypothesis of (2.1), every strongly flat module has an H (/j,} -family of strongly flat submodules. Proof. Starting from (2.1) use [FS, XVI. 8. 11] to obtain the stated conclusion.

D

3. The class of projective by divisible modules The class C of ^-modules that are extensions of projective modules by divisible torsion-free modules is evidently contained in the class of strongly flat J?-modules. As the module N in (1.1) belongs to the class C, it is clear that the class of strongly flat modules coincides with the class of summands of modules in C. In general, the class C is not closed under taking direct summands. An easy counterexample has been provided by Bazzoni-Salce: the additive group Jp of the p-adic integers is a summand of the Z-completion of Z, but it is not a member of the class C. However, we wish to show that for Matlis valuation domains R the answer is in the affirmative. Bazzoni and Salce [BS3, Thm 3.15] show that if R is a valuation domain, then the answer is 'yes' up to cardinality HI, but have no answer for larger cardinalities. We are now going to use a completely different approach that avoids any need for

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transfinite induction, so we do not have to refer to a sort of singular compactness theorem in our proof. Recall that if R is a valuation domain, then p. d.R Q = I implies genQ = H 0 . The next lemmas are important steps in the proof of (3.3). Lemma 3.1 (Bazzoni-Salce [BS3, Lemma 3.14]). Let R be a valuation domain. Suppose M is a torsion-free R-module which is, for some cardinal K, the union of a continuous well-ordered ascending chain

0 = M0 < MI < • • • < MCT < • • • of submodules MCT. // all the factors Ma+i/Ma M as well.

(a < K)

belong to the class C, then so does D

We quote a basic result from [BS3]. Lemma 3.2 (Bazzoni-Salce [BS3, Lemma 3.13]). If R is a valuation domain, then a strongly flat R-module of countable rank has a free dense basic submodule. D We can now derive the following conclusion. Theorem 3.3. Let R be a Matlis valuation domain and M a strongly flat reduced R-module of any rank. Then M has a free dense basic submodule, i.e., M is the extension of a free module by a divisible module. Proof. By (2.1), M has a continuous well-ordered ascending chain (2) with countable rank strongly flat factors M^+i/Mo-. Hence (3.2) shows that all these factors have free dense basic submodules. It only remains to invoke (3.1) to complete the proof. D 4. A test module for strong flatness The class SJ- of strongly flat modules over a fixed domain R and the class WC of weakly cotorsion /^-modules are, respectively, the cotorsion-free and the cotorsion classes of the cotorsion theory cogenerated by Q; see Salce [S] or Trlifaj [T3]. We are wondering whether or not the cotorsion theory CQ can also be generated by a single module. In other words, we are looking for a weakly cotorsion test module to recognize strong flatness, i.e., for a weakly cotorsion C such that M is strongly flat exactly if Ext}j(M, C) = 0 holds. If this happens, then we will have where ±C = {M j Ext]j(M,C) = 0} and (^C)1- = {N

Ext]j(M,/v) = 0 for all

Note that such a module C always exists in the case when Sf coincides with the class of all flat modules (that is, in'case R is almost perfect, see [BS2]). Indeed, by the Flat Test Lemma, a module M is flat if and only if Torf (M, R/I) = 0 for all ideals / of R if and only if Ext^(M, C) = 0 where C = UI
On Strongly Flat Modules over Matlis Domains

211

We first show that, for every module M over any domain there is a single weakly cotorsion test module CM for strong flatness, and then for Matlis domains we establish the existence of a single test module CK for all modules M of cardinality < K. For the first part of the next result, see Bazzoni-Salce [BS3], proof of Theorem 2.1. The second part was observed by S. Bazzoni. Lemma 4.1. For every torsion-free R-module M, there exists an exact sequence 0 —» CM —> N —* M —» 0, where N is strongly flat and CM is weakly cotorsion. M is strongly flat if and only z/Ext#(M, CM] = 0. Proof. Imitating the construction [BS3, Theorem 2.1], we have the pushout diagram: 0 0

H

> F

> M H

M

0 0 with exact rows and columns, where F is a free -R-module of rank gen M, D is a torsion-free divisible module, and N is strongly flat in view of (1). Since the kernel H = CM is .R-complete, and so weakly cotorsion, the exact sequence splits whenever M is strongly flat. Conversely, if the sequence splits, then M is a summand of the strongly flat module N, so itself strongly flat. D We need an upper estimate for the number of generators of H where CM = H. F can be chosen with K = genM generators, so gen# < K\R\. Let XK henceforth denote a free .R-module of rank K for an infinite cardinal K.. For convenience, we will assume that K > \R\. By Matlis [M], for a reduced torsionfree module the properties of 'weakly cotorsion' and '^-complete' are equivalent. We continue with a simple lemma. Lemma 4.2. Let R be a Matlis domain. Any torsion-free weakly cotorsion Rmodule C that is the completion of a module B with at most K generators is an epic image of the complete R-module XK. Proof. Evidently, there is an epimorphism <j> : XK —> B. As C is the completion B,

C. Over Matlis domains, the image of an ^-complete torsion-free module is again complete, so Im/ — C. D

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We are now ready to prove the existence of a single test module for strong flatness for all modules with at most K generators, provided R is a, Matlis domain. Proposition 4.3. Suppose R is a Matlis domain, and K > \R\ is an infinite cardinal. A torsion-free R-module M of projective dimension < I with gen M < K is strongly flat if and only if Proof. Evidently, it is enough to verify that the condition is sufficient, so assume that the indicated Ext vanishes for M. By the proof of (4.1) and its remark it is enough to show that Extjj (M, H ) = 0 for every torsion-free module H with gen H < K. By (4.2) there exists an exact sequence 0 —> N —> XK —> H —> 0. This induces the exact sequence

ExtlR(M,XK) -> E^R(M,H) -» Ext2R(M,N), where the first term is 0 by assumption, while the last term is 0 since p. d. M < 1. Hence the middle term also vanishes. D We shall need an upper estimate of the cardinality of X^. Proposition 4.4. // A = \R\, then for every cardinal IJL>\

Proof. Clearly, \X^\ < fi\. Recall that for a torsion-free module X^ we have toRXn), thus |XM < \X^K\ < (A M ) A = /A D 5. Strongly flat 'modules in the constructible universe We continue with Matlis domains, so in this section R stands for a Matlis domain. We wish to establish the existence of a universal test module for strong flatness, i.e., one which would work for any module, no matter how large it is. We assume that we are working in the constructible universe in order to be able to use arguments that require additional set theoretical hypotheses. We will imitate the discussion in Fuchs-Salce [FS, XVI. 10] where Whitehead modules were dealt with. First we fix the cardinal /j, as the smallest regular cardinal with \R\\R\ < /j,, and concentrate on /^-modules M satisfying

where X, denotes — as above— the test module for strong flatness for all E-modules with < fj, generators. Let us denote by LX^ the class of these modules M. It is clear that the class SF of strongly flat modules is contained in the intersection PI n ~*~Xp, where P\ stands for the class of modules of projective dimension < 1. We are going to show that ifV = L, then the equality SF = PiC\-LXti holds. Next we formulate results needed for the proof of the main theorem (5.3) infra. They follow the presentation by Becker-Fuchs-Shelah [BFS]; for their proofs we refer to the original source or to [FS, XVI. 10].

On Strongly Flat Modules over Matlis Domains

213

The proof of (5.3) is via induction on the minimal cardinality of generating sets of M. In the transfinite induction, the argument for regular cardinals is taken care of by the following lemma. Lemma 5.1. Assume V = L. Let R be a Matlis domain of cardinality A, and K a regular cardinal > Xx. Suppose the R-module M with genM = K admits a continuous well-ordered ascending chain (3)

0 = MO < MI < • • • < Ma < ...

(a < K)

of submodules such that (ii) gen Ma < K for each a < K,; (iii) each Ma (a < K) belongs to ^X^. Then: M G ^X^ if and only if E = {a < K. | 37 > a such that M7/Ma 4. ^X^} is not a stationary subset of K.

D

The singular case requires a form of Shelah's singular compactness theorem. To be ready for that, we need a lemma. Lemma 5.2. Let K be an infinite cardinal with \R\^R\ < K and assume that the R-module M has a filtration (3) with genM a +i/M a < K. Then M admits an H(K)-family C of submodules such that every C & C is the union of a continuous well-ordered ascending chain of submodules in C with factors isomorphic to certain quotients MQ+1/Ma. D Now we have the main result whose proof makes use of a version of (1.4) for modules of projective dimension < 1 in the class -LXM. Theorem 5.3. Assume V = L. Let R be a Matlis domain and ^ the smallest regular cardinal such that \R\\R\ < /j,. An R-module M of projective dimension < 1 belongs to the class X^ if and only if it admits a chain (3) such that (i) M = (J MQ; (ii) genMQ+1/Ma < p. for each a + l < K; (iii) each Ma+i/Ma (a < K) belongs to ^X^. If the cardinal /x has the same meaning as above, then we can state the existence of a universal test module for strong flatness for modules of projective dimension < 1 over any Matlis domain; in fact, (5.3) actually covers the strongly flat modules: Theorem 5.4. Assume that V = L and R is a Maths domain. An R-module M of projective dimension < 1 is strongly flat if and only if it satisfies

Proof. It suffices to verify the 'if part,' so assume Ext#(M,X^) = 0. If genM < H, there is nothing to prove. Suppose genM > /j,, so by (5.3) it has a chain (3) satisfying conditions (i)-(iii). Then all the factors Ma+i/Ma are strongly flat modules, whence it follows that M is strongly flat as well (see, e.g., [FS, VI.2.5]). D

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Needless to say, for Matlis domains of global dimension 1 (= Dedekind domains) there is a universal test module for strong flatness available in ZFC, since such domains are almost perfect; see §4. In order to improve on (5.4) for domains of global dimension > 1, we search for a universal test module for projective dimension < 1. It was proved in [BS1] that, in case V = L, K is a universal test module for projective dimension < 1 for a valuation IC-domain R. (A valuation domain is an IC-domain if it is a Matlis domain of global dimension 2 that satisfies a topological incompleteness condition.) Over an arbitrary Matlis domain of global dimension 2 we can prove— again in case V = L— that there is a test module, viz. K^ , where Theorem 5.5. Assume V = L. Let R be a Matlis domain of global dimension 2, and set Y = K^ with X — 1\R\. Then an R-module M has projective dimension < 1 if and only if it satisfies Ext}j(M,y) =0. Proof. Since the module Y is (/indivisible, we have Ext}j(M, Y) = 0 for any module M of projective dimension < 1; see [FS, VII.2.5(i)]. For the proof of the converse, we can restrict ourselves to torsion divisible modules. Indeed, for an arbitrary module M, there is an exact sequence 0 —» M —-> E —> P —> 0 where E is divisible and P has projective dimension < 1 [FS, VII. 1.4]. If M satisfies ExtJj(M, Y) - 0, then the exact sequence 0 = Ext^(P, Y) -» ExtlR(E,Y) -» Extfl(M,y) = 0 implies Ext^(£,y) = 0, thus every M embeds in a divisible module E satisfying the same condition. If we can verify that E has to be of projective dimension < 1, then the same will follow for M. Note that if E is divisible, then E = F(&T where F is a Q-module (so of projective dimensionl) and T is torsion divisible [FS, VII.2.2] , thus Ext^E, Y) = 0 if and only if ExtlR(T, Y) = 0. A torsion divisible R- module M will be called "free" provided that p. d. M < 1; equivalently, M is a direct sum of countably generated torsion divisible modules of projective dimension < 1; cf. [L] or [FS, VII.2.9]. By induction on K — \M\, we will prove that if M is a torsion divisible -R-module satisfying Ext^(M,y) = 0, then M is "free". First, assume K < A. By [FS, VII.2.10], there is an exact sequence Q-*D->C ->M -»0 a

where C = K^ \ a = Hom^(-K', M), and D is torsion divisible. We have a\ < \MR < A, so \D\ < A. If p. d. M — 2, then p. d. D = I and another application of [FS, VII. 2. 10] shows that D is a direct summand of a direct sum of < A copies of K. Then D is isomorphic to a direct summand in y, and the exact sequence above splits by hypothesis, a contradiction. This proves that in the present case p. d . M < 1. Next, suppose that K is a regular uncountable cardinal > A, and A is a torsion divisible module of cardinality K. A will be called "K-projective" if each subset of A of cardinality < K is contained in a "free" submodule of A of cardinality < K. By the inductive premise and [FS, VH.2.5(ii)], all torsion divisible submodules of M of

On Strongly Flat Modules over Matlis Domains

215

cardinality < K are "free", hence M is "K-projective". Jensen's Diamond Principle then provides for a continuous well-ordered ascending chain {Ma a < K} of "free" submodules of M such that \Ma\ < K for all a < K, M = \Ja A. For each "free" module A and each decomposition A = ® i 6 j Ai into a direct sum of countably generated modules, set i&X

and call it a "basis" of A. The notions of "free" and "basis satisfy conditions (a)-(e) of [EM, IV. 3. 6], so by Shelah's Singular Compactness Theorem and the inductive premise we can conclude that M is "free", cf. [EM, IV.3.7]. D It is now clear that (5.4) and (5.5) together yield a universal test module for strong flatness for Matlis domains of global dimension 2. Corollary 5.6. Assume V = L. Let R be a Matlis domain of global dimension 2, and let X^ and Y be as in (5.4) and" (5.5). Then an R-module M is strongly flat if and only if it satisfies where Z ~ X^ ® Y.

D

6. Strongly flat modules under uniformization In this section, wa will prove that in a model of ZFC with Shelah's Uniformization Principle adjoined, there are no universal test modules for strong flatness for any Matlis domain that is not almost perfect, in particular, for any Matlis valuation domain of global dimension > 1. Combined with the results of §5, this asserts that the existence of universal test modules for strong flatness for Matlis domains of global dimension 2 which are not almost perfect is independent of ZFC. Following [T3, 2.3], UPK will denote Shelah's Uniformization Principle for K (where K is a singular cardinal of cofinality w), and UP will denote "UPre for all singular cardinals K of cofinality u>." Shelah proved that UP is consistent with ZFC + GCH. More specifically, let K be a singular cardinal of cofinality w and E a subset of K+ consisting of ordinals of cofinality u>. Then {no, a £ E} is called a ladder system provided that for each a £ E, na = {na(i} \ i < LU} is a strictly increasing sequence consisting of non-limit ordinals in K+ such that supi A such that for each a € E, f ( n a ( i ) ) = ha(i) for almost all i < w."

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In the balance of this section, R will denote a Matlis domain. We will be interested in the cotorsion theories (P0,ModR) and (Pi,2?) where Pi denotes the class of all .R-modules of projective dimension < i, and T> is the class of all (h)divisible .R-modules, cf. [FS, VII.2.5]. If C = (A, B) is a cotorsion theory, then a module M is a local splitter for C provided that M^ € A n B (hence Ext)j(M, M (w) ) = 0), and there is a non-split embedding v : M^ —> M^ which is locally split, that is, for each n < w, there is a submodule Cn C M^ such that v(M^} ®Cn = M^ and v(M^^} C Cn. Here, for a subset A C w, the symbol M^ A ^ denotes the direct summand of M( W ) consisting of all x £ M (w) such that m(x) = 0 for all i e u>\A (where ^ : M (w) —> M is the ith canonical projection). Example 6.1. The module M = R is a local splitter for the cotorsion theory (Po.ModR). Indeed, take any sequence S = {a,i \ i < w} of elements of R such that {Ran • • • HQ \ n < w} is a strictly decreasing chain of principal ideals of R, and consider the embedding vs : R^ —> R^ denned at the canonical basis of R^ by vs(li) = li — Ij+ifli) and Cn = ©j> n J-i-R (ft < w). By a classical result of Bass, in this case Coker v$ is a non-projective-flat module. D Example 6.2. If R contains a strictly increasing infinite chain of principal ideals, then M = K is a local splitter for the cotorsion theory (Pi, T>). In order to see this, let {riR \ i < u>} where 0 7^ r^ & R and riR C rj+ifi for all i < u. Put J = (Ji
0 -> RM ^ R(") ^^ J _» 0, where ^5 is as in (6.1), and 775(1,) = r» for all i < u>. In particular, J is a flat module of projective dimension 1. Tensoring the last exact sequence with K, we obtain 0 _> #(<") -^ Jf( w ) -> ^ ®R J -> 0. It is easy to see that the embedding /j,$ — IK <S>R 1/5 splits locally (cf. [T3, Lemma 5]). Since .R ®.R J = RJ = J and Q <&R J = QJ = Q canonically, we have K®RJ = Q/J. Because of p. d. Q/ J = p. d. R/J = p. d. J +1 = 2, (j.s is non-split. This establishes our claim that K is a local splitter for (Pi,T>}. D It is worthwhile pointing out that the domains satisfying the hypothesis of Example 6.2 are exactly those domains which are not almost perfect. This fact can easily be checked by recalling the following property that characterizes almost perfect domains R: given any 0 / r e R, the factor ring R/rR has the descending chain condition on principal ideals. Any almost perfect domain is a Matlis domain ([M], [BS2]); consequently, there exist Matlis domains of global dimension > 1 possessing universal test modules in ZFC, cf. §4 (observe that a valuation domain is almost perfect exactly if it is a DVR, i.e., it has global dimension 1). Our next result shows that it is consistent

On Strongly Flat Modules over Matlis Domains

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with ZFC + GCH that, among Matlis domains, only the almost perfect ones admit universal test modules for strong flatness. Theorem 6.3. Assume UP, and let R be a Matlis domain that is not almost perfect (e.g., a Matlis valuation domain of global dimension > 1). For every R-module C, there exists a flat, but not strongly flat R-module F of projective dimension 1 such that Proof. We use the notations of (6.2). Take a singular cardinal K of cofmality w such that K > | End(<2/J)|, \R\, \C\. UPK provides for a stationary subset E C K+ and a ladder system {na \ a € E} which has the uniformization property with respect to all cardinals < K. Let {PQ | a < K+} be a sequence of free modules defined as follows: PQ = R^ for a € E, and Pa = R otherwise. For a £ E, denote by la the canonical generator of Pa. For a G E, let {la.i i < w} be the canonical basis of Pa. Define For a £ E and i < ui, let gai = !„,„,(,;) — lQ.i + l a ,i+iaj G P. Setting La = @i |(7|, the uniformization property yields Ext}j(F, C) — 0, see [Tl, 2.4]. Applying [T3, 1.7] (for C = (P0,ModR), K = P, and N = J) we have p.d.F = 1. Moreover, F = (Ja 1. Tensoring the exact sequence 0 —> R^ —> R^ -» J —» 0 with K, we get the exact sequence 0 —> K^ —> /C^) —> Q/J —> 0, see (6.2). Similarly, the exact sequence 0 — * L — > P — > F — > 0 tensored with JiT yields the exact sequence

0 -» ffi # ® R L,, -> ffi K ® B Pn -> K ® R F -> 0. Another application of [T3, 1.7] (this time to C = (Pi,£>) and A^ = Q/J) yields K <S> F ^ PI , as we wished to prove. D We have just completed the proof of an independence result: Corollary 8.4. Let R be a Matlis domain of global dimension 2 that is not almost perfect. Then the assertion "There is a universal test module for strong flatness in Mod R" is independent of ZFC + GCH. D References [BSl] [BS2]

S. Bazzoni and L. Salce, An independence result on cotorsion theories over valuation domains, J. Algebra 243 (2001), 294-320. S. Bazzoni and L. Salce, Almost perfect domains, Colloq. Math, (to appear).

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S. Bazzoni and L. Salce, On strongly flat modules over integral domains, Rocky Mountain J. Math, (to appear). [BS4] S. Bazzoni and L. Salce, Strongly flat covers, J. London Math. Soc. (2) 66 (2002), 276-294. [BFS] T. Becker, L. Fuchs and S. Shelah, Whitehead modules over domains, Forum Math. 1 (1989), 53-68. [EFS] P. Eklof, L. Fuchs and S. Shelah, Baer modules over domains, Trans. Amer. Math. Soc. 322 (1990), 547-560. [EM] P. C. Eklof and A. H. Mekler, Almost Free Modules (Revised Ed.), (North-Holland, Amsterdam, 2002). [FS] L. Fuchs and L. Salce, Modules over non-Noetherian Domains, Math. Surveys and Monographs, vol. 84 (Amer. Math. Soc., 2001). [L] S. B. Lee, On divisible modules over domains, Arch. Math. 53 (1989), 259-262. [M] E. Matlis, Cotorsion modules, Mem. Amer. Math. Soc. 49 (1964). [S] L. Salce, Cotorsion theories for abelian groups, Symposia Math. 23 (1979), 11-32. [Tl] J. Trlifaj, Whitehead test modules, Trans. Amer. Math. Soc. 348 (1996), 1521-1554. [T2] J. Trlifaj, Cotorsion theories induced by tilting and cotilting modules, Abelian groups, rings and modules (Perth, 2000), Contemporary Math. 273 (Amer. Math. Soc., 2001), 285-300. [T3] J. Trlifaj, Local splitters for bounded cotorsion theories, Forum Math. 14 (2002), 315-324.

Group Identities on Unit Groups of Group Algebras A. Giambruno Dipartimento di Matematica, Universita di Palermo, Via Archirafi 24, 90123 Palermo, Italy a.giambrunoOunipa.it C. Polcino Milies Institute de Matematica e Estatistica, Universidade de Sao Paulo, Caixa Postal 66.281, 05315-970 Sao Paulo, Brazil polcinoSime.usp.br

1. Introduction Let FG be the group algebra of a group G over a field F. A general problem in the theory of group algebras is that of relating properties of the group of units U(FG) of FG to properties of the group algebra or more generally to the structure of G. This is quite a wild problem in this generality. What we have in mind is the following question. If the unit group satisfies a group identity, what can we say about the group Gl What kind of information can we get on the structure of the group algebra or, more generally, can we characterize such groups G somehow? Recall that a group identity for a group U is a nontrivial word w = w(x\,..., xn) of the free group on xi,... ,xn vanishing in U i.e., such that for every choice of elements ui,...,un e U, W(HI, ... ,un) = I. For the group of units of a group algebra to satisfy a group identity is a quite strong condition. For instance, we shall see that if G is a finite non abelian group such that the corresponding group algebra is semisimple and F is not algebraic over a finite field, then the group of units U(FG) does not satisfy any group identity. In the late 70's Hartley conjectured that if G is a torsion group and U(FG) satisfies a group identity, then FG must satisfy a polynomial identity. Motivated by the solution of this conjecture given in [5], [7] and [14] much progress has been recently made towards the understanding of group algebras whose group of units satisfies a group identity. The best results obtained so far actually give a 2000 Mathematics Subject Classification. Primary 16U60; Secondary 16W10, 20C07. Research partially supported by MURST (Italy) and FAPESP and CNPq (Brazil).

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classification of such groups under some mild hypotheses related to well known unsolved problems. After this classification, other generalizations were considered taking into account the natural involution of the group algebra induced by the map g \—> g~l, g € G. A classification of the torsion groups G such that the symmetric units satisfy a group identity was obtained in [8]. More general groups were considered in [10]. Actually the involution on FG determines the subgroup Un(FG) = {u e U(FG) uu* = 1} of unitary units of U(FG). Inspired by the case of n x n matrices where the orthogonal and the symplectic groups play an important role, the study of the group of unitary units of a group algebra has been recently undertaken. Motivated by a result of Gongalves and Passman ([12]), the authors started in [6] the investigation of group algebras FG such that Un(FG] satisfies a group identity obtaining positive results.

2. Group identities on groups of units In the late 70's interest focused on obtaining characterizations of group algebras whose group of units satisfies some special group identity like nilpotency or solvability. The first of these results is due to Bateman and Coleman ([2]) who characterized group algebras of finite groups whose group of units U(FG) is nilpotent. For infinite groups this was proved by Khripta in [13] and Fisher, Parmenter and Sehgal in [4]. The case when [/(FG) is solvable is due to Bateman ([!]). The corresponding characterization for integral group rings such that [/(ZG) is nilpotent is due to Polcino Milies ([19]) for finite groups and to Sehgal and Zassenhaus ([24]) for the general case. The solvability of [/(ZG) was characterized by Sehgal in [22, § VI.4]. As we mentioned in the introduction, the problem of determining when the group of units of a group algebra satisfies a group identity was inspired by Hartley's conjecture that whenever G is a torsion group and the group of units of the group algebra FG satisfies a group identity then FG must satisfy a (non-trivial) polynomial identity. The interest in this conclusion was motivated by the fact that group algebras FG satisfying a polynomial identity were classified in two subsequent papers of Passman and Isaacs-Passman ([16, Corollary 3.8, Corollary 3.10]). Recall that a group is p-abelian if its derived group is a finite p-group. The classification is the following. Let F be a field such that charF = p > 0. Then FG satisfies a ^ob^omial identitv if and onlv if G contains a ??-abelian sub^rou1"* of finite index. We remark that in Hartley's conjecture, the hypothesis of G being torsion cannot be removed since in general we cannot expect that a group identity on U(FG) forces FG to satisfy a polynomial identity. In fact if G is a torsion free nilpotent group, then G can be ordered and, by [22, Proposition 1.6], [/(FG) has only trivial units i.e., [/(FG) = F* x G. It follows that [/(FG) satisfies a group identity but, in view of the above characterization of group algebras satisfying

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a polynomial identity, it is easy to see that FG does not necessarily satisfy a polynomial identity. On the other hand even the converse is not true. In fact, a polynomial identity on a group algebra does not force in general a group identity on the unit group. In order to see this, recall that by Wedderburn's theorem, if R is any non commutative finite dimensional simple algebra over a field F, then R satisfies a polynomial identity (since it is finite dimensional). Now, U(R), the group of units of R, contains either GL^(F] or a finite dimensional division algebra over a finite extension of F. Hence, if F is not algebraic over a finite field, by invoking [11] or [18] it follows that U(R) cannot satisfy any group identity. As a consequence, if F is a field not algebraic over a finite field and G is a finite non abelian group such that FG is semisimple, then U(FG) does not satisfy any group identity. Hartley's conjecture was proved by Giambruno, Jespers and Valenti in [5] for semiprime group algebras and in the general case by Giambruno, Sehgal and Valenti in [7], when F is an infinite field. Later Passman extended this result in [17] obtaining a complete classification of the groups G satisfying the hypotheses of Hartley's conjecture. This result was proved in [14] and [15] in the case when F is a finite field. For a group G let G' denote its derived subgroup. The complete result is the following. Theorem 2.1. Let F be any field and G a torsion group. Then U(FG) satisfies a group identity if and only if one of the following conditions holds. (i) If char F = 0, U(FG) satisfies a group identity if and only if G is abelian. (ii) // charf = p > 0, U(FG] satisfies a group identity if and only if G has a normal p-abelian subgroup of finite index, and 1) either G' is a p-group of bounded period (and U(FG) satisfies (x,y)p = 1 for some k > 0). 2) or G has bounded period and F is finite (and U(FG] satisfies xn = 1 for some integer n). One may wonder what kind of units are needed in order to relate a group identity on the units to the structure of the algebra. For torsion groups G the group algebra FG has many nilpotent elements and these ensure the existence of units. In fact, if a is a nilpotent element. 1 + a is a unit. It turns out that at the basis of the proof of the above theorem there is a purely ring theory result relating the nilpotent elements of a ring to the existence of nilpotent or algebraic ideals. The proof of this result was given in [5, Proposition 1] for algebras over an infinite field and modified in [14, Lemma 3.1] to hold for algebras over a finite field. Here we state and give a sketch of this result in the more general setting of algebras over a commutative integral domain. Proposition 2.2. Let R be an algebra over a commutative integral domain C with prime subring P. If U(R) satisfies the group identity w = 1, then there exists a polynomial f ( x ) e P[x], with zero constant term, whose degree d depends only on the word w, such that for all a, b, c € R with a2 — be — 0, we have f(bacR) = 0.

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Moreover, if \C\ > d, then f(x) = xd and bacR is a nil right ideal of bounded exponent. Proof. Since any free group can be embedded in a free group of rank two, we may assume that w — w(xi,X2) is a word in only two variables (see, for example, [20, Theorem 6.1.1]). Suppose for simplicity that charC r= 2. If we make a suitable change of variables (x\ — xy, x% = yx), then we obtain a group identity v = v ( x , y ) = I for CG of the form x^yjl • • • x i k y j k = I where ji,i2 • • • , jk-i, ik £ {±1,±2} and ii,jk G {0,±1}. Since a 2 = be — 0 then, for every r € R, the elements 1 + a and 1 + crb are units of R, Moreover, for any integer m, (1 + a) m = 1 + ma and (1 + crb)m = 1 + mcrb. It follows that, if we now compute 1 = v(l + a, 1 + crb) = (1 + a)*1 (1 + crb)h • • • ( ! + a)ik (1 + crb)jk, we obtain a linear combination of terms of the type (crba)k, a(crba)k, (crba)kcrb and a(crba)kcrb with integral coefficients. Suppose for instance that i\ ^ 0 and jfe ^ 0. Then, by multiplying the above-linear combination on the left by b and on the right by acr we obtain a linear combination of terms of the type (bacr)k, k>l, with integral coefficients. Notice that the term of highest degree has coefficient ±1. Hence we obtain f(bacr) = 0 for some monic polynomial f(x) e P[x] such that /(O) = 0. This proves the first part of the proposition. It is clear that if C has enough elements then for every A e C, the element Aa is still square-zero. Hence a Vandermonde determinant argument proves that bacr is nilpotent of bounded exponent. Thus bacR^is a nil right ideal of bounded exponent. D Notice that in case C is infinite, the above proposition has an immediate application to semiprime rings. In fact, by Levitzki's Theorem ([21, Corollary 1.6.26]), a semiprime ring cannot have nontrivial one-sided ideals nil of bounded exponent, so bac = 0 whenever a, b,c e R are such that a 2 = be = 0. It easily follows that all the idempotents of R are central. In fact, if e € R is an idempotent, then e(l — e) = (1 — e)e = 0 and for all r e R, the elements er(l — e) and (1 — e)re are both square-zero. Hence 0 = eer(l — e}(l~e) = er(l — e) and similarly (l — e)re = 0. Thus er = re. We summarize this in the following Proposition 2.3. Let R be a semiprime algebra over an infinite commutative integral domain C. IfU(R) satisfies a group identity, then all idempotents of R are central. The conclusion of the above result is applied to semiprime group algebras as follows. If F is an infinite field, char F = p > 0 and g € G is a p'-element of order n, then ^(1 + g + • • • + gn~1) is an idempotent. Hence, if FG is a semiprime group algebra whose units satisfy a group identity, then every p'-element generates a normal subgroup. This implies that the p'-elements form an abelian or a Hamiltonian group. It turns out that if G is a torsion group, then G must actually be abelian (see [7, proof of Theorem 1]).

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Once the classification of Theorem 1 is obtained, one wonders if the hypothesis of G being torsion can be removed and still one can obtain a characterization of groups G such that the group of units U(FG) satisfies a group identity. The main obstruction towards this project is the fact that we have very little knowledge of the unit group of the group algebra of a torsion free group. A well known conjecture states that if G is a torsion free group then U(FG) = F* x G i.e., all units of FG are trivial in this case. This conjecture has been proved in case G is a nilpotent group or more generally when G is an ordered group ([22]). Because of this difficulty in constructing units in the torsion free case, it is natural to impose some kind of condition on the torsion free part of G that allows one to get hold of the units. Adopting this point of view in [9] it was possible to obtain a characterization of non torsion groups G such that the unit group U(FG) satisfies a group identity over any field F. In case charF = p > 0 and G is a group, we denote by P the set of p-elements of G and by T = T(G) the set of torsion elements of G. The result is the following. Theorem 2.4. Suppose that G is a group with an element of infinite order and let F be a, field of characteristic p > 0. We have the following a) If U(FG) satisfies a group identity then P is a subgroup. b) // P is of unbounded exponent and U(FG) satisfies a group identity then i) G contains a p-abelian subgroup of finite index. ii) G' is of bounded p-power exponent. Conversely, if P is a subgroup and G satisfies i) and ii) then U(FG) satisfies a group identity. c) If P is of bounded exponent and U(FG) satisfies a group identity then 0) P is finite or G has a p-abelian subgroup of finite index. 1) T(G/P) is an abelian p'-subgroup and so T is a group. 2) Every idempotent of F(G/P) is central. Conversely, if P is a subgroup, G satisfies 0), 1), 2) and G/T is nilpotent then U(FG) satisfies a group identity. 3. Symmetric units and words If one extends linearly the mapping g —> g~l, for all g £ G, to FG then one obtains an involution (an antiautomorphism of order 2) of the group algebra FG, denoted *. This involution has some interesting properties that have been useful in the study of the group of units. Since the case of characteristic 2 is quite different and needs a separate treatment, one usually assumes that F is a field of characteristic different from 2. Also, let FG+ = {x 6 FG \ x = x*} and FG~ = {x € FG x = -x"} denote the sets of symmetric and skew elements of FG, respectively. Notice that, in general, these two sets are not associative subalgebras of FG. Nevertheless, FG+ is a Jordan subalgebra of FG under the Operation a o b — ab + ba and FG~ is a Lie subalgebra of FG under the bracket operation [a, b] = ab — ba. It turns out that Hamiltonian groups play a role in the study of group identities on symmetric units. Recall that a group G is a Hamiltonian group if G is not abelian

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but every subgroup of G is normal. It is well known that Hamiltonian groups are of the form G = O x E x K& where O is an abelian group with every element of odd order, E is an elementary abelian 2-group and /C8 = (a, b | a4 = 1, a2 = b2, bab~l = a~l) is the Quaternion group of order 8. Hamiltonian 2-groups are important when studying the natural involution on a group algebra. In fact it can be easily checked that if B is any commutative ring and G is a Hamiltonian 2-group, then in the group algebra BG all the symmetric elements commute. For a ring with involution R, let U+ (R) denote the set of symmetric units of R. The general result characterizing group algebras of torsion groups whose symmetric units satisfy a group identity was obtained in [8] as follows. Theorem 3.1. Let F be an infinite field and G a torsion group. If charF = 0, U+(FG) satisfies a group identity if and only if G is either abelian or a Hamiltonian 2-group. If char F = p > 2 then U+ (FG] satisfies a group identity if and only if FG satisfies a polynomial identity and either K$ g G and G' is of bounded exponent pK for some k > 0 or Kg C G and 1. the p-elements ofG form a (normal) subgroup PofG and G/P is a Hamiltonian 2-group; 2. G is of bounded exponent 4ps for some s > 0. As in the general case of the previous section, also here there is a ring theory result which is the starting point for the general investigation. This result generalizes the above Proposition 2.2 to rings with involution. In this case one essentially constructs symmetric units out of square-zero elements: if x € R, s 6 R+ and x2 = s2 = 0, then (1 + x)(l + x*) and (1 + s) are symmetric units. With this in mind the previous proposition generalizes to the following [8, Lemma 2] Lemma 3.2. Let R be an algebra over a commutative infinite domain. IfU+(R) satisfies a group identity then there exists a positive integer N > 1 such that 1. ifa&R and a2 = 0, then (aa*)N = 0. 2. I f s , t e R+ are such that s2 — t2 = 0, then (stsd)N - 0 for all d e R+. With some more work one obtains an analogue of the previous proposition. In fact, let R be a semiprime algebra with involution over a commutative infinite domain and suppose that U+(R) satisfies a group identity. Let e = e* = e2 € R be a symmetric idempotent. Since (1 — e)e = 0, for all r G R it follows that (er(l - e))2 = 0. Hence by Lemma 3.2, (er(l - e)r*e)N = 0. This says that (er(l - e)r*)N+l = 0 and by [8, Theorem 1], e is central in R. We have the following. Proposition 3.3. Let R be a semiprime algebra with involution over an infinite commutative integral domain C. IfU+(R) satisfies a group identity, then all symmetric idempotents of R are central.

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The general case when the group G is not a torsion group was considered in [10]. Here we state a characterization in the case of semiprime group algebras. Recall that for a group G, T = T(G) is the set of torsion elements of G. Theorem 3.4. Suppose that FG is a semiprime group algebra where F is an infinite field and G has an element of infinite order. If U+ (FG) satisfies a group identity then 1) all idempotents of FT(G) are central. 2) J/char-F = 0, T(G) is either abelian or a Hamiltonian 2-group. 3) I/charF =p > 2, T(G) is an abelian p'-subgroup ofG. Conversely, if G satisfies 1),2),3) and G/T is nilpotent then U+(FG] satisfies a group identity. For integral group rings it turns out that one can get easily the following result ([8])Corollary 3.5. For a torsion group G the following conditions are equivalent. 1. U+(1,G) satisfies a group identity. 2. C/(Zu) satisfies a group identity. 3. U(LG] is a nilpotent group of class < 2. 4. Group identities on unitary groups In this section, we shall investigate the case when the group of unitary units of a group algebra satisfies a group identity. We start with some general results for finite dimensional algebras from [6]. We recall that a field F is called nonabsolute if either char(f) = 0 or char(F) = p > 0 and F is not algebraic over its prime field. Theorem 4.1. Let R be a semisimple finite dimensional algebra with involution over an algebraically closed field F with char(F) ^ 2. Then (i) Un(R) satisfies a group identity if and only if R" is commutative. (ii) If, furthermore, F is nonabsolute, then Un(R) contains no free group of rank 2 if and only if R" is commutative. Theorem 4.2. Let F be an algebraically closed field which is nonabsolute, with char(F) ^ 2 and let R be a finite dimensional algebra with involution. Then Un(R) does not contain a free group of rank 2 if and only if Un(R) satisfies the group identity (x\, x^)m = 1, for some positive integer m. As we remarked above, since every free group of finite rank can be embedded in a free group of rank 2, every group which satisfies a group identity also satisfies a group identity in two variables. Let w = 1 be a group identity for a group U in two variables. If we make a suitable change of variables one can always assume that w is of the form (1)

w(x,y) =yxeiy~1x>l2yxC3y-'i • • • y r > x e t ,

where ei,...,et e {±1,±2} and 77 e {!,—!}. In fact, if w = w(xi,x2), we first apply the transformation x\ —» yiy?,, £2 —> 2/23/1 where yi,y^ are new variables in

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order to get a word v(yi,y^) ^ 1 in which j/i and y% appear with exponent ±1, ±2. If, say, yi is the leftmost variable appearing in v, then map y\ —> yxy~l and 2/2 —> x where x and y are new variables. In this way we obtain a non-trivial word of the desired form. Notice that if in (1), e i , . . . , et € {±1}, then w has the further property that it does not become trivial when evaluated on elements of order 2. In general we say that a word w(x,y) in the free group on two generators x,y is 2-free if it does not become trivial when evaluated on elements of order 2. This is the same as saying that if w(x,y) = x^y^x^y?2 • • -xlny^ with ji,iz,... ,in > 1, ii > 0, jn > 0, then after reducing the exponents ii,ji,... in,jn modulo 2, w (x, y) does not become the trivial word. Examples of groups satisfying 2-free words are n-Engel groups and nilpotent groups. It turns out that there is a relationship between the behaviour of the Lie algebra of skew elements and the structure of the group of unitary units. For example, we have the following. Theorem 4.3. Let F be a field of characteristic 0. Then (i) If G is finite, then FG~ is Lie nilpotent if and only if Un(FG) contains no free group of rank 2. (ii) // Un(FG) satisfies a group identity which is 2-free, then T = T(G) is a subgroup and FT~ is commutative. The best results so far obtained in case Un(FG) satisfies a group identity are the following. Theorem 4.4. Let F be a field of characteristic 0 and G a torsion group. Then Un(FG) satisfies a 2-free group identity if and only if one of the following conditions holds. (i) G is an abelian group. (ii) A = (g e G o(g) ^ 2} is a normal abelian subgroup of G and (G \ A) 2 = 1. (iii) G contains an elementary abelian 2-subgroup B such that [G : B] = 2. // this is the case, then (x\, x\) = I is a group identity for FG. The above characterization of torsion groups extends to a result about arbitrary groups as follows. Theorem 4.5. Let F be a field of characteristic 0, G any group and T(G) the set of torsion elements ofG. Suppose that Un(FG) satisfies a group identity which is 2-free. Then T(G) is a subgroup satisfying one of the conditions (i), (ii) or (iii) of Theorem 4.4. References [1] J. M. Bateman, On the solvability of unit groups of group algebras, Trans. Amer. Math. Soc. 157 (1971), 73-86. [2] J. M. Bateman and D. B. Coleman, Group algebras with nilpotent unit group, Proc. Amer. Math. Soc. 19 (1968), 448-449. [3] C.L. Chuang and P.H. Lee, Unitary elements in simple Artinian rings, J. Algebra, 176 (1995), 449-459.

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[4] J. L. Fisher, M. M. Parmenter and S. K. Sehgal, Group rings with solvable n-Engel unit group, Proc. Amer. Math. Soc. 59 (1976), 195-200. [5] A. Giambruno, E. Jespers and A. Valenti, Group identities on units of rings, Arch. Math., 63 (1994), 291-296. [6] A. Giambruno and C. Polcino Milies, Unitary units and skew elements in group algebras, (preprint). [7] A. Giambruno, S. K. Sehgal and A. Valenti, Group algebras whose units satisfy a group identity, Proc. Amer. Math. Soc., 125 (1997), 629-634. [8] A. Giambruno, S. K. Sehgal aad A. Valenti, Symmetric units and group identities, Manuscripta Math., 96, (1998), 443-461. [9] A. Giambruno, S. K. Sehgal and A. Valenti, Group identities on units of group algebras, J. Algebra 226 (2000), 488-504. [10] A. Giambruno, S. K. Sehgal and A. Valenti, Symmetric units and group identities in group algebras, (preprint). [11] J. Z. Gongalves, Free subgroups of units in group rings, Canad. Math. Bull. 27 (1984), 309— 312. [12] J. Z. Gongalves and D. S. Passman, Unitary units in group algebras, Israel J. Math., 125 (2001), 131-155. [13] I. I. Khripta, The nilpotence of the multiplicative group of a group ring, Math. Notes 11 (1972), 119-124. [14] C. H. Liu, Group algebras with units satisfying a group identity, Proc. Amer. Math. Soc. 1999, 127, 327-336. [15] C. H. Liu and D. S. Passman, Group algebras whose units satisfy a group identity II, Proc. Amer. Math. Soc., 127 (1999), 337-341. [16] D. S. Passman, The algebraic structure of group rings, Wiley-Interscience, New York, 1977. [17] D. S. Passman, Group algebras whose units satisfy a group identity, Proc. Amer. Math. Soc., 125 (1997), 657-662. [18] V. P. Platoriov, Linear groups with identical relations, (Russian) Dokl. Acad. Nauk. BSSR, 11 (1967), 581-582. [19] C. Polcino Milies, Integral group rings with nilpotent unit group, Canad. J. Math. 28 (1976), 954-960. [20] D. J. S. Robinson, A course in the theory of groups, Springer-Verlag, New York, 1982. [21] L. H. Rowen, Polynomial identities in ring theory, Academic Press, New York, 1980. [22] S. K. Sehgal, Topics in Group Rings, Marcel Dekker, New York, 1978. [23] S. K. Sehgal, Units in Integral Group Rings, Longman Scientific &; Technical Press, Harlow, 1993. [24] S. K. Sehgal and H. J. Zassenhaus, Integral group rings with nilpotent unit groups, Comm. Algebra 5 (1977), 101-111.

Forty Years of Commutative Ring Theory Robert Gilmer Department of Mathematics, Florida State University, Tallahassee, Florida 32306-4510 gilmerSmath.fsu.edu

0. Introduction This paper is based on a 55-minute talk.given at Algebra Conference Venezia 2002. For the six significant developments that are discussed, the paper faithfully represents the substance of that talk. On the other hand, time constraints for the talk permitted little exposition on newly developed subareas or on the problem area of finite element factorization in integral domains, so the paper is an extension of the talk for those topics. The talk was given at a plenary session where a majority of the audience consisted of algebraists who do not work in commutative ring theory. It was planned for such an audience, as is this paper. Forty years of commutative ring theory is a broad subject, covering tens of thousands of pages of published research. Hence many restrictions have been necessary in regard to this paper. Let me mention a couple. First, we focus primarily, but not exclusively, on rings that need not be Noetherian. Second, our concentration is on core cases, and largely omits discussion of extensions/generalizations/ramifications of core problems. The paper includes a selected bibliography; where they exist, it contains references to survey sources that would provide an interested reader with additional detail. In preparing to give the Venice talk I did extensive reading in the relevant literature. That was both enlightening and generally encouraging — encouraging because I found great breadth in the field, that worthy open questions are easy to locate, and that there are a number of good expository articles that would enable a person outside the area to get a good overview of activity and developments in the field. In regard to the last of these, the source with the highest concentration of such articles that I have found is a book Non-Noetherian Commutative Ring Theory published by Kluwer in 2000 [0.1]. It consists of twenty largely-expository articles, followed by 100 open problems if| commutative ring theory. In the sequel, "the Kluwer volume" refers to [0.1]. 229

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In this paper I cite six significant developments in commutative ring theory that have occurred during the last forty years, four subareas whose theories have been developed during the same period, plus an older problem area that has undergone active investigation in new directions during the past twenty years. I claim no superiority for these choices. It seemed advisable to speak/write about subjects of which I am reasonably knowledgeable. There is much in the area that is worthwhile, but where my own acquaintance is quite limited. Without further ado, here are the lists, each in approximate chronological order within the list. Six significant developments 1. 2. 3. 4. 5. 6.

Every abelian group is the (ideal) class group of a Dedekind domain. All sequences realizable as dimension sequences are determined. A basic theory of Krull dimension of power series rings is developed. The integral closure of a local domain need not be catenary. Finitely generated ideals of a Priifer domain may require many generators. Significant progress on the content formula for polynomials.

Four subareas that have been developed

1. 2. 3. 4.

Overrings of -D[-X]. D + M-constructions and pullbacks. Rings of integer-valued polynomials. Commutative monoid rings.

An older problem area with significant investigations in new directions

1. Finite element factorization in integral domains. The three lists above contain eleven topics — six developments, four subareas, and one older problem area. Generally speaking, there has been only a little recent research activity in developments 1, 2, and 4, while there is ongoing work on 3, 5. and 6. Each of the other five topics is currently active, with varying levels of activity, led by rings of integer-valued polynomials and investigations related to finite element factorization. The remainder of the paper consists of eleven sections, one for each topic. Early in each section there is a diagram indicating the limited line of development within the section that is being covered. The horizontal dashed line on each diagram represents the year 1962 — forty years ago. The region above the dashed line represents pre-1962, the area below, after 1962. Throughout this paper all rings are assumed to be commutative and to contain an identity. We use quasilocal ring to refer to a ring with a unique maximal ideal; a local ring is a Noetherian quasilocal ring. The dimension of R, denoted dim R, refers to the Krull dimension of R — that is, to the maximal length of a chain of proper prime ideals of R. If D is an integral domain with quotient field K, an overring of D is a subring of K containing D. Finally, R(X} denotes the quotient ring of the polynomial ring .Rpfj'with respect to the multiplicative system of monic polynomials, while R(X) is the quotient ring of R[X] with respect to the multiplicative system of polynomials whose coefficients generate the unit ideal R.

Forty Years of Commutative Ring Theory

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1. Every abelian group is the class group of a Dedekind domain

i 1

C1(OK), where OK is the ring of algebraic integers of the finite algebraic number field K

!

,

Claborn, 1966 Claborn, 1968 Claborn (Possum), 1973 i

Leedham-Green, 1972 I

Grams, 1974

If D is an integral domain, the (ideal) class group of D, denoted Cl(D), is defined to be Inv(£>)/Prin(D), where Inv(£>) is the multiplicative group of invertible fractional ideals of D and Prin(D) is its subgroup of nonzero principal fractional ideals. Since D is commutative, Cl(D) is abelian. It has long been an object of study in algebraic number theory, especially the case where D = OK is the ring of all algebraic integers of the finite algebraic number field K. In that case OK is a Dedekind domain, and C\(OK) played a central role in, for example, Kummer's work in the nineteenth century on Fermat's Last Theorem. Two fundamental facts from algebraic number theory are that C\(OK) is finite and each of its classes contains infinitely many prime ideals of OKIn a 1966 paper [1.2], Luther Claborn proved the following result. Theorem C (Claborn, 1966). Each abelian group can be realized as the class group of a Dedekind domain. There appears to have been little in the existing literature to have suggested consideration of Theorem C to Claborn. For example, I've found almost nothing in that literature concerning finite abelian groups realizable as C\(OK) for some algebraic number field K. One can speculate, however, on how Claborn was led to ask about abelian groups that are realizable as class groups. In the 1965 paper [1.1], he considered the relation between C1(D) and C\(E), where D is a Dedekind domain and E is an overring of D. From the theory of Dedekind domains it follows that the map / —> IE is a surjective homomorphism from Cl(D) onto Cl(E) with kernel N generated by {P \ P e Spec(£>) and PE = E}. Hence Cl(£) ~ Cl(D)/N

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is a homomorphic image of C1(.D). It is also known that an arbitrary subset of Spec(Z?) can be realized as {P e Spec(£>) | PE = E} for some overring E of D. It follows that if group G is the class group of a Dedekind domain, then so is G/H for each subgroup H of G that can be generated by a set of classes, each representable by a prime ideal. Moreover, Claborn also proved in [1.1] that if D is Dedekind, then D(X) is Dedekind, Cl(D) ~ Cl(D(X)), and each class of Cl(D(X)) contains a prime ideal. Thus, each homomorphic image of the class group of a Dedekind domain is also the class group of a Dedekind domain. Prom this observation it is natural to ask about the status of Theorem C, for a proof of that result follows from its special case for free abelian groups. This is essentially, but not precisely, how Claborn proved Theorem C in [1.3]; the variations being that he considered, more generally, Krull domains and their divisor class groups rather than restricting to Dedekind domains. Theorem C has been used in work on a wide variety of questions concerning overrings of integral domains, frequently to produce examples. But to make the result more useful, Claborn recognized the desirability of specifying more precisely how an abelian group could be realized as the class group of a Dedekind domain. He succeeded in this goal through two results that we refer to as Theorems C* and C**. Before stating C*, note that for D Dedekind, Spec(D) is a free basis for Inv(D). Moreover, the weak approximation theorem for Dedekind domains shows that for any finite subset {Mi}"=1 of Spec(£>) and for any finite set {fc»}"=1 of positive integers, there exists x G D such that for 1 < i < n, Mi occurs with exponent fc, in the factorization of the principal ideal (x) — xD into prime ideals of D. Theorem C* shows that a form of the converse is also true. Theorem C* (Claborn, 1968). Suppose J is the free abelian group (written additively) on a countably infinite set X = {xi}f and I is a subset of J satisfying (i) each coefficient in the expression of an element of I in terms of the basis X is nonnegative, and (ii) for any nonnegative integers fci,...,fcn, there exists an element x €. I such that ki is the coefficient of Xi in the expression of x as a linear combination of elements of the basis X. Then there exists a Dedekind domain D such that Spec(£>) is in bijective correspondence with X in such a way that nonzero principal ideals of D correspond precisely to elements of the subgroup of J generated

by I. Theorem C* has been applied extensively both in the theory of overrings and in questions concerning finite element factorization. Claborn died in a tragic automobile accident in 1967, so Theorem C* was published posthumously. Theorem C** was published five years later in Robert Fossum's book The Divisor Class Group of a Krull Domain [1.4] (see Theorem 15.18, p. 78); Possum indicates that the proof of Theorem C** came from unpublished notes of Claborn supplied to him by Claborn's wife. We refer the reader to [1.4] for the statement of C**; essentially it differs from C* by removing the hypothesis that a basis for J is countable. If G is the class group of a Dedekind domain, one is frequently interested in the classes that contain prime ideals. In a twist on Theorem C*, Grams in [1.5] used

233

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that result to determine equivalent conditions on a subset S of an abelian group J in order for there to exist a Dedekind domain D with class group J such that 5 is the set of classes that contain prime ideals. A form of one of her results states that this occurs if and only if S generates J as a monoid; hence for a torsion group J, iff S generates J as a group. 2. Determination of all sequences realizable as dimension sequences Krull, 1951 Seidenberg, 1953-4 Jaffard, 1960

Theory of the Classical D + M Construction 1965-73

Arnold and Gilmer 1974 Eakin, 1976 Suppose R is a finite-dimensional ring of dimension no. The sequence no, n\ , . . . , nk, . . . , where nk = dim R[XI, . . . , £&], is called the dimension sequence of R and {di}£2j, where di = ni — n^i, is called the difference sequence of R. The following theorem due to Seidenberg [2.7] implies that each n; is an integer and each di is positive. Theorem (Seidenberg, 1953). If dim R — n0 < oo, then n0 + 1 < dimfl[xi] < 2no + 1 and these bounds are sharp: If HQ + 1 < k < 2no + 1, there exists an UQ -dimensional domain D such that dimDxi = k. Krull [2.5] in 1951 had shown that if R is Noetherian, then dim R[x\, . . . , xm] = dim R+m for all m, so R has dimension sequence no, no + 1, no + 2, ... and constant difference sequence 1,1,1,.... Seidenberg In [2.8] showed that Krull's result extends to the case of a Priifer domain R. These results of Krull and Seidenberg represent significant contributions to what came to be known as the dimension sequence

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problem, which is the problem of determining all sequences of nonnegative integers that can be realized as dimension sequences. This problem came to the fore and was enunciated through the 1960 monograph Theorie de la Dimension dans les Anneaux de Polynomes [2.4] of P. Jaffard, who made major contributions toward its solution. Among those contributions, we mention three: 1. Introduction of the notion of the valuative dimension of a domain (extended to a ring) and development of a basic theory of valuative dimension. 2. A proof of the Special Chain Theorem for Prime Ideals. 3. A proof that the difference sequence of R is eventually constant and is bounded above by dim R + 1. Because a connection between the dimension sequence problem and the Special Chain Theorem is apparent, I will state the latter. Jaffard called a chain C : PQ C PI C . . . C Pf. of prime ideals of R[XI, . . . , xn] special if for each Pi in C, the prime ideal (Pi n R) • R[XI, ...,xn] is also in C. Special Chain Theorem (Jaffard, 1960). If R is a finite-dimensional ring and m is a positive integer, then dim R[XI, ... ,xm] is the length of a special chain of prime ideals of R[XI ,..., xm]. Arnold's and my solution of the dimension sequence problem in [2.2] drew on a broad range of results from commutative algebra, including the results of Krull, Seidenberg, and Jaffard that have already been mentioned. But as the preceding diagram indicates, two resources were crucial. One was a 1969 paper [2.1] of Arnold that came from his doctoral dissertation. It contains a formula for dim D[XI, X 2 , . . . , xm] in terms of the dimensions of certain overrings of D1. Theorem A (Arnold, 1969). If D is an integral domain with quotient field K, then dimD[x 1 ) ...,a: n ] = sup{dini£>[ti ) ...,t n ] | {tj? C K}. The other crucial resource was the theory of the classical D + M-construction that had been developed during the period 1965-73, and in particular a 1973 paper [8.1] of Bastida and Gilmer that was devoted to D + M-constructions. There were basically three aspects of the problem of determining the set T> of dimension sequences — determining necessary conditions on elements of T> (Jaffard made significant contributions here), formulation of a conjecture of what sequences constitute T>, and the problem of realizing the conjectured elements of V as dimension sequences. The following theorem summarizes the main results from [2.1] concerning T>. Theorem (Arnold and Gilmer, 1974). Let S be the set of sequences {ni}^0 of nonnegative integers whose associated difference sequence {di}^.l satisfies the following conditions (i) and (ii): (i) n0 + 1 > d\ > ...; (ii) there exists a positive integer k such that 1 < d^ = d^+i = . . . . Let S* be the set of all sequences lT

The proof of Theorem A in [2.1] depended on a preliminary result whose proof contained a gap. That gap was filled by Eakin in [2.3].

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235

of the form sup{si}^.lt where each s, € <5 and where the supremum is taken in the cardinal order. Let T> be the set of all dimension sequences. (1) P C S*. (2) If s = supfsi}^ £ S*, where each s» 6 S, there exists an integrally closed domain D with at most m maximal ideals such that D has dimension sequence s. Hence S* CT> and equality holds: T> = S*. Parker and Bastida-Pomerance independently transformed the description of V in the preceding theorem into purely arithmetic form. These results appear in [2.6]. The characterization of Bastida and Pomerance states that a sequence {ai}^0 of nonnegative integers is a dimension sequence if and only if nan < (n+ l)a n _i + 1 for each positive integer n. 3. A basic theory of Krull dimension of power series rings is developed

Fields, 1970 1 Arnold 1973-82

Coykendall 2000 1

i 1

Several other contributors

i j i

Polynomial rings play a more basic role in commutative algebra than power series rings, and at almost any stage of development their theory has been further advanced. This was, and remains, the situation in regard to the dimension theory of these two classes of rings. Before 1962* there was little in the literature concerning dimJ?[[xi,... ,xn]]; for a field R, however, the dimension was known to be n. Nonetheless, the prominence of power series rings had been steadily increasing. For example, the fifth paper in Krull's illustrious Beitrdge series was devoted to power

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series rings, they played a central role in I. S. Cohen's seminal 1946 paper on the structure and ideal theory of complete local rings, and Nagata frequently used power series rings throughout the 1950's in constructing examples and counterexamples. The dimension theory of power series rings was patterned after the alreadydeveloped theory for polynomial rings, though the two turn out to be quite different. The first paper devoted exclusively to the power series case seems to have been that of D. Fields [3.6]. It includes a proof, pointed out to Fields by W. Heinzer, that dim.R^.X'i,.... Xn]] = dimR + n if R is Noetherian. In considering the case of a Priifer domain, he restricts to the case of a valuation domain and n = 1. In that case, he proved the following theorem. Theorem (Fields, 1970). Let V be a valuation domain of finite rank n. If V is discrete, then 6imV[[X]] — dimV + I , if V is rank-one nondiscrete, then 6imV[[X]] > 3 . The result for V rank-one nondiscrete already represents a variation from what is true for polynomial rings; moreover, Fields was unable (for good reason, it turns out) to prove in this case that V[[X]] is finite-dimensional. Two inclusion relations that are equalities for polynomial rings, but which may be inequalities for power series rings, contribute significantly to differences in the two theories; each of these arises in the case of V[[-X]] just alluded to. They are: (1) The inclusion £>[[X]];v C .Djvjj.X']] may be proper for a multiplicative set N in the domain D. (2) The inclusion / • D[[X}} C I[[X]\ may be proper for an ideal / of D. Jimmy T. Arnold proved a goodly share of what is known today about the dimension theory of power series rings in eight papers that he authored or coauthored during the period 1973-82. One of the significant concepts that Arnold introduced was that of an SFT-ring (SFT abbreviates strong finite type). The definition is as follows. The ideal / of a ring R is an SFT-ideal if there exists a finitely generated ideal J C / and a positive integer k such that xk G J for each x in I; R is an SFT-ring if each ideal of R is an SFT-ideal. In a 1973 paper [3.1], Arnold proved: Theorem AI (Arnold, 1973). 7/dim.R and dim/Z[[X]] are finite, then R is an SFT-ring. Moreover, an SFT-ring that is either von Neumann regular or a one-dimensional Priifer domain is Noetherian. It follows from this theorem that there exist zero-dimensional (von Neumann regular) rings R such that -R[[^"]] is infinite-dimensional. Moreover, V[[X]] is infinite-dimensional if V is a rank-one nondiscrete valuation domain. In a separate 1973 paper [3.2], Arnold investigated questions concerning the Krull dimension of power series rings in one variable over a Priifer domain. The main result of [3.2] is the following theorem. Theorem A 3 (Arnold, 1973). // D is an n-dimensional Priifer domain, the following conditions are equivalent. (1) D is an SFT-ring. (2) D[pf]] is finite-dimensional.

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237

(3) dimD[[X}]=n + l. He subsequently extended results of [3.2] to power series rings in finitely many variables over a Priifer domain, obtaining a reasonably complete theory in this case. In particular he showed in [3.3] that if D is an n-dimensional Priifer domain and an SFT-ring, then dimD[[Xi,..., Xm]] = mn + 1 for each positive integer m. Theorem AI shows that the SFT-property in R is a necessary condition for finite-dimensionality of -RQ-X"]], and A2 shows that for Priifer domains it is also sufficient. The question of whether it is sufficient for any ring R was open for many years, but in 2000 J. Coykendall proved in [3.5]: Theorem (Coykendall, 2000). There exists a one-dimensional SFT-domain V\ such that Vi\[X]\ is neither an SFT-ring nor finite-dimensional. Because the domain V\ is closely related to several topics discussed in this paper, we outline its construction. Let v be a valuation on a field L with value group
finite-dimensional2?

(Q2). If .R[[^]] is finite-dimensional, is its dimension at most 2(dim.R) + 1?

2

We remark that the analogue of (Ql) for unique factorization has been open for more than 30 years.

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4. The chain conjecture is disproved

Krull, 1937, Beitrage III Nagata, 1956

Ogoma, 1980 Heitmann, 1982, 1993

i -| i

Ratliff 1975 (LNM # 647) 1981 (AMM survey)

In regard to the section diagram, the box with the dashed boundary indicates that there are two general resources for this topic, both written by L. J. Ratliff, Jr. One is a monograph in Springer's Lecture Notes in Mathematics series [4.7] and the other is a survey article that appeared in the American Mathematical Monthly [4.8]. While [4.7] is not a survey article, it is written in a style that allows comprehension by nonspecialists. To discuss the Chain Conjecture, we begin with two definitions. A chain PQ < PI < • • • < Pn of prime ideals of a ring R is saturated if no prime ideal of R lies properly between Pj_i and Pi for 1 < i < n. The ring R is said to be catenary if for all primes P < Q of R, all saturated chains of primes between P and Q have the same length. In the third paper in his Beitrage series, Krull proved that an integral domain finitely generated as an algebra over a field — a so-called "finite integral domain" — is catenary. And in his 1946 paper [4.1], Cohen proved that a complete local ring is catenary. Questions concerning the catenary property can usually be reduced to the case of quasilocal integral domains, and most work in the literature on the property restricts to the case of Noetherian rings. After the appearance of Cohen's paper, a question was raised that came to'be known as the chain problem for prime ideals. It asked whether every Noetherian ring (or equivalently, every local domain) is catenary.

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Using power series rings, Nagata in [4.5] gave an example of a non-catenary family of local domains. Each domain E of this family was not integrally closed, and its integral closure was catenary. Thus was the chain problem for prime ideals transformed into the Chain Conjecture, which states: The integral closure of a local domain is catenary. Work on the Chain Conjecture continued over a period of about twenty years, with Ratliff as one of the leading researchers in the area. In fact, there came to be a family of interrelated conjectures that were collectively known as the Catenary Chain Conjectures, and one of the purposes of Ratliff's LNM volume was to present the state of knowledge re the Catenary Chain Conjectures at the time it was written. T. Ogoma in 1980 [4.6] provided a strong counterexample to the Chain Conjecture: A 3-dimensional, Henselian, pseudo-geometric, integrally closed local domain that is not catenary. Ray Heitmann gave a simplified presentation of Ogoma's example in a 1982 paper [4.2]. In a separate context, Heitmann in 1993 gave a characterization of complete local rings that can be realized as the completion of a unique factorization domain (UFD), but his result had implications for all of the Catenary Chain Conjectures. His characterization theorem is: Theorem (Heitmann, 1993). The complete local ring R is the completion of a UFD if and only if either (1) R is a field, (2) R is a rank-one discrete valuation ring, or (3) R has depth at least 2 and no element of its prime subring is a zero divisor in R. By means of this theorem, Heitmann obtains a 3-dimensional catenary local domain A that is not quasi-unmixed. The domain A provides a counterexample to what Ratliff in [4.7] calls the Normal Chain Conjecture (NCC), the weakest of the Catenary Chain Conjectures, so each of these conjectures is false3.

3

Counterexamples to several individual conjectures had been given earlier by other authors, but Heitmann's was the first counterexample to the NCC in the literature.

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5. Finitely generated ideals of Priifer domains may require many generators

I

Priifer, 1932

Sally & Vasconcelos, 1974 Heitmann, 1976 1 ' Schiilting, 1979 I Swan, 1984

] i 1

'

1^

, | ^•'i

Roquette A. Dress, Becker

-, i '

i _ | " ~-r ~ kucharz,~1989, f991~

Many omitted contributors

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A Priifer domairi is a domain in which each nonzero finitely generated ideal is invertible. Equivalently, D is a Priifer domain if Dp is a valuation domain for each P S Spec(.D). Priifer domains have been extensively studied. Many characterizations — probably 100 or more — of Priifer domains are known. One reason for the central role they play is that they arise in many and diverse contexts4. The domains now called Priifer domains5 were first considered by Heinz Priifer in a classical 1932 paper [5.7]. In current terminology, Priifer's paper deals with various star operations on the ideals of an integral domain and with properties of their associated systems of star ideals. In Priifer's terminology, a Priifer domain was called a domain with property £B; here L refers to a particular system of ideals and B refers to a property of that system. Relevant to the topic of this section, Priifer proved: 4 A personal anecdote. I once had a student who repeatedly asked me what multiplicative ideal theory was. Initially I gave him the same answer each time he asked. When he persisted in asking, I began to rephrase the answer in attempts to reach him. Finally (he asked no more), one day in exasperation I gave him, approximately, the following inaccurate response. "Look at a paper. If you find the term 'Priifer domain' in it, it's pultiplicative ideal theory. Otherwise, it isn't." Inaccurate, but not completely offbase. 5 The term Priifer ring seems to have been first applied to such domains in Cartan and Eilenberg's book [5.1, p. 133].

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Theorem (Priifer, 1932). The domain D is a Prufer domain if each nonzero 2-generated ideal of D is invertible. Based on this theorem and the well-known fact that each ideal of a Dedekind domain (which is the same as a Noetherian Prufer domain) is 2-generated, the following question was raised6 in the early 1960's. Question. Does each Prufer domain have the 2-generator property? By 1970 there were several results in the literature to suggest this question might have an affirmative answer, but the first positive result in that direction appeared in a 1974 paper [5.9] of Sally and Vasconcelos. They proved that a one-dimensional Prufer domain has the 2-generator property. Heitmann [5.4] then proved a broad generalization of their result: Theorem (Heitmann, 1976). An invertible ideal in an n-dimensional domain can be generated by n + 1 elements. In particular an n-dimensional Prufer domain has the (n + I)-generator property. Using results of Roquette, A. Dress, Becker and others on real holomorphy rings, H.-W. Schiilting in 1979 [5.10] proved that the real holomorphy ring of the field R(X, Y), R the real field, is a two-dimensional Prufer domain with a finitely generated ideal requiring 3 generators. To review the definition, a field F is formally real if —1 is not a sum of squares in F. If K is a formally real field, the real holomorphy ring H(K) can be defined as the intersection of the family of all valuation domains V on K such that the residue field of V is formally real; alternately, H(K) is the subring of K generated by {s~l \ s 6 K, s ^= 0, and s is a sum of squares in K} [5.8]. The domain H(K) is a Prufer domain and its class group is an elementary abelian 2-group since,(ai, 02, • • • , an)2 = (a\ + a% + • • • + a^) f°r a-^ ai *=• &• After Schiilting's paper appeared one could ask whether every Prufer domain has the 3-generator property. In the opposite direction, one could ask whether the bounds given in Heitmann's theorem are best possible for Prufer domains. General sentiment was in the direction of the latter, and R. Swan established this fact in a 1984 paper [5.11] through the following theorem. Theorem S (Swan, 1984). For each n £ J.+, there exists an n-dimensional Prufer domain Dn with a finitely generated ideal requiring n + l generators. There exists an infinite-dimensional Prufer domain D such that there is no bound on the number of generators required for all finitely generated ideals of D. Though Swan did not view the domains he constructed as real holomorphy rings, it is interesting to note that Dn is, in fact, -H"(R({Xj}™T11) and that D = H ( R ( { X i } f ) ) . Schiilting or others familiar with his work may have asked whether, for example, the fractional ideal (1, A"i,... ,Xn} of H(R(Xi,... ,Xn)) requires n + 1 generators, but it was not until after the appearance of Swan's paper that W. Kucharz [5.5] involved real holomorphy rings in a proof that this is the case. Swan's paper uses instead many tools, mostly topological, that are outside those 6

My name is associated with the question, but the first place I've found the question in writing is in a 1970 paper [5.3] with W. Heinzer.

R. Gilmer

242

usually employed in commutative ring theory. This is also true of Kucharz's alternate proof of Theorem S in [5.5]. Activity in the area of generating sets for finitely generated ideals of Priifer domains continues today. It is generally directed toward one of two goals: (a) a more ring-theoretic proof of Theorem S, or (b) a determination of the minimum number of generators required to generate all finitely generated ideals of either a particular Priifer domain D or all Priifer domains of a given class. 6. Significant progress on the content formula for polynomials

Gauss, Kronecker, Dedekind, Hurwitz, Mertens, Priifer, Northcott, . . .

Tsang , 1965

1

•i 1

Anderson & Kang, 1996

Much activity related to the content formula

i '

Glaz & Vasconcelos, 1997-E

Heinzer & Huneke, 1997-98

Work on contents of polynomials and power series is a large area with many extensions/generalizations/ramifications. We focus here on a specific problem, the best-known problem in the area. There is, however, an excellent survey source in this case on the larger area; that is D. D. Anderson's article [6.1] in the Kluwer volume, particularly Section 8 of [6.1]. Moreover, [6.3] is a more detailed survey of the content formula problem than the one we present here. Suppose/ € R[x}. The content of/, denoted c(/), is the ideal of-R generated by the coefficients of /. It is easy to see that c(fg) C c(f)c(g] for all f , g £ R{x\. This inclusion may be proper. While the diagram above lists seven names associated with work in this area before 1962, some are included because of the larger problem area, and not because of work directly related to the specific question under consideration. For example, Gauss is listed because of Gauss's Lemma, which may be stated as "the content of the product of two polynomials over a UFD is the product of their

Forty Years of Commutative Ring Theory

243

contents", but in that statement, "the content of /" means the greatest common divisor of the coefficients of /, which is not what content refers to in the definition above. There is a connection between the two involving system of ideals, but that is a digression for us. The main result proved before 1962 that has relevance for the content formula problem is what Krull in his 1935 monograph Idealtheorie calls the Hilfssatz von Dedekind-Mertens: Dedekind-Mertens Lemma. For f , g € R[X], there exists a positive integer n such that c(fg}c(g)n = c(f)c(g)n+l. There are many different versions of this lemma in the literature. Both [6.7, p. 1306] and [6.1, Sect. 8] contain interesting commentary on these versions and their history. Some versions specify n, with different versions specifying different values. With n unspecified, as above, the conclusion of the Dedekind-Mertens Lemma is valid for polynomials in several variables over R. A corollary to the Lemma is the following. Corollary. If c(g) is invertible, then c(fg) = c(f)c(g)

for each f in R[X].

The content formula problem asks about the converse — that is, if g ^ 0 and (/ff) = c(f}c(g) f°r each / € R[x], must c(g) be invertible? In that generality one can easily give a negative answer and the question will soon be refined. The question comes from the 1965 doctoral dissertation, titled Gauss's Lemma, of Hwa Tsang. Her degree is from the University of Chicago, and I. Kaplansky was her thesis advisor. Her thesis was not published; she published a paper entitled Gauss's Lemma in 1972, but there is little overlap between material in that paper and that in her thesis. In her dissertation Tsang writes that Kaplansky was the source of all its questions, so we may attribute the (appropriate modification of) the above question to him. Tsang called a nonzero polynomial g & R[x] Gaussian if c(fg] = c(f)c(g) for each / 6 -R[-X]. Her thesis contains the following conjecture. c

Conjecture KT/GV. If D is an integral domain and g 6 D[x] is Gaussian, then c(g) is invertible. In naming the conjecture KT/GV, K and T stand for Kaplansky and Tsang, while G and V stand for Glaz and Vasconcelos, who repeated the conjecture in their 1998 paper [6.5]. Glaz and Vasconcelos considered primarily integral domains. They used Hilbert functions and the theory of prestable ideals to prove: Theorem (Glaz-Vasconcelos, 1998). Conjecture KT/GV is correct if either (1) D is integrally closed of nonzero characteristic, (2) D is Noetherian and integrally closed, or (3) D has the 2-generator property. There has traditionally been some inconsistency in regard to ring vs. domain in the content formula problem. It was so from the beginning: in much of [6.8], the coefficient ring is allowed to contain zero divisors, but Conjecture KT/GV posits an integral domain as the coefficient ring. In [6.1], Anderson states Kaplansky's question as follows. Question K. If g 6 R[x] is Gaussian and c(g) is a regular ideal of R, is c(g) invertible?

R. Gilmer

244

Here "regular ideal" means an ideal containing a regular element — that is, an element that is not a zero divisor. It is known that the regularity hypothesis is necessary; that is the refinement needed in the earlier phrasing of the converse of the Dedekind-Mertens Lemma. For example, if (R, M) is a zero-dimensional quasilocal ring with M2 = (0), each nonzero polynomial over R is Gaussian, but no element of M[x\ has invertible content. A finitely generated regular ideal is invertible if and only if it is locally principal. Hence Question K can be reduced to the case where R is quasilocal; the end of the question then becomes "is c(g) principal?". Much of the work that has been done on Question K has dealt with this reduction. Inspired by work of Glaz and Vasconcelos, Heinzer and Huneke considered Question K and several variants thereon. Focusing here on the relation of their work to K, they used the concept of an approximately Gorenstein local ring to prove: Theorem (Heinzer-Huneke, 1997). Question K has an affirmative is locally Noetherian, hence in particular if R is Noetherian.

answer if R

The status of Conjecture KT/GV and Question K remain open. There is no apparent reduction of K to the domain case, so the conjecture could be correct while Question K has a negative answer.

7. Overrings of D[X]

Kronecker Function Rings

D(X)

D(X)

Composites A + XB[X]

In fact, there is not enough cohesiveness within the topic of overrings of D[X] to call this a subarea, though there are subtopics (for example, rings of integer-valued polynomials, which will be treated separately) where cohesion exists. Nevertheless, there has been much work during our 40-year window that fits beneath an umbrella of overrings of D [X].

Forty Years of Commutative Ring Theory

245

Before 1962 the theory of Kronecker function rings was developed by Krull in Beitrage II. Without going into details, these rings were a transfer mechanism for passing from an integrally closed domain D to a Prufer domain D*, an overring of -D[-X"], in such a way that certain valuation overrings of D are in bijective correspondence with a subset of Spec(Z?*). Further development of the theory of Kronecker function rings continued until the mid-1970's. One notable achievement here was the extension of the concept to rings with zero divisors, replacing valuation domains with Manis valuation rings; J. Huckaba and some of his students were major players in that extension. The domain R(X) was denned in the introduction. Krull considered this ring in Section 1 of Beitrage VIII, but the notation R(X) was introduced by Nagata in his 1962 book Local Rings. For that reason, R(X) is sometimes referred to in the literature as "the Nagata ring". Again R(X) serves as a transfer mechanism and it serves well in this role: each ideal of R is the contraction of its extension to R(X), prime and primary ideals of R extend to prime and primary ideals of R(X), and the maximal ideals of R(X) are precisely the extensions to R(X) of maximal ideals of R. Moreover, all residue fields of R(X) are infinite and the indeterminate X is frequently useful in constructing elements of R(X) with desired properties. Because of its significant transfer role, R(X) has been thoroughly investigated as a ring in its own right. The ring R(X) was also defined in the introduction, and its role in Claborn's proof that each abelian group is the class group of a Dedekind domain has been noted. There are also good transfer properties between R and R(X), though not as sharp as for R(X). Interest in the ring R(X) surged after the appearance of Quillen's 1976 paper which gave a proof of Serre's Conjecture. His proof made prominent use of Hoirock's Theorem, which can be stated as follows. Horrock's Theorem. Let R be a quasilocal ring, let M be a finitely generated projective R[X]-module, and let N be the multiplicative set of monic polynomials over R. If MN is free as an R(X)-module, then M is free as an R[X]-module. The rings R(X) and R(X) are often studied in tandem in current research. The paper [7.2] is a survey of work on certain other types of overrings of D[X].

246

R- Gilmer

8. D + M-constructions and pullbacks For this section a few definitions are needed before displaying its diagram. Suppose V is a valuation domain of the form K + M, where K is a field and M is the maximal ideal of V. If D is a subring of V, the domain D\ = D + M is called a (classical) D + M-construction. Without the assumption that V is a valuation domain — or even quasilocal — but retaining the assumption that M is a maximal ideal of V, D\ is a generalized D + M-construction. For each of these, the domain DI is a special case of a pullback construction, defined as follows. Let I be an ideal of a ring S, let

S/I be the canonical map, and let R be a subring of S/I. The subring ip~1(K) of S is called the pullback of R along (p. Classical D+M Krull, 1936 Seidenberg, 1953

Generalized D +M Priifer, 1932 Nagata, 1956

Pullbacks Zariski-Samuel Volume II, 1960

Gilmer-Ohm, 1965 i Extensive theory i ] developed 1965-76 ] i and beyond i Brewer-Rutter, 1976 Costa-Mott-Zafrullah, 1978 1978-80 Hedstrom-Houston, Anderson-Dobbs, Fontana Articles by Gabelli-Houston and by Lucas in the Kluwer volume

In the diagram above, Krull's use of a classical Z?+M-construction in Beitrage II is veiled; he presents an example of a one-dimensional quasilocal integrally closed domain that is not a valuation domain. The example can be viewed as K + K(X)[Y](Y), but that is not the way Krull presents it. Each of Seidenberg, Priifer, and Nagata also used D + M-constructions to produce examples. (Nagata's examples provided a negative answer to the chain problem for prime ideals that

Forty Years of Commutative Ring Theory

247

was mentioned in Section 4.) On the other hand, a limited theory of pullbacks is developed in Commutative Algebra, Vol. II [8.6] as a part of Zariski-SamueFs presentation of valuation theory. The concept of a valuation ideal is considered in Appendix 3 of [8.6]. Here the ideal J of a domain D is a valuation ideal of D if I is contracted from a valuation overring of D. If D is a Priifer domain, each primary ideal of D is a valuation ideal. In 1963 J. Ohm and I sought to determine whether this property characterizes Priifer domains. With the additional hypothesis that D satisfies a.c.c. on prime ideals, we were able to show that D is Priifer. However, we used a classical D + M-construction to show that in the absence of a.c.c. for prime ideals, a domain in which primary ideals are valuation ideals need not be a Priifer domain. This work was published in [8.4] in 1965. The significant aspect of this paper is that its Section 5 is devoted not just to a presentation of some counterexamples, but also contains several theorems that fall within the framework of a general theory of classical D + M-constructions. This theory was extensively developed 1965-76, and further development has continued to the present. As has already been suggested, D + M-constructions are often used to provide examples. One advantage they have in this regard is that they are tractable. In case a counterexample to a question can be achieved via a classical D + M-construction, it may well be the most accessible counterexample available. The generalized D + M-construction was initially investigated in depth in a 1976 paper [8.2] of Brewer and Rutter. Some of the earlier theory of the classical construction carries over to this case, but the generalized construction provides greater flexibility and admits examples not achievable through its counterpart. The fact that D+M-constructions are a special type of pullback was recognized at an early stage. However, because the additional generality that pullbacks provide is frequently unnecessary in constructing examples, they were not studied separately in detail until around 1978. There are differences of significance between the theory of general pullbacks and of D + M-constructions. Articles by S. Gabelli and E. Houston [8.3] and by T. Lucas [8.5] in the Kluwer volume are good sources for additional information on this subarea.

R. Gilmer

248

9. Rings of integer-valued polynomials Polya, 1919 Ostrowski, 1919

Skolem, 1936

Cahen and Chabert, 1971 many other contributors

We begin with two definitions. If D is an integral domain with quotient field K, then {/ e K[x] \ f ( x ) £ D Va e D} is called the ring of integer-valued polynomials on D. It is denoted by Int(D). More generally, if E is a nonempty subset of D, then Int(E,D) is defined to be {/ e K[x\ \ f ( E ) C D}. Thus Int(D) = Int(D,£>). For any E1, Int(E,.D) is an overring of D[X], and its intersection with K is I?. It has been known essentially since the 17th century that, in current terminology, Int(Z) is free as an abelian group and the set {(^)}^0 ig a basis for Int(Z). The study of integer-valued polynomials in their own right stems from papers of G. Polya and of A. Ostrowski published in 1919. These papers were concerned with Int(£?#•), where OK is the ring of integers of the finite algebraic number field K. Primary emphasis was on Int(Ojf) as an O^-module, whether it is free, and if so, whether it admits a basis {fi}°^0 with deg/j = i for each i. T. Skolem was first to consider Int(D) as a ring. He did so in a 1936 paper. To describe his result, we use the following notation: If J is an ideal of Int(£>) and a & D, then /(a) = {/(a) | / e /}. Skolem proved that if / is a finitely generated ideal of Int(Z) such that /(a) = Z for each a e Z, then / = Int(Z) 7 . This led to subsequent investigations into what were referred to as Skolem rings and strong Skolem rings. The definitions are as follows. The domain D is a Skolem ring if lnt(D) is the only finitely generated ideal / of Int(.D) with /(a) = D for each a E D; D is a strong Skolem ring if the equality /(a) — J(a) for each a 6 D implies / — J for all finitely generated ideals /, J of Int(.D). Thus, Skolem's result cited above can be rephrased as: Z is a Skolem ring. In 1979 D. Brizolis proved that Z is, in fact, a strong Skolem ring. Current work on rings of integer-valued polynomials has its origin in papers of P.-J. Cahen and J.-L. Chabert published during the period 1971-73. Cahen and Chabert worked under the direction of P. Samuel, with 1973 theses from Universite Paris XI, Orsay. They have continued to be leading figures in research in this area 7

In fact, he proved this result for integer-valued polynomials in finitely many variables over 2.

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and have coauthored an American Mathematical Society monograph [9.2] on the subject. Historically, the following is a list of some questions/problems that have been considered in regard to integer-valued polynomials. 1. Under what conditions a. does Int(D) properly contain Z?[X]? b. is Int(Ds) = Int(D)s for a multiplicative system S in D? c. does D have the (strong) Skolem property? d. is Int(£>) a Priifer domain, and if Int(D) is Priifer, does it have the 2-generator property? 2. Describe Spec(Int(D)). In particular, what is dim(Int(D))? My own estimation is that there is relative satisfaction within the field with the current state of knowledge re la,b,d while one hopes for further progress on Ic and 2. In regard to la, if E — r\{DP \ P 6 Spec(£>) and D/P is infinite}, then Int(D) C -E[X]. Hence one usually assumes that D has enough finite residue fields to assure that the inclusion D C E is proper. One important case where equality holds in Ib is that where D is Noetherian. In regard to dim(Int(D)), the situation is reminiscent of that for power series rings at the time Fields did his work. It is known that max{dim.P[.X"] — l,dim.D + 1} < dim(Int(Z))) and that dim(Int(D)) = dim£>[Jf] = dini-D + 1 if the dimension of D is equal to its valuative dimension (hence if D is Noetherian or a Priifer domain). No example is known where dim(Int(.D)) > dim(£>[Jf]), but neither is there a proof that Int(D) is finite-dimensional if D is finite-dimensional. The paper [9.4] is a good source for information on the Krull dimension of rings of integer-valued polynomials. Recent work in this area has focused more on the rings Int(.E, D) than on Int(.D), including cases where E is not a subset of D (see [9.3]). The notion of a P-ordering, introduced by M. Bhargava in [9.1], has led to advances in the state of knowledge concerning the additive structure of the rings lnt(E, D).

250

R. Gilmer

10. Commutative monoid rings

Banaschewski, 1961

Cohn-Livmgstone _ f, Connell

rn ., 0

„_ 1963-67

„ Berman

Parker, 1973-75

The mathematical literature before 1962 contains only fragmentary results concerning commutative monoid rings. These usually appeared in another context or as part of a larger investigation. There were a few exceptions. G. Higrnan published a well-known paper in 1940 on units of group rings. Hewitt and Zuckerman published two papers in 1954-55 in which semigroup rings had a nontrivial role, and Banaschewski in'1961 investigated conditions under which a semigroup ring has no zero divisors. None of this work was restricted to the commutative case. During the period 1963-67 there was activity on the topic of group rings. This included papers of Cohn and Livingstone, Connell, and Berman. Of these, Berman restricted to the commutative case, while the others did not. During this same time period, Budach, Drbohlav, Redei, and others developed a theory of finitely generated commutative semigroups that was later seen to have applications to Noetherian semigroup rings. Commutative monoid rings as an area of investigation began in earnest with the 1973 dissertation of T. Parker8. It was concerned with nilpotent elements, the nil and Jacobson radicals, and zero divisors of commutative semigroup rings. A polynomial ring over Ris a, monoid ring over R, and initially there was a tendency 8

Another anecdote. Parker was my doctoral student. Early in 1972 he was ready to choose a thesis topic. He was free to choose his own topic or to have me suggest one. He chose the latter, but indicated that he preferred a problem in field theory. He accepted an ill-defined problem of developing a useful concept of normality for transcendental field extensions. After working on this for 2-3 weeks, he had a change of mind, saying that instead he wished to work on commutative semigroup rings. I was mystified by his choice because there was precious little on the topic in the literature at the time, and his only exposure from coursework that I recalled were a couple of exercises from D. G. Northcott's book Lessons on Rings, Modules, and Multiplicities.

251

Forty Years of Commutative Ring Theory

to consider for monoid rings questions that had already been investigated for polynomial rings. Almost always there was more richness (and difficulty) in the case of general monoid rings since the monoid associated with a polynomial ring is torsionfree and cancellative. For many ring-theoretic properties E, equivalent conditions are known for a monoid ring to have property E. Among properties where such a characterization theorem was challenging, we mention the Noetherian property, unique factorization, and the property of being an arithmetical ring. The problem of determining equivalent conditions for a noncommutative semigroup ring to be (say) right Noetherian remains open. 11. Finite element factorization in integral domains

Narkiewicz, Czogola, Sliwa, . . . 1964-88

Zaks, 1976, 80

Factorization in rings of algebraic integers

Factorization in integral domains i 1

Many contributors and contributions

i '

Finite element factorization has held an important position in commutative algebra since the inception of the subject. Our focus in this section is not on the broad area, but rather on some new directions of investigation in the area that developed from a 1960 paper of L. Carlitz. In that paper Carlitz proved the following theorem. Theorem (Carlitz). Let OK be the ring of integers of the finite algebraic number field K. Then K has class number 1 or 2 if and only if, in OK, 3/13/2 • • - 3/m = z\z-i • • • zn with yi, zt irreducible implies ~ni — n. This theorem represents an arithmetic characterization of finite algebraic number fields of class number at most two. Soon efforts were underway to extend the

252

R. Gilmer

characterization to other number fields with small class number or with specified class group of small order. W. Narkiewicz was a prominent figure in this extension effort. Generally speaking, if K is a finite algebraic number field of class number h > 2 and if OK is the ring of integers in K, then one loses uniqueness of factorization into finite products of irreducible in significant ways. To elaborate, for a nonzero nonunit a e OK, let L(a) = {M 6 2+ a is a product of m irreducible elements in OK}- Sliwa in 1982 proved that for each positive integer m, there exists a 6 OK such that |L(a)| = TO. On the other hand, Geroldinger in 1988 proved that for almost all elements a € OK (for an appropriate density function), there exist k, n 6 / + such that L(a) = {n, n + 1,..., n + k — I } . In a 1980 paper [11.4], A. Zaks abstracted the condition in Carlitz's theorem and investigated it for arbitrary integral domains. To wit, Zaks used the term half-factorial domain (abbreviated HFD) for an integral domain D in which each nonzero nonunit is a finite product of irreducible elements (that is, D is atomic) and any two factorizations of a fixed element into irreducibles have the same number of factors. Any UFD is an HFD, but not conversely. Among results proved by Zaks was the following theorem. Theorem (Zaks, 1980). Let D be a Krull domain with divisor class group C(D). (a) // |C(£>)| < 2, then D is an HFD. (b) D[X] is an HFD if and only if \C(D)\ < 2. Zaks uses Claborn's Theorem C* to give examples of Dedekind HFD's and nonHFD's with various properties. In particular he shows that for each prime q there exists a Dedekind HFD with class group of order q. The topic of factorization in integral domains has been very active since the appearance of Zaks's paper. A problem of basic concern has been that of measuring how far a given domain is from being an HFD. The term elasticity has been applied to such a measure; it is denoted by p(D) and is defined to be sup{m/n | J/iJ/2 • • • J/m = ziZ2---zn for yi, Zj € D irreducible}. An HFD has elasticity 1. To some degree work in the other branch of the diagram above serves as a model for the investigation in arbitrary domains, and Claborn's work is often used in constructing pertinent examples. S. Chapman has pointed out that many questions involving finite element factorization in integral domains turn out to actually be problems in additive number theory or combinatorial problems on certain types of arithmetically defined semigroups. Their solutions are likely to involve very little ring theory. Two questions in the area where a more ring-theoretic solution is to be expected are these: (Ql) (Zaks, 1980). Is every abeiian group the class group of a Dedekind domain that is an HFD? (Q2). Which HFD's are such that their integral closure is also an HFD? There has been recent progress on (Ql) by Geroldinger and Gobel, who proved that every Warfield group is the class group of a Dedekind HFD. In regard to (Q2), Coykendall has shown that not every HFD is such that its integral closure is also an HFD.

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There are several good survey sources on finite element factorization in integral domains in the literature. One is a 1997 article on elasticity by David Anderson in [11.1], and the Kluwer volume contains two such articles — one by Chapman and Coykendall [11.2], the other by Chapman, Freeze, and Smith [11.3]. Acknowledgement The author acknowledges with thanks input and/or other assistance he has received from Paul-Jean Cahen, Jean-Luc Chabert, Scott Chapman, Jim Coykendall, Robert Fossum, and Bill Smith in the preparation of this paper and/or the Venice talk on which it is based. Selected Bibliography 0. General 0.1 Non-Noetherian Commutative Ring Theory, edited by S. Chapman and S. Glaz, Kluwer, Dordrecht, 2000. 1. Every abelian group is a class group

1.1 L. Claborn, Dedekind domains and rings of quotients, Pacific J. Math. 15 (1965), 59-64. 1.2 , Every abelian group is a class group, Pacific J. Math. 18 (1966), 219-222. 1.3 , Specified relations in the ideal group, Michigan Math. J. 15 (1968), 249-255. 1.4 R. Fossum, The Divisor Class Group of a Krull Domain, Springer-Verlag, New York, 1973. 1.5 A. P. Grams, The distribution of prime ideals of a Dedekind domain, Bull. Austral. Math. Soc. 11 (1974), 429-441. 1.6 C. R. Leedham-Green, The class group of Dedekind domains, Trans. Amer. Math. Soc. 163 (1972), 493-500. 2. Dimension sequences

2.1 J. T. Arnold, On the dimension theory of overrings of an integral domain, Trans. Amer. Math. Soc. 138 (1969), 313-326. 2.2 J. T. Arnold and R. Gilmer, The dimension sequence of a commutative ring, Amer. J. Math. 96 (1974), 385-408. 2.3 P. Eakin, On Arnold's formula for the dimension of a polynomial ring, Proc. Amer. Math. Soc. 54 (1976), 11-15. 2.4 P. Jaffard, Theorie de la Dimension dans les Anneaux de Polynomes, Gauthier-Villars, Paris, 1960. 2.5 W. Krull, Jacobsonsche Ringe, Hilbertsche Nullstellensatz, Dimensionstheorie, Math. Ann. 54 (1951), 354-387. 2.6 T. Parker, A number-theoretic characterization of dimension sequences, Amer. J. Math. 97 (1975), 308-311. 2.7 A. Seidenberg, A note on the dimension theory of rings, Pacific J. Math. 3 (1953), 505-512. 2.8 , A note on the dimension theory of rings. II., Pacific J. Math. 4 (1954), 603614.

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3. Krull dimension of power series 3.1 J. T. Arnold, Krull dimension in power series rings, Trans. Amer. Math. Soc. 177 (1973), 299-304. 3.2 , Power series rings over Priifer domains, Pacific J. Math. 44 (1973), 1-11. 3.3 , Power series rings with finite Krull dimension, Indiana Univ. Math. J. 31 (1982), 897-911. 3.4 J. W. Brewer, Power Series over Commutative Rings, Led. Notes Pure Appl. Math. Vol. 64, Marcel Dekker, New York, 1981. 3.5 J. Coykendall, The SFT-property does not imply finite dimension for power series rings, J. Algebra (to appear). 3.6 D. E. Fields, Dimension theory of power series rings, Pacific J. Math. 35 (1970), 601-611. 3.7 B. G. Kang and M. H. Park, A localization of a power series ring over a valuation ring, J. Pure Appl. Algebra 140 (1999), 107-124. 4. The chain conjecture 4.1 I. S. Cohen, On the structure and ideal theory of complete local rings, Trans. Amer. Math. Soc. 59 (1946), 54-106. 4.2 R. C. Heitmann, A non-catenary, normal, local domain, Rocky Mountain J. Math. 12 (1982), 145-148. 4.3 , Characterization of completions of unique factorization domains, Trans. Amer. Math. Soc. 337 (1993), 379-387. 4.4 W. Krull, Beitrdge zur Arithmetik kommutativer Integritatsbereiche. III. Zum Dimensionsbegriff der Idealtheorie, Math. Z. 42 (1937), 745-766. 4.5 M. Nagata, On the chain problem of prime ideals, Nagoya Math. J. 10 (1956), 51-64. 4.6 T. Ogoma, Non-catenary pseudo-geometric normal rings, Japan. J. Math. 6 (1980), 147-163. 4.7 L. J. Ratliff, Jr., Chain Conjectures in Ring Theory, Lect. Notes Math. No. 647, Springer-Verlag, New York, 1978. 4.8 , A brief history and survey of the catenary chain conjectures, Amer. Math. Monthly 88 (1981), 169-178. 5. The two-generator property in Priifer domains 5.1 H. Cartan and S. Eilenberg, Homological Algebra, Princeton Univ. Press, Princeton, 1956. 5.2 M. Fontana, J. A. Huckaba, and I. J. Papick, Priifer Domains, Marcel Dekker, New York, 1997. 5.3 R. Gilmer and W. Heinzer, On the number of generators of an invertible ideal, J. Algebra 14 (1970), 139-151. 5.4 R. C. Heitmann, Generating ideals in Priifer domains, Pacific J. Math. 62 (1976), 117-126. 5.5 W. Kucharz, Invertible ideals in real holomorphy rings, J. P^eine Angew. Math. 395 (1989), 171-185. 5.6 , Generating ideals in real holomorphy rings, J. Algebra 144 (1991), 1-7. 5.7 H. Priifer, Untersuchungen iiber Teilbarkeftseigenschaften in Korpern, J. Reine Angew. Math. 68 (1932), 1-36. 5.8 P. Roquette, Principal ideal theorems for polynomial rings in fields, J. Reine Angew. Math. 262/263 (1973), 361-372.

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5.9 J. Sally and W. Vasconcelos, Stable rings, J. Pure Appl. Algebra 4 (1974), 319-336. 5.10 H.-W. Schiilting, Uber die Erzeugendenanzahl invertierbarer Ideale in Priiferringen, Comm. Algebra 7 (1979), 1331-1349. 5.11 R. Swan, n-generator ideals in Priifer domains, Pacific J. Math. Ill (1984), 433-446. 6. The content formula for polynomials 6.1 D. D. Anderson, GCD-domains, Gauss's Lemma, and contents of polynomials, the Kluwer volume, pp. 1—31. 6.2 D. D. Anderson and B. G. Kang, Content formulas for polynomials and power series and complete integral closure, J. Algebra 181 (1996), 82-94. 6.3 A. Corso and S. Glaz, Gaussian ideals and the Dedekind-Mertens Lemma, Lect. Notes Pure Appl. Math. Vol. 217, Marcel Dekker, New York, 2001, pp. 131-143. 6.4 S. Glaz and W. Vasconcelos, Gaussian polynomials, Lect. Notes Pure Appl. Math. Vol. 186, Marcel Dekker, New York, 1997, pp. 325-337. 6.5 , The content of Gaussian polynomials, J. Algebra 202 (1998), 1-9. 6.6 W. Heinzer and C. Huneke, Gaussian polynomials and content ideals, Proc. Amer. Math. Soc. 125 (1997), 739-745. 6.7 , The Dedekind-Mertens Lemma and the contents of polynomials, Proc. Amer. Math. Soc. 126 (1998), 1305-1309. 6.8 H. Tsang, Gauss's Lemma, Ph.D. dissertation, Univ. Chicago, 1965. 7. Overrings of D [X] 7.1 M. Fontana and A. Loper, Kronecker function rings: a general approach, Lect. Notes Pure Appl. Math. Vol. 220, Marcel Dekker, New York, 2001, pp. 189-205. 7.2 M. Zafrullah, Various facets of rings between D[X] and K[X], Lect. Notes Pure Appl. Math. Vol. 231, Marcel Dekker, New York, 2002, pp. 445-460. 8. D + M-constructions and pullbacks i 8.1 E. Bastida and R. Gilmer, Overrings and divisorial ideals of rings of the form D + M, Michigan Math. J. 20 (1973), 79-95. 8.2 J. Brewer and E. A. Rutter, D + M constructions with general overrings, Michigan Math. J. 23 (1976), 33-42. 8.3 S. Gabelli and E. Houston, Ideal theory in pullbacks, the Kluwer volume, pp. 199-227. 8.4 R. Gilmer and J. Ohm, Primary ideals and valuation ideals, Trans. Amer. Math. Soc. 117 (1965), 237-250. 8.5 T. Lucas, Examples built with D+M, A + XB[X] and other pullback constructions, the Kluwer volume, pp. 341-368. 8.6 O. Zariski and P. Samuel, Commutative Algebra, Vol. II., Springer-Verlag, New York, 1976. 9. Integer-valued polynomials 9.1 M. Bhargava, P-orderings and polynomial functions on arbitrary subsets of a Dedekind domain, J. Reine Angew. Math. 490 (1997), 101-127. 9.2 P.-J. Cahen and J.-L. Chabert, Integer-Valued Polynomials, Amer. Math. Soc. Surveys and Monographs 48, Providence, RI, 1997. 9.3 , What's new about integer-valued polynomials on a subset?, the Kluwer volume, pp. 75-96. 9.4 M. Fontana, L. Izelgue, S.-E. Kabbaj, and F. Tartarone, On the Krull dimension of domains of integer-valued polynomials, Exposition. Math. 15 (1997), 433-465.

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R. Gilmer

10. Commutative monoid rings 10.1 R. Gilmer, Commutative Semigroup Rings, Univ. Chicago Press, Chicago, 1984. 11. Finite element factorization 11.1 D. F. Anderson, Elasticity of factorizations in integral domains, a survey, Lect. Notes Pure Appl. Math. Vol. 171, Marcel Dekker, New York, 1995, pp. 115-123. 11.2 S. T. Chapman and J. Coykendall, Half-factorial domains, a survey, the Kluwer volume, pp. 97-115. 11.3 S. T. Chapman, M. Freeze, and W. W. Smith, On generalized lengths of factorizations in Dedekind and Krull domains, the Kluwer volume, pp. 117-137. 11.4 A. Zaks, Half-factorial domains, Israel J. Math. 37 (1980), 281-302.

Modules Induced from a Normal Subgroup of Prime Index S. P. Glasby Department of Mathematics, Central Washington University, WA 98926-7424, USA GlasbyS8cwu.edu Abstract. Let G be a finite group and H a normal subgroup of prime index p. Let V be an irreducible F/f-module and U a quotient of the induced FG-module V f . We describe the structure of U, which is semisimple when char(F) ^ p and uniserial if char(F) = p. Furthermore, we describe the division rings arising as endomorphism algebras of the simple components of U. We use techniques from noncommutative ring theory to study EndFG(VT) anc^ relate the right ideal structure of to the submodule structure of V\.

1. Introduction • Throughout this paper G will denote a finite group and H will denote a normal subgroup of prime index p. Furthermore, V will denote an irreducible (right) WHmodule, and VI = V ®FH IFG is the associated induced FG-module. Let a be an element of G not in H, and let A := End¥H(V) and T := EndFG(^T)This paper is motivated by the following problem: "Given an irreducible WHmodule V, where F is an arbitrary field, and a quotient U of Vf, determine the submodule structure of U and the endomorphism algebras of the simple modules." By Schur's lemma, A is a division algebra over F, so we shall need techniques from noncommutative ring theory. We determine the submodule structure of U by explicitly realizing Endfci(U) as a direct sum of minimal right ideals, or as a local ring. It suffices to solve our problem in the case when U — V\. Henceforth U = V\. In the case when F is algebraically closed of characteristic zero, it is well known that two cases arise. Either V is not G-stable and V\ is irreducible, or V is Gstable and V| is a direct sum of p pairwise nonisomorphic irreducible submodules. 2000 Mathematics Subject Classification. 20C40, 16S35.

257

S. P. Glasby

258

In [GK96] the structure of Vt is analyzed in the case when F is an arbitrary field satisfying char(F) ^ 0. The assumption that char(F) ^= 0 was made to ensure that A is a field. The main theorem of [GK96] states that the structure of V| is divided into five cases when V is G-stable. In this paper, we drop the hypothesis that char(F) 7^ 0, and even more cases arise in the stable case (Theorems 5, 8 and 9). Fortunately, all these cases can be unified by considering the factorization of a certain binomial tp — A in a twisted polynomial ring A [t; a], which is a (left and right) principal ideal domain. As we will focus on the case when F need not be algebraically closed, a crucial role will be played by the endomorphism algebra A = EndpH(^). In [GK96] the submodules of V\ are described up to isomorphism. As this paper is motivated by computational applications we will strive towards a higher standard: an explicit description of the vectors in the submodule, and an explicit description of the matrices in the endomorphism algebra of the submodule. This is easily achieved in the non-stable case, which we describe for the sake of completeness. 2. The non-stable case Let CQ, e i , . . . , &d-i be an F-basis for V and let a: H —> GL(V) be the representation afforded by the irreducible Fff-module V relative to this basis. The g-conjugate of a (g 6 G) is the representation g i-> (ghg~l)a, and we say that a is G-stable if for each g £ G, a is equivalent to its g-conjugate. In this section we shall assume that a is not G-stable. Let a\: G —> GL(yf) be the representation afforded by V\ relative to the basis Note that G/H = (aH) has order p, and we are writing e^a? rather than Then (ha / 0 0\

i aj.

ha\ = \afa

\

where h 6 H and a h = alha *. The elements of EndFc(^T) are the matrices commuting with Ga], namely the p x p block scalar matrices diag(<5,..., <5) where 5 e A. We shall henceforth assume that V is G-stable. In particular, assume that we know a 6 Autip(V) satisfying (1)

(aha-l)a = a(hff)a~l

for all h € H.

There are 'practical' methods for computing a. A crude method involves viewing (ahia~l)
Modules Induced from a Normal Subgroup of Prime Index

259

[P84], [HR94], [ILOO], and [NP95]. There is also a recursive method for finding a which we shall not discuss here.

3. The elements of EndFG(^t) In this section we explicitly describe the matrices in F = EndFG(^T) an(i giye an isomorphism F —> (A, a, A) where (A, a, A) is a general cyclic F-algebra, see [L91, 14.5]. It is worth recalling that if char(F) ^ 0, then the endomorphism algebra A is commutative. Even in this case, though, F can be noncommutative. Lemma 1. If i & Z, then a~lal: V —» Va? is an ¥H -isomorphism between the submodules V and Val o/ Proof. It follows from Eqn (1) that

Replacing v by va~l gives va~lalh = vha~lal. Hence a~lal is an Fif-homomorphisra, and since it is invertible, it is an FH-isomorphism. D Conjugation by a induces an automorphism of A, which we also call a. [Proof: Conjugating the equation h5 = 5h by a and using (1) shows that a~1Sa G A.] We abbreviate a~l5a = 5a by a(5). The reader can determine from the context whether the symbol a refers to an element of Autp(l / ), or Aut F (A). It follows from Eqn (1) that

for all h£ H. Hence a~ p (a p
(5 6 A).

In summary, we have proved Lemma 2. The element a €E Autp^ satisfying Eqn (1) induces via conjugation an automorphism of A = Endp^^), also called a. There exists A e A x satisfying Oi-p(cf(T) = A,

(2a,b,c)

a(A) = A,

of(S) = XSX~l

for all 5 e A. Theorem 3. The representation a]: G —> GL(T/f) afforded ¥-basis 60,61, • • • , fid—it • • • i Goce 'a1,

by V\ relative to the

S. P. Glasby

260

for V\ is given by

/ 0 (4a,b)

0\

a

0 0 \aA 0

a.

\

haj

where h G H. Moreover, there is an isomorphism from the general cyclic algebra (A, a, A) ioEnd FG (V7)

P-I (A, a, A) -> End FG (^T) : t=0

i=0

where /O

(5a,b)

/

0 0

A o \

G\ J

,

D(5) =

a(6)

Q)

€ A. Proof. By Lemma 1, a lat: V —» Va1 is an F.£f-isomorphism. Hence ha*\ is the p x p block scalar matrix given by Eqn (4b). Similarly, aa] is given by Eqn (4a) as >+i) a «+i

an(j

(voT(p-V a?-l}a = va(a~pap] =

where the last step follows from Eqn (2a). We follow [J96] and write R = A[t; a] for the twisted polynomial ring with the usual addition, and multiplication determined by t6 — a(5)t for S € A. The right ideal (tp — X)R is two-sided as t(tp - A) = (tp - X)t and 5(tp - A) = (tp - A)A- 1 5A by virtue of Eqns (2b) and (2c). The general cyclic algebra (A, a, A) is defined to be the quotient ring R/(tp — \)R. Since R is a (left) euclidean domain, the elements of (A, a, A) may be written uniquely as X)f=o ^^ wnere x = t + (tp — X)R, and multiplication is determined by the rules xp = A and x6 = a(5}x where 8 6 A. The matrices commuting with Ho~\ are precisely the block matrices (oi,j)o
Modules Induced from a Normal Subgroup of Prime Index

261

To see that D(S) e T, we show that (a(r1)D(6) = D(S)(aa^). The first product equals 0 KT)£W =

0 \aA(5

5a

0 0

0

\

0

/

a~u a

and the second product is identical if aX6 = S aX. However, this is true by Eqn (2c). To see that X e F, write aa'] = AX where A — diag(a,... ,a). It follows from Eqn (2b) that A and X commute. Therefore aal = AX and X commute. In summary, elements of F may be written uniquely as Y^=o D^i^X* where 6i 6 A. Since X? = XI and XD(5) = D(a(5))X it follows that the map defined by Y^=o $ixl ^ !Cf=o D(Si)Xl is an isomorphism (A, a, A) —> F as claimed. D A consequence of Eqn (2c) is that a has order p or 1 modulo the inner automorphisms of A. It follows from the Skolem-Noether theorem [CR90, 3.62] that the order of a modulo inner automorphisms is precisely the order of the restriction a Z, where Z = Z(A) is the centre of A-

4. The case when a\Z has order p In this section we determine the structure of F := EndFcC^T) in the case when a induces an automorphism of order p on the field Z(A). Of primary interest to us is Part (a) of the following classical theorem. Although this result can be deduced from [J96, Theorem 1.1.22] and the fact that tp — A is a 'two-sided maximal' element of A[t; a], we prefer to give an elementary proof which generalizes [L91, Theorem 14.6]. Theorem 4. Let F be the general cyclic algebra (A, a, A) where A ^ 0 and a(A) = A. Suppose that a\Z(A) has order p, and fixed subfield ZQ. Then (a) F is a simple Zo-algebra, (b) C r (A) = Z(A), (c) Z(T] = ZQ, and (d) |F : Z0| = (pDeg(A)) 2 where |A : Z(A)| = Deg(A) 2 . Proof. The following proof does not assume that p is prime. Let 7 = Jix11 + • • • + 7r£v be a nonzero element of an ideal / of F, where 0 < n < • • • < v < p, 7j e A, and r is chosen minimal. By minimality, each 74 is nonzero. To prove Part (a) it suffices to prove that r — 1. Then / = F as 71 xn € / is a unit because 71 and x are both units. Assume now that r > 1. Then

k=2 ,

lies in I for each <5 G A. By the minimality of r, each coefficient of xlk is zero. This implies that a n equals alk modulo inner automorphisms for k = 2 , . . . , r . This contradiction proves Part (a).

262

S. P. Glasby

The proofs of Parts (b) and (c) are straightforward, so we shall omit their proofs. Part (d) follows from |r : A| = |Z(A) : Z0| = p, and |A : Z(A)| = Deg(A) 2 is a square. D Before proceeding to Theorem 5, we define the left- and right-twisted powers, /P* and /z tC , where n £ A and i e Z. These expressions are like norms, indeed Jacobson [J96] uses the notation Ni(fj,) to suggest this. These "norms", however, are not multiplicative in general. Consider the twisted polynomial ring A[t; a] and define ijp't*, and for fi 6 A and i € Z. It follows from the power laws (j^t)1 (fjLt)i ((^tyy = (/-it)1! that

— (/rf) l+J and

for i, j e Z. Similar laws hold for right-twisted powers. The left-twisted powers of nonnegative integers can be defined by the recurrence relation (6)

fj,

= 1,

and

fj, ^

' = p, Q Z (^) = /na(/i )

for i > 0,

and negative powers can be defined by ^Jp~l = a^//,-12)"1. It is important in the sequel whether or not A"1 has a left-twisted pth root. Theorem 5. Let V be a G-stable irreducible ¥H-module where H<sG and \G/H\ — p is prime. Let a and A be as in Lemma 2. Suppose that a Z has order p where Z = Z(A) and A = End FG (K). (a) // the equation ^v = A"1 has no solution for fj, € A x , then V] is irreducible, and EndFo(^T) *s isomorphic to the general cyclic algebra (A, a, A) as per Theorem 3. (b) If/j, e A x satisfies ^ = A"1, then V^ = U(p,Q} + • • • + ?/(/ P-I = V^^ia~iai (j = 0 , l , . . . , p - l ) i=0

are isomorphic irreducible submodules satisfying U(^j)\. = V, and where fj^p = A"1. Moreover, if p: G —> GL(U(fi)) is the representation afforded by U(fj) relative to the basis e'Q,..., e'p_1 where

P-I e} = e,-5>^crV

(j = 0 , 1 , . . . ,d - 1),

i=0

iften ap = afj,~l, hp = ha for h £ H, and End FG (E%)) = C^afi-1) = {5 e A <5a = <5^}. Proof. By Theorem 4(a), (A, a, A) is a'simple ring. In Part (a) more is true: (A, a, A) is a division ring by [J96, Theorem 1.3.16]. By Theorem 3, (A, a, A) is isomorphic to EndpcC^T) and so we have proved that V\ is irreducible as desired.

Modules Induced from a Normal Subgroup of Prime Index

263

Consider Part (b) . Let s = /it be an element of the twisted polynomial ring A[t; a], then si = (ntf = M3^ and s5 = fjitd = (p,o.(5)fji~l)iJLt = fj,a(5)n~ls.

Therefore the map A[t;a] -> A^a/r 1 ]: Ef=o ^ ^ E?=o ^O^T1^ is an isomorphism. We are abusing notation here by identifying a/i"1 in AutF(V) with <$ ^ s^'1 in Aut F (A). Ify = /zx, then p p Dp : y = (^x) = /i A = A- A = i. By taking quotients we get an isomorphism (A, a, A) —> (A, a//"1, 1) given by P-I p-i i=o »=o p where x = t + (t - A) and y = s + (s - 1). As yp — 1 = (y — l)(y p ~ 1 + • • • + y + 1), and A[s. a/j~ a ] is right euclidean it follows that (y — 1)(A, a^~l, 1) is a maximal right ideal of (A, a/i"1, 1). Now y — 1 corresponds to /xx — 1 which corresponds to D(u)X — 1 whose kernel gives rise to the irreducible submodule £7(/i) of VI in the statement of Part(b). We shall reprove this, and prove a little more, using a more elementary argument. Let U be a submodule of V] satisfying Ul = V. Let : V -»• V| be an Fffhomomorphism such that V(f> = U{. Let TT^: V\ —> Va* be the F7?-epimorphism given by (Ef=o vi(*~lQ'%)'Ki = Via~lal. Then <5^ = ^7rja~ l a l is an F7f-homomorphism V —> V, or an element of A. Since TTQ + TTI + • • • + TT P _I is the identity map 1 : V] -»• FT, it follows that P-I <£ = ! =(TT O +TTI H ----- h7r p _i) = ^^Q!~V. p

i=0

We now view as a map V —* U and note that U = Ua. Then a~ 1 a: V —> T^a, a~l<j)a: Va —» C/a and ^~ 1 : (7a —> V are each Fff -isomorphisms. Hence their composite, (a~ 1 a)(a~ 1 c/!>a)<^~ 1 is an isomorphism V —» V, denoted /^-1 where /z e A x . Rearranging gives <^a = ap,~l 0. Furthermore /^p*_i = A"1

S. P. Glasby

264

implies that fjpp = A J . In summary, any submodule U of V| satisfying U\, = V equals U(n) for some fj, satisfying /jpp = A"1. Furthermore, by retracing the above argument, if /Jpp = A"1, then U(fi) is an irreducible submodule of V\ satisfying As EndFG(Vt) is a simple ring, V] is a direct sum of isomorphic simple submodules. Therefore, V\ = f/(/uo) + • • • + U(fj,p-i) as desired. It follows from Lemma 1 that the representation p: G —> GL(V) satisfies ap — ajjT1 and hp = ha for h 6 H . Consequently, the matrices commuting with Gp equal the elements of A centralizing ap. Hence EndrG([/(/i)) = C^(apTl) as claimed. D 5. The case when a is inner In this section assume that a|Z(A) has order 1, or equivalently by the SkolemNoether theorem, that a is inner. Fix e G A x such that a is the inner automorphism a(S) = £-l5e. Clearly a(e) = e and by Eqn (2c) £- p fc p = of (5} = \5\~l. Therefore, 77 = e p A € Z(A). If y = gx, then yp = E-PXP = ep\ = -r/ and yS = ex6 = eSex = 5ex = 6y. Hence P-I p-i (8) (A, a, A) -> (A.1,7?)': i=0

i=0

is an isomorphism. Thus we may untwist Theorem 6. Let V be a G-stable irreducible WH -module where H<\G and \G/H\ = p is prime. Suppose that a induces the inner automorphism a(6) = 5£ of the division algebra A = EndWH(V). Then 77 = £PA 6 Zx where Z = Z(A). Suppose that sp — TJ = v(s)n(s) inhere n(s) = Y^iLoHisl an<^ z/ ( s ) = Si^o"^*1' are mon*c polynomials in A[s]. Then WM = YT=Ql VY%=™ Vjei+ja-(i+j)ai+j is a submodule of VI . Let p: G GL(W M ) 6e the representation afforded by W^ relative to the basis (9)

4, . . .

where p—m

p—m

= ek

e

'k =

j=o

and X is given by Eqn (5a). Then

(10)

/ 0

1

0

0

a/9 = ae -i

0

\

1 -Mm-1/

and hp = di&g(ha,..., ha) where h 6 H.' Moreover, fm-l

End FG (W M ) = < Y" i=0

Modules Induced from a Normal Subgroup of Prime Index

265

If n(s) e Z[s], then * A[s]//i(s)A[s] s A ®z K = Z[s]///(s)Z[s]. Proo/. Arguing as in Theorem 5, we have a series of right ideals: i/(s)A[s] C A[s], D

Ky)(A,M)C(A,M),

i/(£:r) (A, a, A) C (A, a, A),

and Er=o- (^)(£^) is a right ideal of T = End FG (V r T). This right ideal corresponds to the submodule V]Y^=oD(vi)(£XYY of ^T- It follows from Eqn (5a) and (eX)p - 77 = 0 that the minimum polynomial of eX equals sp - 77. Let v' = vi/(eX) where v £ V. Then (11)

ir

v'n(sX) = w(eX)v(£X)

= v((sX)p - rf) = vQ = 0.

This proves that (9) is a basis for m—1

n

W» = imi/(eJT) = ker i/(e*) = ^ V ^^e^a^^V^' t=0

j=0

It follows from Lemma 1 that hp = diag(/iu, . . . , ha) is a block scalar matrix (h € H). Since a = aX, (12)

v'(eXYa = v'(eXyaX = v'ae~1(eX)i+1.

It follows from Eqns (11) and (12) that the matrix for ap is correct. It is now a simple matter to show that < ^™^ ^i(aP)1 <5i G A ^ is contained in EndpcCW/i)- A familiar calculation shows that an element of EndFc(WM) is determined by the entries in its top row. As this may be arbitrary, we have found all the elements of EiidyoW. D It follows from Theorem 6 that a necessary condition for W^ to be irreducible is that fj,(s) is irreducible in A[s]. Lemma 7 describes an important case when EndnrG(Wfi) is a division ring, and hence W^ is irreducible. The following proof follows Prof. Deitmar's suggestion [D02]. Lemma 7. Let A be a division algebra with center ¥, and let /j,(s) £ F[s] be irreducible of prime degree. Suppose that no S € A satisfies fj,(S) = 0. Then the quotient ring A[s]//j,(s)A[s] is a division algebra. Proof. Let K = F[s]//j(s)F[s]. Then K is a field and |K : F| = deg^(s) is prime. Clearly /i(s)A[s] is a two-sided ideal of A[s], and A[5]///(s)A[s] is isomorphic to AK = A CS>F K.. By [L91, 15.1(3)], AK is a central simple K-algebra, and hence is isomorphic to Mn(D) for some division algebra D over F. The degree of D and the Schur index of AK are defined as follows Deg(D) = (dim F D) 1/2

and

Ind(A K ) = Deg(D).

By [P82, Prop. 13.4], Ind(A K ) divides Ind(A), and Ind(A) divides |K : F| Ind(A K ). Thus either Ind(A K ) = Ind(A) = Deg(A) = Deg(Ac)

266

S. P. Glasby

and AK is a division algebra by [P82, Prop. 13.4(ii)], or Ind(A) = |K : F| Ind(A K ). If the second case occurred, then by [P82, Cor. 13.4], K is isomorphic to a subfield of A, and so /j(s) has a root in A, contrary to our hypothesis. D If 7? ^ A p , then r) $ Zp and so sp - 77 is irreducible in Z[s], and it follows from Lemma 7 that V^ = WsP-r, is irreducible. Note that End F G(^T) = A ® Z[nl/p] is a division algebra.

6. The case when a is inner and £p = rj In this section we shall assume that £ e A x satisfies £p — i] — 0. Let y = ex and z = £~ly = £~lex. It is useful to consider the isomorphisms (A, a, A) —> (A, 1, TJ) —> (A, 1, 1) defined by x i-+ e~ly and y i-> £z. Note y and z are central in (A, 1,77) and (A, 1, 1) respectively, and yp = rj and zp = I . Theorem 8. Let V be a G-stable irreducible WH -module where H
where

P-I U(&) = V ]£(&>)- VcTV t=0

is irreducible, and f/(^w) = U(^ui') if and only if u and ui' are conjugate in A. // /j,(s) is an irreducible factor of sp — rj in Z[s], then W^ defined in Theorem 6 is a Wedderburn component of V j, and W^ = U(9i) + • • • + U(9n] where di, . . . ,9n are the roots of n(s) in the field Z(£, w). In addition, the representation pg : G —» GL(U(S)) afforded by U(9) relative to the basis e'0,... ,e'd_1 where

satisfies (12a,b)

ape = a.e~l6

and

hpg = ha

for h e H, and '. (b) // char(F) = p, then uj — 1 and V\ is uniserial with unique composition series {0} = W0 c Wi C • • • C Wp = V] where k

p—i

Moreover, Wk-\/Wk = [/(£) for k = l,...,p and EndFc(^(0) = ^-

Modules Induced from a Normal Subgroup of Prime Index

267

Proof. Since z8 = 8z, we see that (£ lex)5 = 6£~l and so £ €. Z. = 77, hence Case (a): Now

5—

(13)

EX). This implies that £

= yp -

yp -

Therefore Vf E P =o(^ w ) P ~ 1 ~ i ( e - X ") ir is a submodule of Vf where X is given by Eqn (5a). We show directly that U(£w) is a submodule of Vf. This follows from (w(£w)~Va~V)a = ^

va(a~l(S,uj)~'l£la)a~^+l'lal+l

-*

_1

j.

/ ^

\_/'oJ-l'l

1-L1

_^i-i-1\

i_L1

and setting i = p — 1 in the right-hand side of Eqn (14) gives

'1 ef 'A = As 17(£w)! = I/, we see that U(£u) is an irreducible FG-submodule of Vf . Setting 9 = £LO establishes the truth of Eqns (12a,b). We may calculate Hom(ty(£a;),[/(£u/)) directly by finding all 5 in Endp(Vj that intertwine p^u and p^>. As <5 intertwines hp^ and hp^>, it follows that 5 commutes with If a, and hence S € A. Also so 5ae & — £uj'6. Since £ e Z x and <5Q£ = <5, this amounts to 6u = u>'5. Setting i = j shows that EndrG(U(£w)) — C&(u>). The Galois group Gal(Z(w) : Z) is cyclic of order dividing p — 1. Also u> and w' are conjugate in Gal(Z(u;) : Z) if and only if they share the same minimal polynomial over Z. The latter holds by Dixon's Theorem [L91, 16.8] if and only if u; and u/ are conjugate in A. Note that u> and u>' share the same minimal polynomial over Z precisely when £u and £o/ share the same minimal polynomial. This proves that WM is a Wedderburn component of Vf . Case (b): Suppose now that char(F) = p. Then w = 1 and Eqn (13) becomes

yp - r, = (y - C)p = (y - 0(E£o (^^(-O^^V). As = (A, a, A) =* (A, 1, 77) ^ (A, 1, 1) =* A[«]

has a unique composition series, so too does Vf. By noting that z — £~lex and D(£~l£) = £-le, we see that Wi = V](£,~leX - 1)P~T defines the unique composition series for Vf where X is given by Eqn (5a). Let R be the diagonal matrix dia.g(!1^1e; . . . . (^e)?-1). and let S be the matrix whose (i,j)th block is the binomial coefficient ( l ) where 0 < i,j < p. A direct calculation verifies that R(£~1eX)R~1 = C and S~1CS = J where /O

1

0\

/I and

0 0 \1 0

1

\

J= 1 1 V

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Therefore ^eX -1 = T~l(J - 1)T where T = S^R, and hence Wk = V\ (r1^ - l) P ~ fc = V]T~l(J - l)P~kT = V](J - l)p~kT. It is easily seen that im( J — l)p~k = ker( J — l)k is the subspace ( 0 , . . . , 0, V , . . . , V) where the first V is in column p — k. The (i,j)th entry of T = S~1R is (-l) i+J '(j)(£~ l£ ) j - The last row of T gives

More generally, the last k rows of T give k

p—i

i=l

j=0

Since p — i — £ = - (i + I) in a field (such as F) of characteristic p, we see that (p~l) = (—!)•'( t+i '-~ 1 ) and the formula for Wk simplifies to k

p-i

^ = I>E Setting fc = 1 shows Wi = U(£). A direct calculation shows that We showed in Part (a) that Endfo(U(^)) equals C&(£) = A.

D

In Case (a), CA(£W) equals A precisely when u> & Z. If A is the rational quaternions, and u is a primitive cube root of unity, then CA(W) equals Q(w). There are infinitely many primitive cube roots of 1 in this case, and they form a conjugacy class of A by Dixon's Theorem (as they all satisfy the irreducible polynomial s2 + s + 1 over Q). Thus isomorphism of the submodules U(£w) is governed by conjugacy in A, and not conjugacy in Gal(Q(w) : ). Finally, it remains to generalize Theorem 8(a) to allow for the possibility that A may not contain a primitive pth root of 1. Theorem 9. Let V be a G-stable irreducible WH-module where H
VI = WM1 + • - • + W^ where sp —r\~ f^i(s) • • • /J,r(s) is a factorization into monic irreducibles over Z, and where W^ defined in Theorem. 6. If j-i,(s) is a monic irreducible factor of sp —r/, and n(s) = i'i(s) • • -vn(s) where the Vi(s) are monic and irreducible in A[s], then W^ is a Wedderburn component ofV\, and W^ = W®™ where WVn is an irreducible WG-module and Endpc (WVn ) is given in Theorem 6. In addition, where B = {5(s) €. A[s] | 6(s)vn(s) e i/ n (s)A[s]} is the idealizer of the right ideal

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Proof. Since char(F) ^ p, the monic polynomials m(s),... ,nr(s} are distinct and pairwise coprime in Z[s]. Prom this it follows that V\ equals WMl -i (- W Mr . By Theorem 6, End FG (W M ) £* AK where AK = A[s]///(s)A[s] S A ®z K, and K is the field Z[s]/n(s)Z[s\. By [L91, 15.1(3)], AK is a simple ring. Therefore /j(s)A[s] is a two-sided maximal ideal of A[s], and so /j,(s) is called a two-sided maximal element of A[s]. By [J96, Theorem 1.2.19(b)], AK = Mn(D) where D is the division ring B/i/ n (s)A[s]. Moreover, Z(A K ) = Z(Mn(D}) so K = Z(D). Thus VFM S W«n where W^n is an irreducible submodule of V] and EndFG(W^ n ) = D. In addition, v\,...,vn are similar [J96, Def. 1.2.7], and WVl,... ,WVn are isomorphic. If ju(s),/Lt'(s) are distinct monic irreducible factors of s p —77 in Z[s] and z/(s),i/(s) in A[s] are monic irreducible factors of /u(s) and //(s) respectively, then it follows from [J96, Def. 1.2.7] that v(s) and v'(s) are not similar. This means that an irreducible summand of WM is not isomorphic to an irreducible summand of W^i. Hence the W^ are Wedderburn components as claimed. D Acknowledgment I am very grateful to Prof. A, D. H. Deitmar for providing a proof [D02] of Lemma 7 in the case when sp - TJ has no root in A. I would also like to thank the referee for his/her helpful suggestions. References [CR90] C. W. Curtis and I. Reiner, Methods of Representation Theory: with Applications to Finite Groups and Orders, Vol. 1, Classic Library Edn, John Wiley and Sons, 1990. [D02] A. D. H. Deitmar, sci.math.research, September 6, 2002. [GK96] S. P. Glasby and L. G. Kovacs, Irreducible modules and normal subgroups of prime index, Comm. Algebra 24 (1996), no. 4, 1529-1546. (MR 97a:20012) [ILOO] G. Ivanyos and K. Lux, Treating the exceptional cases of the MeatAxe, Experiment. Math. 9 (2000), no. 3, 373-381. (MR 2001j:16067) [HR94] D. F. Holt and S. Rees, Testing modules for irreducibility, J. Austral. Math. Soc. Ser. A 57 (1994), no. 1, 1-16. (MR 95e:20023) [J96] N. Jacobson, Finite-Dimensional Division Algebras over Fields, Springer-Verlag, 1996. [L91] T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics 131, Springer-Verlag, 1991. [NP95] P. M. Neumann and C. E. Praeger, Cyclic matrices over finite fields, J. London Math. Soc. 52 (1995), no. 2, 263-284. (MR 96j:15017) [P84] R. A. Parker, The computer calculation of modular characters (the meat-axe), Computational group theory (Durham, 1982), 267-274, Academic Press, London, 1984. (MR 84k:20041) [P82] R. S. Pierce, Associative Algebras, Graduate Texts in Mathematics 88, Springer-Verlag, 1982.

Uniquely Transitive Torsion-free Abelian Groups Rudiger Gobel Fachbereich 6, Mathematik und Informatik, Universitat Essen, 45117 Essen, Germany R. Goebel8Uni-Essen.De

Saharon Shelah Department of Mathematics, Hebrew University, Jerusalem, Israel, and Rutgers University, New Brunswick, NJ, USA Shelahflmath.huj i.ac.il Abstract. We will answer a question raised by Emmanuel Dror Farjoun concerning the existence of t6rsion-free abelian groups G such that for any ordered pair of pure elements there is a unique automorphism mapping the first element onto the second one. We will show the existence of such a group of cardinality A for any successor cardinal A = /i+

1. Introduction We will consider the set pG of all non-zero pure elements of a torsion-free abelian group G. Recall that g € G is pure if the equations xn — g for natural numbers n ^ 1 have no solution x € G. Clearly every element of the automorphism group Aut G of G induces a permutation on the set pG and it is natural to consider groups where the action of Aut G on pG is transitive: for any pair x, y e pG there is an automorphism tp € AutG such that x

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can chose a bijection a: X —> Y with xa = y which extends to an automorphism a G AutG. Hence free groups are T-groups and a similar argument holds for a wide class of abelian groups. There are T-groups like Z N l /Zl N l l with Z'NI! the set of all elements in Z N] of countable support, which are Ki-free but not separable, see Dugas, Hausen [4]. The existence of Ni-free, indecomposable T-groups in L for any regular, not weakly compact cardinality was also shown in Dugas, Shelah [5, Theorem (b), p. 192]. This was used to answer a problem in Hausen [11], see also [12] and [5, Theorem (a), p. 192]. Thus we may strengthening the action of Aut G on pG and say that G is a U-group if any two automorphisms
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T for UT-groups are a problem. Note that in many earlier constructions G has a free dense and pure .R-submodule of rank > 1 mostly of rank \G\. But T-groups must obviously be cyclic over their endomorphism ring R, hence [3, 2] do not apply in principle. Inspecting the proof in [5] it is easy to see that G is not torsion-free over its endomorphism ring. This comes from the list of new variables x,y,... added to R in the construction in order to make R acting transitive on all pairs of pure elements. Even refining the list of pure pairs in [5] it seems hard to avoid clashes of related pairs such that x — y for example has a proper annihilator. Thus the groups in [5] are T-groups and not UT-groups (even modifying the arguments). Thus a new approach is need, which will be established in Section 3. We will use a geometric argument choosing carefully new partial automorphisms for making G transitive but with very small domain and image in order to preserve the U-property for the new monoid. Then we will feed the partial maps with pushouts to grow them up and become real automorphisms without destroying the UT-property. At the end we will have a suitable subgroup F of automorphisms of some group G, thus G becomes an .R-module over the ring ZF = df R. Finally we have to fit these approximations to ideas getting rid of the endomorphisms outside R, see Section 4. We need the Strong Black Box as discussed and proven in terms of model theory in Eklof, Mekler [6, Chapter XIV]. Note that this prediction principle is stronger then (Shelah's) General Black Box, see [2, Appendix]. The Strong Black Box is also restricted to those particular cardinals mentioned in the theorem. However, here we will apply a version of the Strong Black Box stated and proven on the grounds of modules in ordinary, naive set theory, which can be found in a recent paper by Gobel, Wallutis [10], see also [9]. In order to show End G = R well-known arguments for realizing rings as endomorphism rings must be modified because the final ring and its action are only given to us at the very end of the construction: We will first replace the layers Ga from the construction by a new filtration only depending on the norm. Note that the members of the new filtration of the right .R-module G must be right Rasubmodules for suitable subrings Ra of R to have cardinality less than \G\. But they are still good enough to kill unwanted endomorphisms referring to the prediction used during the construction. Moreover note that the two tasks indicated in the last two paragraphs must be intertwined and applied with repetition. While the final G is an Hj-free abelian group, hence torsion-free, it is not hard to see that G is torsion as an -R-module: If 0 ^ g € G, then we can choose distinct elements g',x,y £ pG such that ng' = g, and x — y e pG. Hence there are distinct unit elements rx, ry, rxy e U(R) = ±F such that g'rx = x,g'ry = y,g'rxy = x + y. The endomorphism rx + ry does not belong to +F, in particular it can not be rxy. Hence r = rx + ry — rxy ^ 0 but g'r = x + y— (x + y) = 0 and g is torsion. It is worth noting that the result can be strengthened under V = L, where we get strongly-A-free groups of cardinality A as in Theorem 1.1 for each regular, uncountable cardinal A which is not weakly compact. In this case the approximations in Section 3 can be improved, replacing 'Hi-free' by 'free' at all obvious places. The main result of this section will then be a theorem on free groups G with a free (non-abelian) group

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F C Aut G acting uniquely transitive on G. Also Section 4 must be modified: The Strong Black Box 4.2 must be replaced by <0> following arguments similar to [3].

2. Warming up: Construction of a special group We begin with a particular case of an old theorem and discuss extra properties of the constructed group. Part of this proposition will be used in Section 3. Proposition 2.1. Let K be a cardinal with K K ° = K, F be a free (non-abelian) group of rank < K and R = ZF be its integral group ring. Then there is a group G with the following properties. (i) G is an Ki-/ree abelian group of rank K with EndG = R. (ii) G is torsion-free as an R-module. (iii) Aut G = ±F (iv) // (f e End G, then f is injective. Proof. Note that the integral group ring R = IF has free additive group R+ with basis F. We can apply a main theorem from [2] showing the existence of an Ki-free abelian group G with End G — R. The free group F is orderable (i.e. has a linear ordering which is compatible with multiplication by elements from the right), see Mura, Rhemtulla [16, p. 37]. However note, that torsion-free groups may be nonorderable, see [16, pp. 89 - 95, Example 4.3.1.]. The integral group ring ZF of any orderable group F satisfies the unit conjecture, this is to say that the units of R = ZF are the obvious ones, hence C/(ZF) = ±F, see Sehgal [17, p. 276, Lemma 45.3]. Moreover, any group ring which satisfies the unit conjecture also satisfies the zero divisor conjecture, hence R has no zero-divisors, see Sehgal [17, p. 276, Lemma 45.2]. Therefore R is torsion-free as an .R-module. Now the remaining part of the proof is easy: Aut G = U(R) = i-F and if 0 7^ g 6 G, then g 6 (&R C G because G is also an tti-free .R-module by construction in [2]. If we consider multiplication of g by some r € R on a non-trivial component of g in this free direct sum, then r = 0 because R is torsion-free as an .R-module. Hence G is torsion-free as an .R-module. Any ip G End G = R is scalar multiplication by a suitable r & R hence injective because G is a torsion-free R-module. D We will use Proposition 2.1 in Section 3. We get more out of it if we know that a particular endomorphism is pure: Lemma 2.2. Let F be the free (non-abelian) group and EndG = ZF be the endomorphism ring of the Ki-/ree abelian group G given by Proposition 2.1. //

is a monomorphism and not onto. If Q ^ (p e ZF, then f is pure in ZF+ if and only if Imp is pure in G. Proof. All endomorphisms of G in Proposition 2.1 are monomorphisms as shown there. If ip e R = ZF = End G would be onto, then (p must be an automorphism, thus if 6 U(R), which is ±F; and this was excluded. We come to the last assertion. We shall write 0 ^

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under multiplication by n. Hence Gr is not pure in G. Conversely let r be pure in R and consider any g G G such that gp £ Gr for some prime p. Hence gp = g'r and by construction of G (just note that G is Ki-free as .R-module) there is h e G such that g' e hR and hR is a pure subgroup of G. Hence also g £ hR and we can write g — hrg,g' — hrg> which gives hprg = hrg>r and prg = rg>r because G is R- torsion-free. Using that p cannot divide r by purity in R and that r, rg> are elements of the group ring "LF we can write ry = r'p for some r' € R. Finally gp = g'r = (hr'p)r, hence g = hr'r € Gr and Gr is pure in G. D If we replace [2] in the proof of Proposition 2.1 by [3], then we can strengthen Proposition 2.1 in the constructible universe L. We get a Corollary 2.3. Let K be a regular, uncountable cardinal which is not weakly compact such that OK holds and let F be a free (non-abelian) group of rank < K and R = 1,F be its integral group ring. Then there is a strongly-K-free abelian group G of rank K with EndG = R and properties (ii), (Hi) and (iv) of Proposition 2.1. Recall that G is K-free if all subgroups of cardinality < K are free, and G is strongly K-free if also any subgroup of cardinality < K is contained in a subgroup U of cardinality < n such that G/U is «-free as well.

3. Growing partial automorphisms Besides the set pG of pure elements of a group G we consider a particular subset pAut G of all partial automorphisms tp of G. Here tp is an isomorphism with domain Dom ip and range Im tp subgroups of G. The inverse isomorphism will be denoted by tp~l. However note that ~1, then only domain and image are interchanged, thus trivially ip~1 e pAut G. It remains to consider domain and range of tpijj. Passing to an inverse, as just noted, it is enough to deal with We already observed that (3.1)

Dom(tpip) = (Im'tp n Dom i i From ip e pAutG follows that GfDoimp is Ni-free, hence Im
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is Ni-free. We apply tp l and (3.1) to see that Dom ip/ Dom(y>VO is Ki-free. Moreover tp € pAutG, and therefore G/Domcp is ttj-free, hence G/Dom(y>i/>) is Hj-free as desired. We arrive at our first Lemma 3.1. The set pAutG of all partial automorphism tp of G with G/Domy> and G/ Im ip both K! -free abelian groups is a submonoid of all partial automorphisms with 1 = idc and —1 = — idc acting as multiplication by I and —1 respectively, which is closed under taking (weak] inverses. Moreover, if 3 C pAut G, then (3) C pAut G is the submonoid of all products taken from the set {±1} U 3 U 3"1, where 3"1 = {ip~l \ ijj & 3}. We begin with an observation which allows us to consider induced partial automorphisms on a factor group. Observation 3.2. If U C G are abelian groups and (p £ pAutG with (Imy n U)tp~l ^ U ana (Dom(p fi U)(p C U, then w induces a partial automorphism ip of G where G = {g=g + U\g& G} taking ~g to ~g!p for any g 6 Dom (p. Moreover Domip = Domtp and Imp = Im
D

In order to show Proposition 3.9 we relate elements of free (non-abelian) groups and elements in pAut G. It is important to be able to work with elements of pAut G acting on a partial free basis of G. To be precise, we will need the following definition extending freeness frdm G to pAut G. Definition 3.3. Let $ = {ipt \ t € u] C pAutG be a finite set of partial automorphisms. Then (G, 5?) is called 'Ri-free if any countable subset of G belongs to a countable subgroup X C G with basis B and the following properties for any y e (5-). (i) G/X is Ki-/ree. (ii) ip induces a partial injection on B, that is, if b e B n Domtp, then also b

, respectively. It also follows that G is HI-free. We can ease arguments in Lemma 3.11 and Lemma 3.12 to note here that we only need a partial basis bffl (a subset of B) with the property (ii) and (&{#)} nDomy = (b($) fiDomy); see Proposition 3.9. Next we relate basis elements of free non-abelian groups and partial automorphisms with care. Suppose the set 5 ='{y>t | t € J} generates a free group (5) and as in Definition 3.5 there is a map TT: ^ —» pAutG (acting on the left), then this map can be extended to (3). The extension is unique if we restrict ourselves

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to reduced elements ip = ( p \ . . . ipn in (5") with ipi € # U # 1 and define naturally Tr((f) = T T ( < P I ) . . . 7r(i)7r(i£>2)) = ^(vi)71"^)- Thus we have equality if also the formal product tpi pAutG as above we say that TT (or 7r(3r) = {TT(^) tp e 5"}) satisfies the U-property ij (f = ip1 for any reduced elements ip, ip' e ($) with xir(ip) = x-rr((f>') and some x 6 Dom(y) n Dam(ip') O pG. The last definition is crucial for this paper because it is the microscopic version of U-groups discussed in the introduction. We also must pass from groups Gf with this U-property to suitable extension G5 with the U-property. All this we encode into our main definition of quintuples y and their extensions. Normally our maps will act on the right, but we allow three exceptions, the maps e, it and h below. Also $Pn0(.7) denotes all finite subsets of the set J. Definition 3.5. Let 8. be the family of all quintuples r = (G,£, £ ,7T,/i).= (G>,y, £ ',7r>,/i') such that the following holds. (i) G is an K^/ree abelian group. (ii) # = {
(3).

(iii) e: J —» {!,—!} is a map. (iv) TT : $ —> pAut G is a map which satisfies the U-property. We shall write 7r r ((^t) = (f>\ and omit y if the meaning is clear from the context. (v) h: ^Pn 0 (J) —> Im(/i) is a partial function from Dom/i C ^3^ 0 (J). If u £ Dom/i and U = h(u), $ = {tpt | t & u}, then the following conditions must hold. (a) U is a countable subgroup of G and (Domip* H U)
t e J ? ) C pAutG f .

Hence Gf/Dom)(&). We will pass from groups Gr to larger groups G5 related to p, rj £ .£ by taking pushouts

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or unions. This is reflected in the next definition (in particular condition (Hi)) of an ordering on .&. This is the final step before we can start working. Definition 3.8. Lefy < t) (y,t) 6 £) if the following holds for y = (Gf ,$*,£*,•**,&) and ^(G'^V,^,/*'). (i) G' C G> and G'/G' is Ki-/ree. (ii) TT'' extends TTJ in i/ie weafc sense (?rr ^ Tr 0 ), i/iai is Jr C J"> (equivalently 3? C 3rt)) and a/so 92* C 93 j? extends for all t G J f . (ui) //t S J r , then one of the following cases holds (a) e*(t) = e'(t) and tf = tf. (b) Gr C D o m ^ n l m v ? ? . (c) e*(t) = 1 = -e»(t) and Gf C (d) e"(i) = 1 = -£ r (t) and Gf C (iv) /V C /i» extends (i.e., ifh*(u) C G r , i/ien V(u) = /^(u) C (v) //u £ Dom/i r and Gr = df G^/hf(u) C G D =df Gt>/h^(u), then any basis B of a countable subgroup X C Gf as in Definition 3.5 extends to a basis B' of some countable subgroup X' of G which also satisfies Definition 3.5. Proposition 3.9. Suppose that p < 5 in ^ and u € Dom/i1', [G^l > Ko,G' = Dom ^ = Im )? | i e u) is a free subgroup of AutG". If also 6 = E^aA £ ZF is as above, then ^ e ZF^ and we may assume that ker 9** =*• 0. It remains to show that 0^ is surjective if and only if I = {0} is a singleton and OQ = ±1- If / = {0}, then it is clear that 0^ is surjective if and only if ao = ±1. Hence we may assume that 0^ is an isomorphism, and |/| > 1 for contradiction. In order to apply Definition 3.5 we pass to the quotient G = G^/h(u) and to the induced maps lpt, which we rename again as G 9 ,^- It follows that ft(w)^1' C h(u) and silently we assume that h(u) is invariant under (fl 5 )" 1 ; otherwise we must enlarge h(u) by a back and forth argument such that the quotient satisfies again Definition 3.5 (v). Also G5 = G°/h(w) ^ 0 because If X ^ 0 is a countable subgroup of G 9 , then X is free. We may assume without restriction that XQ*> C X, XF*> C X and X(ef>)~1 C JY and 0X = df 0' \ X e End A". If x £ A, then x = o^ e G^" = G1', thus 5 = x^")"1 e X and ^x is °*s° surjective (on X). We can start with some X' with a special basis B' / 0 as in Definition 3.5 (v) (the weak version mentioned after the Definition 3.3) and let X be its closure as above. Then X' will be a summand of X because G/X' is Hx-free. Hence B' extends to a basis B of X: The maps ipx = ^ \ X, (
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automorphism of G. In order to distinguish it from the element in F we will call the automorphism ip* €. Aut G and F becomes F* . The mapping * extends naturally to all of ZF, (by the identification IF = End (7), thus 0* = Ei 6 / a A* e EndG, where 0* e F*. From [J| > 1 and Lemma 2.2 it follows that 9* is not surjective. Let y 6 G \ GO* which will help us showing that 9x can not be surjective either, in fact we want to show that B' n Xdx — 0- Fix an element c € B' and define a map $ : B —> G such that c$ = y. lib & B and there is

G. It follows 6
Thus Ox, hence 9** is not surjective.

D

It is convenient to check the U-property by the following simple characterization. Proposition 3.10. Let ($) be the group freely generated by 5" and IT: $ -» pAutG* be a map as in Definition 3.5 with ir(~1 in (&). From x E Domip*^)'1 C it follows x = xOf. We have 9 = 1 by hypothesis, and (p = tp follows. D The last proposition shows that the U-property is a strong restriction on ir($). If only xip~l is not reduced. The next lemma will be used to make the desired group 'more transitive'. We want to isolate the argument on the existence of h(u): If f = (G,-J, £,7T, h) € &, <po € pAutG with 0 ^ J as in Lemma 3.11, then we extend h: ^^.0(J) —> Im(/i) to ft': !PN0(J') -^ Im(/i') where J' = J U {0}. If u e Dom/i, then /i''(u) = h*(u). If x,y and p are from Lemma 3.11, then let h(u') for u' = u U {0} be a countable subgroup [/' C G containing {x, y} U /i(«) such that G = G/U' and the induced maps y satisfy Definition 3.5 (v) (the weak version mentioned after Definition 3.3 will suffice). Note that we only use extensions of groups Gf as in Lemma 3.11 or Lemma 3.12 and unions of ascending chains of such groups. By a back and

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forth argument, and a moments reflection about the action of the extended partial automorphisms by the pushouts, it follows that such a countable subgroup U' exists. Lemma 3.11 (to get more partial automorphisms), //y = (G,^, £,TT,/I) e A and x, y e pG such that xipf ^ ±y for all tp 6 (3), then let tp% : xL —> yZ(x —> y) 6e the natural isomorphism. If $*> — 3~U { = «7U{0},£° = e U {(0, l)},7r" = 7r r U{(o = r/ and

2 • • • r/ek~llPk with 0 ^ £i € Z and i,k- Now we assume that zip** = z for some z £ Dom}~i , and Im??1' n DomTy15 = 1y n Zx = 0 by the choice of x,i/. Hence (T? 5 ) 2 = 0 and if'1 = 0 is a contradiction, because 0 ^ z'& Domip^, so the first claim follows. Next we show that

1rieiw.

(3.2)

We look at the path of z, the set [z] of all consecutive images of z: z0 = z,z1

In order to apply (TJ')£I to z\ we must have z\ & Dom(7/') £l , but Dom(?7t')£l is either Zx or Zy, hence z\ is one of the four elements of the set V = {±£, ±j/} by purity. Inductively we get Zi € V for all 0 < i < 2k — 1. Suppose that rj£2 appears in y>, then z% = z<22 = ±1 which also was excluded. Hence (3.2) follows. Our assumption is reduced to z = ± z < f l ( i f } e i p = \ is possible and t) € ^ follows. D The next lemma will increase domain and image of partial automorphisms, respectively. Lemma 3.12 (growing the partial automorphisms) . 7/p = (G, 3", e, TT,/I) e ^ withy = {ips s e J} andt £ J, then there is y < 5 = (G",# t ',£ 9 ,7r 1 ', ft) € £ suc/i ifte following holds.

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a pushout with D — GO n GI and Gr = GO = GI. (b) Ife*(t) = -1, then G0 = Imp?, GI = Domp? and D = Impji. Proof. The set J and h do not change when passing from y to t}. Thus we consider 7rf next and restrict to £f = 1 (the case ef = — 1 follows if we replace pt by p^~ ). For the pushout we let G" = (G x G)/H with ff = {(xp, -x) x € Dom pt}. If we also say that «70 = (t/ x 0) + H/H C G« and C/i = (0 x U) + H/H for any UCG, then in particular G" = G0 + GI and D =d! G0 n GI = (ImpJ) 0 = (Domp^)i by the pushout. Moreover we identify G0 = G r , hence Domp]; = (Domp^) 0 =df D' and D = Imp*. The canonical map extends p£ because ((x, 0) + F)p? = (0,x) + H = (xpj[, 0) + H for all x G Dom(p\. Clearly GO = Domp? and GI = Imp?. Moreover G'/D = Go/Dompj©Gi/Irnpj is Hi-free, hence also G"5 and G'/G5 = G\/D are Hi-free, and the maps pj = p-j (t ^ s G J) remain the same. It follows that TT" : 5" —> pAutG". The existence of a partial basis satisfying Definition 3-3 was discussed before Lemma 3.11. So for y < y £ £ we only must check the U-property for $ with the new partial automorphisms from p-j (s G J) and apply Proposition 3.10: Consider a reduced product p = pip t 1 p2 . . . (ptn~1(pn, where 1 ^ p$ < except possibly pi = ±1 and pn = 1. Suppose that zip** = z for some z 6 pG^ and let ZQ = Z, Zi =

be the path [z] of z. Thus zn = ZQ by assumption on p. First we note that p? = 1. The equation zop5 = ZQ reduces to ±t'1(p?)5l+1p2(p?)'52 • • • p^_]^ = ij with i^ G GO, which is the first case for a new z = t( G pGoIf ZQ G GO, then also ZQ G Dom p* and z\ = zop* € GQ. We will continue along the path step by step having two subcases each time, but one of them will lead to a contradiction. In the first step either z\ G D' or z\ £ D'. In any case 5± = 1 and in the second case t\ = Zip? G GI \,D, but t\ G Dompj so necessarily p2 — 1 and n = 2. We get t\y>\ = z% = ti, and'ti = ZQ G GO contradicting t\ G GI \ D. We arrive at the other case z\ € -D'. Hence t\ = z : pj G D and we must have ti G Domp 2 . Therefore also Z2 = tip 2 G Go, and <5j = 1. Again we have two cases

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22 G D' and z<2 <£ D' with 5^ = 1, where the second case leads to a contradiction. Hence t% = z ^ f l 6 D and we continue until we reach n and all the y?j? s are replaced by the yjs. Now our first remark applies and ip = 1 follows from the U-property for £. D Lemma 3.13. Let a be a limit ordinal. Then any increasing continuous chain f = (G-7', & , £J , 7r3', hi)jea in (&, <) obtained by applications of Lemma 3.11 and Lemma 3.12 /ios a natural supremum j: = (G, #, e, TT, ft.) in £., where G — Ujga ^J > ' 7 ) ^ = Ujea JJ ant^ ^i h are defined below. Proof. The map h extends uniquely, because h is defined on finite subsets of J. Similarly we can handle TT. We define ?r : £ —> pAut G by taking unions: If t € J, then t e J-3 for all i < j € a and i 6 a large enough. Therefore we can let fl =df Ui<jeo: ^'(Vt)) an<^ Dom^j = (Ji<jea Dom 7T-3 (
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all t £ J f2 = J f l . We continue this way for each n. Prom Lemma 3.13 follows J < ?' 6 &, and by construction (3"'} C Aut G' acts uniquely transitive on pG' such that also (c) of Theorem 3.14 holds. D Now we can restrict elements in & to triples and say that y = ( G * , $ f , i r f ) belongs to A* if and only if G? is an Kj-free abelian group, # = {ft t & J*} freely generates a (non-abelian) group (3) and ?r f : ^ —> Aut Gr is an injective map such that 7rf (-5) = {7rf(<£t) = ¥>£ t € J f } satisfies the U-property. Still we can view &* as a subset of ^ and have an induced ordering, compare Definition 3.8: We have ? < 9 (?) *) € £*) if (v) and the following holds. (i) Gr C G" and G«/G f is Hi-free, (ii) TT* CTT". The other conditions in Definition 3.8 are vacuous. 4. Construction of uniquely transitive groups In this section we will sharpen Theorem 3.14 which shows that a group ring R = ZF for some free group F can be represented as R C EndG of some Ni-free group G (hence G is an ^-module), such that the units U(R) = ±F act uniquely transitive on G. If ±F C AutG is not necessarily transitive but satisfies the uniqueness property (( (p = ip'), then we will also say that the pair (G, F) has the U-property. We will modify G such that equality holds, that is R = End G. Thus we have to kill unwanted endomorphisms, which is done using the Strong Black Box 4.2. We need some preparation to work with this prediction principle. , First we need an §-adic topology. Let S = {qn \ n 6 01} be an enumeration of a multiplicatively closed set generated by 1 and at least one more natural number different from 1 and define SQ = l,s n + i = sn • qn for all n £ w which obviously defines a Hausdorff S-topology on the groups G under consideration. Let G be the S-adic completion of G and G C, G naturally, where purity "Ct" is S-purity. Next we must formulate the Strong Black Box and adjust our notations; see Strong Black Box 4.2. We rely on the version adjusted to and proven for modules using only naive set theory as explained in the introduction, see [10, 9]. Thus we have to fix a few parameters next. To do so we choose an enumeration by ordinals a £ A, with A = fj,+ a successor cardinal such that ^° = p.: Let Fa be a free non-abelian group with a set of n free generators 3a, the $as constitute a strictly increasing, continuous chain in a < A. We will write .Ra = ZFa with R = U a eA R<* and note that .R+ and R+ are free abelian groups. By Theorem 3.14 there is an .RQ-module Ga of cardinality n which is cyclic as .RQ-module and Nj-free as abelian group and Fa acts sharply transitive on pGa. We can choose a free abelian and §-dense subgroup Ba Cf Ga C» Ba. Also the Bas constitute a strictly increasing, continuous chain in a < A. Passing to & + 1 we can let £ a +i = Ba ® Aa with Aa = 0 i < p Zflj (and p = fj,). It helps (when using [10, 9]) not to identify p and p, in the formulation of the Strong Black Box 4.2.

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Thus the construction is based on a free abelian group B of rank A and its Sadic completion P., in fact the desired group G will be sandwiched as B C* G C* B. Put B = 0 a 7 for all 7 with yU < 7 < A where we put g^ = idM and so |7| = \p\ = p, for all such 7's. For technical reasons we also put g7 = gM for 7 < /^. Definition 4.1. Lei i/ie bijections gy (7 < A) 6e 05 afeoue and put 7 a> j = 7a x 7$ /or aH (a, i) e A x p. We de/me P to be a canonical summand of B if P — ©(a i)6/ ^e",i /or some / C A x p wrfft, |/| < NO SMC/I ^ftai: • z/ (a, i) € I, then (i, i) 6 /; if (a, i) e /, a £ p then (i, a) e / and • i/ (a, i) e J, t/ien (J n (/i x JJL)) ga,i = I Da x i. Accordingly, a homomorphism : P —> B is a canonical homomorphism i/ P is a canonical summand of B and Imrf* C P; we pui [] = [P], [i^]A = [P]A and

IHI = Note, by the above definition, a canonical summand P satisfies ||P||A £ Let £ denote the set of all canonical homomorphisms. From assumption yuH° = JJL follows |£| = A. We are now ready to formulate the Strong Black Box: The Strong Black Box 4.2. Let A = ^+ and [i = p,*0 be as before and let E C A° 6e a stationary subset of A. TTien i/iere exists a family £* o/ canonical homomorphisms with the following properties : (1) / / 0 € £ * , toen W e £ . (2) // 0, $ are two different elements of C* of the same norm a, then ||[] A Pi

fe'U<"(3) PREDICTION: For any homomorphism i/>: B —> B and for any subset I of A x p uwt/i |/| < KO tfte set

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is stationary. We will enumerate €* taking care of the order of the norm and can write £* = {4>a | a < A} such that ||[^Q]A|| < ||[0,g]A|| for all a < /3 < A. Note that we distinguish between (p E # and given by the Strong Black Box 4.2. The enumeration is now used to find a continuous, ascending chain Ga (a < A) such that Ba C* Ga C* Ba and the corresponding yQ — (G a ,3a,7r a ) € &*. Let £ C A° be a fixed stationary subset of A° and choose fo = (Go,So,TO) to be any triple given by Theorem 3.14 for cardinality fj,. For any /? < A, let Pp = Dom(j)p and suppose that the triples fp = (Gp,$p,Trp) are constructed for all /3 < a. If a is a limit ordinal, then by continuity we let j:a = sup/3p is scalar multiplication by some r G Rp when restricted to Pp or if Imp e R later on, so in this case <j>p is a good candidate which should survive the massacre. This we must and will see clearly at the marked place near the end of the proof!) Recall that \\p\\ € A", hence there are (/3n,in) £ [<j)p] (n 6 w) such that &Q < 0i < • • • < /3n < • • • and su P«ew/3n = ll^/sll- Without loss of generality we may assume that j3n $. E for all n e w and hence G^n+1 — Gpn ® e@nRpn- We put I = {((3n,in) \ n < u>}. Then I\ n [g]\ is finite for all g e Gp. We apply the Step Lemma 4.4 to /, P — Domfip and H — Gp. Therefore there exists an extension pQ = fp+i °f ?/3 and an element yp £ Ga such that ypfip ^ Ga and \\yp\\ = \\<j>p\\ = \\Pp\\- Thus the chain Ga (a < A) is constructed up to the used Step Lemma 4.4. Finally let G = (JaC\ Gait remains to show R = End G, the Step Lemma 4.4 and the following proposition which ensures that the U-property can be extended when applying the step lemma. We will use obvious simplified notations: Proposition 4.3. Let G = U n ew Gn> xn* 6 G\ be pure (Rn-torsion-free) elements, where G'n+l = Gn ® G^ C» Gn+i and Gn+i is obtained from G'n+1 by Theorem 3.14. IfbeGo,x

= Ene^n^-i + b

and G/

= (G,xRw)*, then U(RU) = ±F

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for the free group F = \Jne^ Fn, the pair (G', F) satisfies the U-property and G' is Proof. It is well-known that G' is Ni-free, see [2] or [6, 9] for instance. We want to show that the pair (G',F) satisfies the U-property. If r 6 RU and tp € F, then r £ Rn and tp e Fn for some n e w large enough. Hence there is no restriction to write any 0 ^ y £ G' as y = xkr + g with xk = 5Z >k '"-1 xn ~^~" i r ^ Em b — b e GQ. g = go ~\~ 5i> 9o € GQ, 0 ^ g\ € Gn and [Go] n [07] = [xkr] n [5] = 0. Suppose yy> = y, hence (xkr)ip + gtp = xkr + g and by continuity / m>k

(xmr)lP ~l~ 5V Sk

==

/ m>k

(Xmr)

i 9'

Sk

We consider any m > n, k and note that xm is a torsion-free element of the Enmodule Gm and also (p e U(Rn}. (hence Gm is invariant under application of y,r) and 0 7^ xmnp,xm(p e G m . Restricting the support [xkr] of the last displayed equation to xm shows that ^-1 (xmr) = s™,~1 (xmr}
@

^ea,i for some I* C A x p and let H be

a subgroup of B with P C* H C* B which is ^i-free and an R-module, where R = df Rf = |J Ra with I' =df [/]A = {a < A 3i < p, (a,i) e /*}. y = (F^.TT) e &* (i/iws Er = Z(5r}). Also suppose that there is a set I = {(a n ,z n ) | n < u)} C [P] = /* SMC/I t/iot «o < ^i < • • • < an < • • • (n < uj) are discrete ordinals and (i) H = (J G Q n , J? = |J E^; where #Qrv = Z{3 r Qn >, yn = (£<,„,&,„,*•„) e

^*

n
n
(ii) 7A n [5] A is finite for all g & H (7A = [I] A).

If 4>: P —* H is a homomorphism which is not multiplication by an element from R, then there exists an element y € P and f < y' = (H',3s',7r') & A* such that H C* H' C*B,y€ H' and y<j> i H1 . Proof. By assumption H is an .R-module and hence the S-completion H is an Rmodule. Let Z be the 8-completion of Z and let x — ^2 snean,in € P. We will use nGuJ

y = o : e P o r y = x + 6 e P f o r some 6 e,P to construct two new groups H C,, Hy: Let F^ = (H,yR)* C B. By Proposition 4.3 and Theorem 3.14 we obtain ty = (Hy, ^j/, TTy) € ^* such that y < y y and y e Hy. Clearly y^) e H. Put z = y0

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and y = x. If z £ Hx, then we choose H' = Hx and have px € A* with <j> £ End Hx. Also RX = Z($X}. If z € Hx, then also z € H'x by an easy limit argument and there exist integers k and n such that SkX<j> = grn + xr'n for some rn,r'n 6 .R and g & H. It follows x(sfc<j!) — r'n) = grn and rn,r'n 6 #n' for some n' £ w. Since (sfc$ - r'n) ^ 0 there is 6' e P such that b =df b'(sk(f> - r'n) ^ 0. Note that b' has finite support. Moreover, by the cotorsion-freeness of H there exists TT € Z such that Trb £ H. Let y = x + irb'. We claim that 0 ^ End(Hy). By way of contradiction assume that Si(x+Trb') = g*r*m + (x+-Kb')r^ for some integer I > k and elements r£,, r^* £ flm/ for some m' G w and g* £ ff. Without loss of generality, we may assume n = m, hence sj(x + 7r6')0 = £*r* + (x + 7r6')r** and if s = s;/Sfc, then > = si(x + Trb')(f> - ssfeX = g*r*n + (x + 7r6')r** - s(grn + xr'n) =• = (9*< - sgrn) + TrbV;' + x(rZ - sr'n}. Since [Trb'j = [&'], [7r6'^>] = [b'(j>] and g* ,g & H an easy support argument shows that n* = sr'm hence ssk(-Kb')4> = (g*r^ - sgrn) + s-rrb'r'n and thus sir(skb'(j) - b'r'n) — (g*rn~syrn] € H. By purity we get ~(skb'(f) — b'r'n} = Trb & H - & contradiction. D

r

Proof of Theorem 1.1. If R = EndG, then also Aut G = ±F, because U(R) = ±F and any r £ R\ U(R) viewed as endomorphism of G is not bijective by Proposition 3.9. Hence it remains to show that EndG C R. First we choose a new filtration of G and define G° =df {g 6 G | \\g\\ < a}

(a < A).

Lemma 4.5. The new filtration on G has the following properties. (a) G n P0 C G^ for all /3 < A; (b) {Ga 1 a < A} is a X-filtration of G; (c) Ifa,(3<\ are ordinals such that \\<j>p\\ = a then Ga C G$. Proof. Inspection of the definitions, or [10, 9].

D

Assume that £ EndG \ R and let ' — ((> \ B, hence 4>' ^ R. Let I = {(an,in) \ n < uj] C A x p such that ao < a\ < • • -an < • • • is a sequence of discrete ordinals and I\ n [g] \ is finite for all g € G. Note that the existence of I can be easily arranged, e.g. let E C A°, a G A° \ E, in 6 p (n < w) arbitrary and (an)n' ^ G'. (The last innocent sentence uses 4>' £ R and not just 0' ^ .R/3 for a suitable /?, see our comment in the construction of G.) By the Black Box Theorem 4.2 the set

E' = {a e E | 3/3 < A : ||^|| = a, ^ C 4>' , [y] C is stationary since ][y]| < H 0 . Note, [y] C [0^] implies that y e P/j. Moreover, the set G = {a < A | Ga0 C Ga} is a cub in A, hence E' n G ^ 0. Let a e S' n G. Then Gac/>' C Ga and there exists an ordinal /3 < A such that ||0^|| = a, 0 C and y £.Pp. The first property implies that G" C G/3 by Lemma 4.5 and the latter

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properties imply that <j)p £ R. Moreover, Pp C B with \\Pp\\ = a and hence Pp, and also P@(j) are contained in Ga C Gp. Therefore <j>p: Pp —> G/j with @ £ Rp and thus it follows form the construction that yp4>Q ^ Gp+i. On the other hand it follows from Lemma 4.5 that yp
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introduction that A-transitive in the sense of [5] is the same as transitive (or just T) in this context. Corollary 5.4. For any cardinal X = n+ with ^° = ^ there is an ^i-free abelian group of cardinality X which is E-transitive, but not transitive. In view of Proposition 5.1 we will strengthen this corollary further and sketch a proof: Theorem 5.5. For any cardinal X = p,+ with /U N ° = fj, there is an Ki-/ree abelian group G of cardinality X with AutG = ±1, which is transitive with respect to Mon( G. From Aut G = ±1 follows immediately that G is not a T-group. Sketch of a proof. The proof is very similar to the proof given for Aut G but simpler because the U-property must go, see Observation 5.2. We must run through the paper once more, essentially replacing Aut G by Mont G. First we will replace the definition pAut G by pMont G =df {tp \ ip: Dom<£> —» Imip C G, (p an isomorphism, G/Domip Hi-free}. If tf L pMont G, then (3) is the submonoid of plvlont G which is generated as a monoid by {±1} U #, hence (#} C pMont G by Proposition 5.1. The Definition 3.3 of an Ki-free pairs (G, ( pMont G is a map which naturally extends to {#) and satisfies the weak U-property: If ) n Dom7r( = xip' for all x e pG n D then tp = if'. (iv) h: y$x0(J) —> Im(/i) is a partial function from Dom/i C ^3 No (J) such that for u & Dom h C G the following holds. (a) h(u) is a countable subgroup of G and (Dom^ n h(u))ipl C h(u) for all_t e u (b) If G = G/h(u) and Tp\ denote the monomorphism induced by ifl, then (G, (vl)teu) is Hi-free. The definition of an order on 8. remains the same, except that condition (iii)(d) must be removed. Following the arguments along Section 3 we note that Lemma 3.11 and a simple version of Lemma 3.12 are crucial for the weak U-property. We obtain a theorem parallel to Theorem 3.14: For each j: = (G,3;,ir,h) £ ^ we can find r/ = (G',^', TT', h') 6 £ with y < y' such that 3" C pMont G' which freely

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generates {#') as a submonoid, {#'} acts transitive on pG', all members of (5") \ {1} are proper monomorphisms and \G'\ + ]#' = \G\ + \3\. As in Section 4 we modify G' and get a new G with endomorphism ring R = Z(^). Recall that (y) is a free monoid generated by #, hence U(R) = ±1. Modification of Proposition 3.9 will show that any e R \ U(R) is not in Mon t (G); thus Mont(G) = ($} and AutG = U(R) = ±1. The group G cannot be transitive. D References [1] A. L. S. Corner, The independence of Kaplansy's notions of transitivity and full transitivity, Quart. Journ. Math. (2) 27 (1976), 15-20. [2] A. L. S. Corner, R. Gobel, Prescribing endomorphism algebras — A unified treatment, Proc. London Math. Soc. (3) 50 (1985), 471-483. [3] M. Dugas, R. Gobel. Every cotorsion-free ring is an endomorphism ring, Proc. London Math. Soc. (3) 45 (1982), 319-336. [4] M. Dugas, J. Hausen, Torsion-free E-uniserial groups of infinite rank, pp. 181-189 in Abelian Group Theory, Proceedings of the 1987 Perth Conference, Contemporary Mathematics 87, Amer. Math. Soc., Providence, RI, 1989. [5] M. Dugas, S. Shelah, E-transitive groups in L, pp. 191—199 in Abelian Group Theory, Proceedings of the 1987 Perth Conference, Contemporary Mathematics 87, Arncr. Math. See., Providence, RI, 1989. [6] P. Eklof, A. Mekler, Almost free modules, Set-theoretic methods, North-Holland, Amsterdam 1990. [7] L. Puchs, Infinite abelian groups — Volume 1,2 Academic Press, New York 1970, 1973. [8] R. Gobel, S. Shelah, Decompositions of reflexive modules, Arch. Math. 76 (2001), 166-181. [9] R. Gobel, J. Trlifaj, Approximation and endomorphism algebras of modules, to appear at W. de Gruyter, Berlin (2003). [10] R. Gobel, S. Wallutis, An algebraic version of the strong black box, submitted [11] J. Hausen, On strongly irreducible torsion-free groups, pp. 351—358 in Abelian Group Theory, Proceedings of the .Third Oberwolfach Conference on Abelian Groups 1985, Gordon and Breach, London 1987. [12] J. Hausen, E-transitive torsion-free abelian groups, J. Algebra 107 (1987), 17-27. [13] P. Hill, On transitive and fully transitive primary groups, Proc. Amer. Math. Soc. 22 (1969), 414-417. [14] I. Kaplansky, Infinite abelian groups, Ann Arbour 1954. [15] C. Meggiben, A nontransitive, fully transitive primary group, J. Algebra 13 (1969), 571-574. [16] R. B. Mura, A. Rhemtulla, Orderable groups, Marcel Dekker, New York 1977. [17] S. K. Sehgal, Units in integral group rings, Pitman Monographs, New York 1993.

Generalized .E-Rings Riidiger Gobel Fachbereich 6, Mathematik und Informatik, Universitat Essen, 45117 Essen, Germany R.GoebelSUni-Essen.De Saharon Shelah Department of Mathematics, Hebrew University, Jerusalem, Israel, and Rutgers University, New Brunswick, NJ, USA [email protected] i.ae.il Lutz Striingmann Fachbereich 6, Mathematik und Informatik, Universitat Essen, 45117 Essen, Germany Iutz.struengmann8uni-essen.de Abstract. A ring R is called an .E-ring if the canonical homomorphism from R to the endomorphism ring End(flz) of the additive group Rz, taking any r £ R to the endomorphism left multiplication by r is an isomorphism of rings. In this case Rz is called an .E-group. Subrings of Q are obvious examples of E-rings. However there is a proper class of examples cpnstructed recently, see [8]. .E-rings come up naturally in various topics of algebra (see the introduction). So it's not surprising that they were investigated thoroughly in the last decade, see [4, 10, 7, 18, 21]. This also led to a generalization: an abelian group G is an E-group if there is an epimorphism from G onto the additive group of End(G). If G is torsion-free of finite rank, then G is an Egroup if and only if it is an E-group, see [14]. The obvious question whether the classes of .E-groups and E-groups coincide in general was raised a few years ago. We will answer it by showing that the two notions differ in the infinite rank case. Applying combinatorial machinery to non-commutative rings we shall produce an abelian group G with (non-commutative) End(G) and the desired epimorphism with prescribed kernel H. Hence, if we let H = 0, we obtain a non-commutative ring R such that End(-Rz) — R but R is not an .E-ring.

2000 Mathematics Subject Classification, primary: 20K20, 20K30; secondary: 16S60, 16W20. Key words and phrases. Indecomposable modules, .E-rings, homotopy theory. This work is supported by the project No. G-0494-081.06/93 of the German-Israeli Foundation for Scientific Research & Development. 291

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1. Introduction Easy examples of E-rings are subrings of Q and further examples up to size 2^° coming from p-a,dic integers have been known for a long time. Details on those rings can be found in the two monographs [12, 13]. Arbitrarily large E-rings were constructed more recently first in [8] and then in [7]. The examples in [7] share additional properties related to their automorphism groups. E-rings are important in connection with (strongly) indecomposable modules, see [1], [20] and [21]. They are also crucial for investigating problems in homotopy theory, see the work of Farjoun [10], [18] and Casacuberta, Rodriguez, Tai [4] or [22]. So it is not surprising that they were studied in detail over the last decade. In this connection, some years ago Feigelstock, Hausen and Raphael extended the notion of E-rings and called an abelian group G an EE-group, (we prefer the shorter script E-group) if there is an epimorphism from G onto the additive group of End(G). The results appeared recently in [14]. In particular the authors show that, if G is torsion-free of finite rank, then G is an E-group if and only if G is an E-group—the two notions coincide for groups of finite rank. The obvious question was clear: It was asked whether this is true in general. Here we want to answer this open question. The nice feature about this problem is the fact that combinatorial arguments are used and applied to non-commutative ring theory to produce the required examples. We recall the two definitions. Definition 1.1. If R is a ring, then 6 : R —> End(Rz) denotes the homomorphism which takes any r € R to the Z-endomorphism 6(r) which is multiplication by r on the left. If this homomorphism is an isomorphism, then R is called an E-ring and RZ is called an E-group. Recall from Schu'ltz [24] that E-rings are necessarily commutative. The rings R we will construct are definitely not commutative and so not E-rings. However they are close to E-rings in the sense that the following definition holds. Definition 1.2. If R is the endomorphism ring of some abelian group G and there is an epimorphism G —> R —» 0 with kernel H, then G is called an E(H)-group. Moreover, G is called an 'E-group if G is an E(H)-group for some abelian group H and G is a strong E-group if G is an E(0)-group, i.e., the epimorphism is an isomorphism. Note that trivially a strong E-group G is in fact a ring R with RZ = G and such that End(-Rz) == R as rings. Thus a non-commutative strong E-group cannot be an E-ring and hence we solve Vinsonhaler's 25-dollar-problem [23, Problem 1] which was asked for the first time in [24]. One of the main results in a paper by Feigelstock, Hausen and Raphael [14] is the following Theorem 1.3. Let G be any torsion-free abelian group of finite rank. Then G is an E-group if and only if G is an E-group. We want to prove the following result which complements the theorem by Feigelstock, Hausen and Raphael and answers their problem.

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Theorem 1.4. For any infinite cardinal A = /u+ with /J.NO = n there is an ^i-free abelian group G of cardinality \G\ = X which is a (strong] E-group. Remark 1.5. A modification of our construction would also ensure that the constructed E(H)-groups are proper in the sense that they are not strong ^.-groups. All one has to do is to satisfy that Ga/H ^ Ga for all a < X where G = (J Ga is the a<X

~E,(H)-group. Then the group G can not be a strong ^,-group. An .R-module is Hi-free if its additive group is Hi-free, i.e., all its countable subgroups are free. Recall that endomorphism rings of Ki-free abelian groups have Hi-free additive group. The key tool of this paper can be found in Section 2, where we investigate non-commutative polynomial rings over rings. The construction of G is based on a strong version of (Shelah's) Black Box as stated in [25] or slightly modified in [17], see also [5]. The paper is based on heretofore informal notes written in 1998. 2. Almost free rings over non-commuting variables Polynomial rings R[X] over a ring R -with commuting variable X are obviously free ^-modules. We will extend this to the non-commutative case, showing that the non-commutative polynomial ring R(X) as abelian group is Ki-free if RZ is so. This will be needed in Section 3. The ring construction is easy and well-known, see Bourbaki [3, pp. 216, 446 ff.]. Let R be a ring of characteristic 0, then M denotes all monomials, the elements (2.1) r-i-X"*1 • • • rnX*», ijt ^ 0 (j < n), rj £ R \ {1} (1< j < n) and rj ^ 0, (j < n). The ring R(X) is formally generated by all sums of the monomials in M. First we assume that RZ = ® Zt is freely generated as abelian group by T = {tj : i £ /} teT and to — 1 without loss of generality. Then the multiplication on RZ is coded into the so-called "constants of structure", see Bourbaki [3, p. 437], which are (2-2)

7jfc

e Z ( i , j , k e Z) with tj • tk = X>jVii€i

The constants are well-defined by independence. We want to use T to find a representation for elements in R(X). Any element in R can be expressed as a sum over T with coefficients in Z. If r 6 R(X) is a sum of monomials in M, then we may substitute any TV e R from (2.1) as indicated, and the polynomial in M turns into a sum of what we will call T-monomials. (2.3)

m = t^Xh • • • tinXjn e TM, with jk + 0 (k < n), tik ^ I (1< k < n).

Hence R(X) is generated as abelian 'group by all T-monomials and multiplication is ruled by the structure constants (2.2) restricted to T-monomials. Hence it seems plausible that the next lemma holds. We use the structure constants on R to define the ring multiplication on R(X). If m, m' e TM, let m = mt, m' = t'm'

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with m,m' £ TM and m ending with Xn, m' beginning with Xm> . Then the product is defined by cases. (2.4)

If t ^ 1 = t', then mm' = mtm' , and ift'^'i—t, then mm' = rnt'm' , (2.5)

if t' = 1 = t, then mm' = mm', with 'middle term' Xm+m'

(2.6) if t' ^ I ^ t, (2.2) for tt' = 5^7J fe ti,t = tj,t' = tk, is mm' = If to = 1 is involved in the last equation, then the summand 7°fcmm' has a 'middle term' xm+m as in case t = t' = I . So clearly mm' is a sum of T-monomials. Lemma 2.1. If R is a ring with RZ = 0 Zt, T = {i, : i € 7} and structure teT constants (2.2) as a&oi>e, then let R'z — 0 Zm be the direct sum taken over all rotTM

T -monomials TM as above. The multiplication on R' is now defined by (2. 4) -(2. 6) and it follows that R' = R(X) as rings with center %R' — 1Z. Proof. Reducing elements in R(X) to sums of T-monomials we have seen that R' = R(X) as sets. Moreover, considering sums of T-monomials it is obvious that R% = R(X)z as abelian groups. We finally must show equality as rings. There are two natural ways to do this. Either we extend the structure constants on R to R', R(X) and check their ring properties (see Bourbaki [3, p. 438]) or we check that ^'-multiplication is 7?{X)-multiplication. We prefer the second way. Hence we must show that (2.2), (2.4)-(2.6) and its linear extension define uniquely the ring structure on 7?' which is the same as R(X). Any case of the ring laws reduces to consider distributivity of a product (2.7)

(zm + z"m")mr

with z,z" e Z, m, m' as in (2.4)-(2.6) and m" = m"t" similar to m above. If t = t" = 1 or t' = I , then (2.7) becomes immediately the unique T-monomial zmm' + z"m"m' by (2.4)-(2.6) and linear extension. If m — m", then (2.7) can be treated with (2.4) and distributivity in T given by (2.2):

m(zt + z"t")t'm? = m(ztt' + z"t"t'}m' = zmtt'm' + z"m"t"t'm' = zmm' + z"m"m' showing the unique 7?(X)-multiplication (by TO') in this case. If m ^ m', then zmm' and z"m"m' are sums of distinct T-monomials, hence (2.7) defines the unique 7?(JSL")-multiplication by linear extension. D Our main interest in Lemma 2.1 is extracted as the following Corollary 2.2. (a) 7/7? is the ring above with R% free, then R(X)z/R% (b) IfRz is Hi-/ree, then R(X)z/Rz is Hi-/ree.

is free.

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Proof. By Lemma 2.1 we have T < TM and TM is a Z-basis of R(X)Z, hence (a) follows. For (b) choose any countable set C of R and let RC = {{G}} be the subring generated by C and Rc(X) the subring of R(X) generated by C and X respectively. The ring RC is free by hypothesis and Rc(X}/Rc is free by (a), hence (b) follows. D If J denotes the sum of all those monomials with at least one factor X, then J is a two sided ideal of R(X) (of 'non-constant' polynomials). We have R(X) = R® J and if R is a field, then J is a maximal ideal of R(X). Separating summands of higher order becomes more complicated and fortunately is not needed. Corollary 2.3. Let R be as above. The set J of sums of monomials which are not constant are a two-sided ideal of R(X) and R(X} = R@ J is a ring split-extension. 3. A class of quadruples for constructing E-groups We want to find an abelian group G with R = End G and u : G —> R an epimorphism with prescribed kernel ker(cr) = H. Hence G is a left .R-module and the epirnorphisrn a induces a Z-homomorphism

cr* :-G -> R such that <j» (x) G R = End G is defined by (3.1)

a,(x)(y)=<j(y)xfoiaQ.x,y

e G.

The construction of (G, R, R is a Z-homomorphism with kera = H, and IR €E Im(cr) C» R, (e) cr» : G —» EndG is a Z-homomorphism defined by (3.1) and a*(g) € R for each g € G, hence cr» : G —* R, (f) Amifl G = 0 and R C + End(G). We often use qa = (Ra,Ga}aa,aa*) £ £# for those quadruples. The next lemma is used to prove Proposition 3.3. Lemma 3.2. (a) If G is a free abelian group of finite rank, then EndG is a free abelian group. (b) If G is Hi-/ree, then EndG is an ^i-free abelian group as well. Proof. Any endomorphism of G can be represented as an element of the cartesian product GG and End G C GG as abelian group. However Hi-freeness is closed under cartesian products and subgroups, hence EndG is Ni-free. If G is freely generated by a finite set E, we may replace GG by GE which is free and (a) follows. D

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The class R.R of quadruples is partially ordered by inclusion, i.e., if q = (R,G,a,af),q' = (R',G',a',a'j

£ &H,

then q < q' if and only if R C R', G C G', a C a' and (hence) cr, C cr^. The following proposition will ensure that our construction of an E(JJ)-group will take place within &H. Proposition 3.3. (a) If qi = (.Ri,G ! t,0t,0'i»), (i & I) is a continuous, ascending chain of quadruples in &H, and G; C* Gj+i, 1^ = l.Ri+i /or oZZ i £ I, tften U 9i = (|J fli, U G 4 , (J ^ U ffi.) € £«. ie/

t€/

t€/

i€/

t6/

(b) (Z, Z® H,a,(7f) £ &# w/iere cr : Z® H ^ %, is the canonical projection. (c) I/ q £ £ff, iften i/iere exists q < q' € RH such that RQ C aq>(Gq') and G, C. G,/. Proof, (a) By continuity and (3.1) we have cr* = ( U CTJ), = U ie/ »e/ .R with pure image, where G = [J Gj and H = U Uj. Moreover, JJ is a unital ring te/ te/ with l fl = 1^ for alH £ / and G is a left .R-module. GZ is Hi-free since G» is pure in GJ+I for all i £ / and therefore R C End^ G is Hi-free as well by Lemma 3.2(b). Note that Ann^ G = 0. Finally, keru = H and IK £ Im(a) is clear. Hence it remains to show that R is pure in Endz G. Assume that tp £ Endz G and n(p £ R for some integer n. Thus there is some i s / such that n

i. Let g £ Gj, then n^(^) = r'^ € Gj and hence ^(5) £ Gj by purity. Thus

i by torsion-freeness. Thus

/?" satisfies kercr' = kercr = H and J? C Imcr' as required. Clearly, Im EndgG', hence /?" must be enlarged for cr^, : G' —> fZ' in Definition 3.1(e). The ring R" acts by scalar multiplication (on the left) on G'. The action is faithful by hypothesis (Definition 3.1(f)), hence R" can be viewed as a subring of Endz G'. Let R' = R"[Im cr^]* be the pure unital subring of Endz G' generated by R" and Im a*. Obviously q < q' for q' — (R1, G',u',cr(,) and a1 = a'q satisfies (c) of Proposition 3.3. It remains to show that q' £ &H. We must check Definition 3.1 (c), (d) and (f). We have kercr' = kera = H. Moreover, if g £ G', then a'*(g)(G') = 0 implies o-'f(g) = 0, hence Annn< G' = 0 and (f) follows. It remains to show that Imcr' is pure in R'. As shown above, Imcr' is pure in R", hence it suffices to show that R"

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is pure in R'. We will even show that R" is pure in Endz G'. Let ip e Endz G' and assume that rap = rx0 ® r'x e R" for some integer n. By torsion-freeness of G' and RZ it follows that

' = r and therefore Endz G is a 'Z-homomorphism and R acts faithfully on the left R-module G by scalar multiplication, then we denote by Ra, C, End z G the pure unital subring of Endz G generated by R C Endz G and Ima*. The next proposition provides the link to our construction in Section 4. Proposition 3.5. Let q = (R,G,a,crt) 6 AH- Then there exists a 'transcendental extension' q < q' £ AH such that the following holds for q' = (R': G',CT',<7»). (a) G' = G ® R(X)e as R(X}-module where X acts as identity on G. (b) ff'(g) = cr(g)Xe, a' (re) —re if r 6 R and a' (re) = rXe otherwise. (c) R' = (R(X))< with R(X) from Section 2. (d) E C Proof. We must show that q' 6 &H. The ring R is Hi-free by hypothesis, hence R(X) is Ki-free by Corollary 2.2(b) and R' is Hi-free by Lemma 3.2. Clearly, keru' = kercr = H and Imcr' C^ R' with R C Im(cr) follow as in the proof of Proposition 3.3. Mdreover, G' is HI -free and Ann#> G' = 0, so the proposition follows. D We have an immediate corollary-definition. Corollary 3.6. If Kfa = {q 6 B-HiGq rnaps onto Rq}, then 8.°H is dense in AHProof. Apply Proposition 3.3(c) u> times and note that Proposition 3.3(a) can be used because the union of the countable sequences of rings and modules respectively are Hi-free by construction. Hence any q e &H is below some q' & Kfa. D 4. Construction of E-groups by black box arguments The combinatorial ideas of Sections 4 and 5 can be found in Shelah [25], see also the appendix Shelah' s Black Box in Corner, Gobel [5] and this Black Box could be used. However, we will use a stronger version of the Black Box as developed in [17]. The Black Box needs, as usually, some minor alterations, which are obvious and the proof is left to the reader. Let A be some infinite cardinal, and <{Xa : a < A} a sequence of transcendental elements which will be used to define ririg extensions. First we define ring extensions 'locally' and let Ra+i 2 Ra (Xa) = Ra (Xa) = Raca ® JaC'a a rmS direct sum with central idempotents ca and c'a where Ja is

298

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defined as in Corollary 2.3. Since there is no danger of confusion we usually will omit the 'place holders' ca and c'a. The sequence of rings Ra with \Ra\ < X (a < A) will be completed during the construction of the abelian group G (with End^G = U Ra)i taking unions at limit steps, i.e., Ra = \J R@ if a is a limit a<\

P
ordinal. Note that |J Ra = \J a<\

Ra(Xa).

ct<\

Similarly we define B = 0 Ra (Xa) eaCGCB a<X

where B denotes the p-adic completion of B (as an abelian group) for some fixed prime p. Since we want to apply the Black Box later on we need a free basis-module inside B. Therefore, we assume that our rings Ra (and hence Ra(Xa}~) (a < A) are Hi-free, hence homogeneous of type Z. By a well-known result [16, Theorem 128] there exists for each a < A a completely decomposable group Fa C* Ra(Xa) such that \Ra(Xa)\ < Fa\*° and fr'a C^ Ka(Xa) c_ t t<'a where Fa is the p-adic completion of Fa (a < A). Note that Fa is a free abelian group since Ra(Xa) is Ni-free. We collect the Fa (a < A) and define F := 0 Faea. a<\

Thus F is a free abelian group of cardinality at most A such that FC*BCf F. Let Fa = 0 Za£ where p = |F|. Writing e( £] «) for a£ea it follows that B' = e
0

t

___

Ze( £)Q ) satisfies B = B' . For later use we put the lexicographic ordering

(e,a)6pxA

on p x A; since p, A are ordinals p x A is well ordered. We are ready to define supports of elements in -B. If 0 ^ g e B then we can write g = £ gaea and / C A, |J| < H 0 , ga e Ra(Xa). Moreover, each ga = g^ a£l

with g'a € Ra and g'^ S Ja. We define two kinds of support. (4.1)

The A-support of g is the set [g}\ = {a < A : ga ^ 0}.

The notion of A-support naturally extends to subsets of B, see again [17]. On the other hand, any element 0 ^ g & B = B' can be written as 9 = (#(e,a)e( £ ,a))( e ,c<)epxA & B' C

and we define the 'usual' support of g by [g] = {(e, a) e p x A | P(e,a) 7^ 0}. Note that |[^]| < HO and that the A-support of g is [g]^ = {a < A ] 3e < p : (e,a) € [g]}. As usual we may define a norm on B' by ||'{o;}|| = a+1 (a < A), ||M[| = sup a6M ||a[| (M C A) and \\g\\ = \\\g]>\\ (g ej?), i.e., ||5|| = min{/? < A [
Generalized E-Rings

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have a ring support and ring norm denned by [g]nng = {£ < P \ 3a < A : (e, a) 6 [g]} and \\9\\ring = \Mring\\-

We will now state a suitable version of the Strong Black Box as developed in [17]. The proof is almost identical with the one in [17, Section 1] and will therefore be left to the reader but we will give all the necessary definitions and results. Fix cardinals K > KQ, p, = JJLK such that A = /K + . We need to say what we mean by a canonical homomorphism. For this we fix bijections 7 for all 7 with p, < 7 < A where we put
0

Ze(e a )

(e,a)e/

for some I C p x A with \I\ < K such that:

• • • •

if (e, a) e / tfien (e, e) e /; i/ (e, a) e I, a 6 p then (a, e) e /; if (e, a) e / then (/ n (/x x /i)) g £>a = / n lmgEia; and ||P|| < A", where A° = {a < A | cf (a) = H0}.

Accordingly, ip: P —> £?' is said io 6e a canonical homomorphism i/P is a canonical summand of B' and Im B' and for any subset I of A with |/| < K the set {a €E \3ip€C*: \\p\\ =a,^C^,I is stationary. For the proof of the Theorem we have to define an equivalence relation on C: Definition 4.3. Canonical homomorphism ['] such that (xf) domt/j' is the unique extension of the R-homomorphism defined by e(e,a)f = e (e , a)/ ((e, a) e [
300

R. Gobel, S. Shelah, and L. Striingrnann

Definition 4.4. Let ipo C n\\ < Pn < \\. Moreover, two admissible sequences (<£ n ) n'n)n are said to be equivalent or of the same type if
is also

an fi

l-

TT,
ement of C. Moreover, if we let T be the set of all possible types of admissible sequences of canonical homomorphisms, then clearly |T| < /J,K = \L. If (f>n)n) and (i) M= U Gan,R= U flan; n
n

(ii) (^a n ,Ga n ,o- Qn ,a Qrl *) 6 ^ff /or all n< u (iii) JA n [5] A is /mi£e /or all g £ M (Ix = [I}\). Moreover, let (p : P —> M 6e a homomorphism which is not multiplication by an element from R. Then there exists an element y € P and an element q C q' = (R', M',a',a'f'} e &# swc/i i/iai M C, M' C» B7, y e M' anrf y

Again we may choose R^ = R^ [Im a [Mi C P]f C R. Continuing this way we obtain a sequence of groups M™ and rings R™ (n 6 w) such that M™+1 is an .R"-module and a(M£) C R™. Taking My = (J M™ and R'y = \J Ry we get that My is an .R^-module and ay = n(Eoi

n£uj

& \My'- My —> E^. By a standard support argument we see that My and (hence) R'y

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R. Gobel, S. Shelah, and L. Striingmann

are still Hi-free. Finally take Ry = (R'y) , C Endz(M y ). By construction Imay is pure in Ry and hence qy = (Ry,My,<Jy,cr*) G AHAt this stage we determine y more specifically in order to obtain that ip g EndzMj,. Let y = £ p"e (enian) and x =

M' = My and R' = Ry and hence qy € &H with

for some rn,r'n e J?™ and g 6 M. It follows that (pk(P - ^) J/ = ^nff. Since (pk(p — r'n} ^ 0 there is b' £ P such that

(pV-O^Vo. Note that b' has finite support. Moreover, by the cotorsion-freeness of R there exists TT £ JJ such that irb £ M with 6 = (pk

$. Endz (My). By way of contradiction assume that pl A; and elements r^,r^ € -R™, and 5* e M. Without loss of generality, we may assume n = m, hence

Let s = pl /pk . Hence TT&') - spkip(y) = r*ng* + r'*(-Kb' + y)- s(rng + r'ny) = = (r*ng* - srng) + r'jirb' + (r* - sr'n)y. Since [ntf] = [b'}, (ip(-Kb')} = \
rn5 e M.

k

By purity we get Tt(p (b') - r'nb') = nb £ M— a contradiction. Finally we put M' = Myl, R' = Ry, and q' = qy e &H. Thus

where j,<*> = E n>fc ^

e (£n , Qn) or j/W = E §e(,n,«n) + n^b. n>fc V

D

Generalized .E-Rings

303

We will now carry on the construction to a and distinguish three cases. Case 1: Suppose a is a limit ordinal. Then Ga = (J GQ is Hi-free by (iii) 0
and hence Proposition 3.3(a) shows that we can take unions, i.e., qa = (J qp. /3
Case 2: Suppose a = 0 + 1, then ||<^g|| G A°. Assume that Imipp <2 Gp or (pp G -R/3. In this case we let Gp+i = Gp © Rp(Xp)ep as in Proposition 3.5 with R'a = Rp (Xp) - Rp (Xp) with X acting as identity on GQ. We let Ra = (-R'Q)CT, and by Proposition 3.5 it follows that (Ra, Ga, aa, a*a) G &H where aa and cr* = craf are taken from Proposition 3.5. Put yp = 0. Case 3: Suppose that a = /3 + 1 and Imipp ^ Gp, ipp £ Rp. In this case we try to 'kill' our undesired homomorphism ) such that @o < Pi < • • • < fin < • • • and sup new /3 n = ||
\\y0\\ = \\v0\\ = \\Pp\\-

Finally put qH = (GH,RH,VH,CH*)

= U 9«

e

&H- Obviously, GH has

a
cardinality A and R C Endg G. Moreover, k
Next we describe the elements of GH • Lemma 5.2. Let GH be as above and let g G GH \ B. Then there are k < LJ and a finite subset N of A* such that g G B+ ^ RpUg ana [ffJA H [J//J]A is infinite if and /3eJV /5 ^ -W- In particular, if \\g\\ is a limit ordinal then \\g\\ = \\yp, \\ = \\f@, \\ for

/?* =

Proof. Let g G GH = B + ^ ^3 Rpyp • Then there exist a finite subset AT' of /3
A*, 6 G B, fc G w, a^n & Rp (0 & N', n
Since y^

— ^y&

£ B' C B this expression reduces to

9=b+ '/3GAT'

for some ap € Rp (0 G A^'), b' G B'. Putting W = {/? G AT' | a^ / 0} (A/' ^ 0 for g ^ B) the conclusion of the lemma follows since [yp}\ n [y/j'JA is finite for / ? / / ? ' by Corollary 4.6(iii). D

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Using the above lemma we prove further properties of GHLemma 5.3. Let GH be as above and define Ga (a < A) by Ga := {g £ GH \ \\9\\ < a, \\g\\ring < a}. Then: (a) GH H Pp C Gp+i for all 0 < A* ; (b) {Ga a < X} is a X-filtration of GH', and (c) if (3 < A*, a < A are ordinals such that \\. We construct qn = (Rn, GH, O~H,&H*) = U qa as in the previous section. Thus RH C* Endz GH and a : GH —* RH with a
kernel kercr# = H. Moreover, GH is Ni-free and RH is obviously non-commutative. We first claim that an is surjective. Therefore let r £ RH, hence there exists a < A such that r £ Ra. Without loss of generality we may assume that a 0 E. By (v) we conclude that r e Ra C Imaa+i- Since cra+i C <JH as functions it follows that r £ Im O~H and thus O~H is surjective. It remains to prove that RH = EndzG/f- Assume that

, hence
Generalized JE-Rings

305

a0 < ai < • • • an < • • • and JA n [g]\ is finite for all g e GH- Note that the existence of I can be easily arranged, e.g., let E C A°, a € A° \ E, £„ € p (n < LJ) arbitrary and (an}n
E' = {a e E | 3/3 < A* : ||^|| = a,^ C y,', [y] C [V0]} is stationary since |[j/]| < KQ. Note, [y] C [1^3] implies that y e dom<^0. Moreover, the set C = {a < X : 0 C B' with || domip /}\\ring < \\6omipp\\ = a and hence domipp, and also (domip^ip are contained in Ga C Op . Therefore Gp with Endz G is an isomorphism, where Endz G is non- commutative. Hence, R := G has a non-commutative ring structure such that Endz(-Rz) = R. D References [1] D. Arnold, R. S. Pierce, J. D. Reid, C. Vinsonhaler, W. Wickless, Torsion-free abelian groups of finite rank projective as modules over their endomorphism rings, pp. 1-10 in J. Algebra 71 (1981). [2] R. A. Bowshell, P. Schultz, Unital rings whose additive endomorphisms commute, pp. 197-214 in Math. Ann. 228 (1977). [3] N. Bourbaki, Algebra I, Hermann, Publ. Paris 1974. [4] C. Casacuberta, J. Rodriguez, Jin- Yen Tai, Localizations of abelian Eilenberg-Mac Lane spaces of finite type, Preprint, Dept. of Math. Universitat Autonoma de Barcelona (1997). [5] A. L. S. Corner, R. Gobel, Prescribing endomorphism algebras — a unified treatment, pp. 447-479 in Proc. London Math. Soc. (3) 50 (1985). [6] M. Dugas, Large B-modules exist, pp. 405-413 in J. Algebra 142 (1991). [7] M. Dugas, R. Gobel, Torsion-free nilpotent groups and E-modules, pp. 340-351 in Arch. Math. 54 (1990). [8] M. Dugas, A. Mader, C. Vinsonhaler, Large E-rings exist, pp. 88-101 in J. Algebra 108 (1987). [9] P. Eklof, A. Mekler, Almost free modules, Set-theoretic methods, North-Holland, Amsterdam 1990. [10] E. D. Farjoun, Cellular spaces, null spaces and homotopy localizations, Springer Lecture Notes 1622 Berlin, Heidelberg (1996). [11] T. G. Faticoni, Each countable torsion-free reduced ring is a pure subring of an B-ring, pp. 2245-2564 in Comm. Algebra (12) 15 (1987). [12] S. Feigelstock, Additive groups of rings, Research Notes in Math. Vol. 83, Pitman, London 1971.

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[13] S. Feigelstock, Additive groups of rings II, Research Notes in Math. Vol. 169, Pitman, London 1988. [14] S. Feigelstock, J. Hausen, R. Raphael, Groups which map onto their endomorphism rings, pp. 231-241 in Proceedings Dublin Conference 1998, Basel (1999). [15] L. Fuchs, Infinite abelian groups — Volumes 1,2 Academic Press, New York 1970, 1973 [16] P. Griffith, Infinite abelian group theory, Chicago Lectures in Mathematics, The University of Chicago Press, Chicago and London 1970. [17] R. Gobel, S. L. Wallutis, An algebraic version of the strong black box, preprint. [18] A. Libman, Localization of groups, G-modules and Chain-complexes, PhD-thesis, Hebrew University 1997 [19] A. Mader, C. Vinsonhaler, Torsion-free ^-modules, J. Algebra 115, 401-411 (1988) [20] A. Paras, Abelian groups as Noetherian modules over their endomorphism rings, pp. 325-332 in Contemporary Mathematics 171 Amer. Math. Soc., Providence, (1994). [21] R. S. Pierce, ^-modules, pp. 221-240 in Contemporary Math. 87 Amer. Math. Soc., Providence 1989 [22] J. Rodriguez, On homotopy colimits of spaces with a single homology or homotopy group, PhD-thesis, Universitat Autonoma de Barcelona (1997) [23] C. Vinsonhaler, E-rings and related structures, pp. 387-402 in Non-Noetherian commutative ring theory, Math. Appl., Kluwer Acad. Publ. Dordrecht (2000). [24] P. Schultz, The endomorphism ring of the additive group of a ring, pp. 60-69 in Journ. Austral. Math. Soc. 15 (1973^. [25] S. Shelah, A combinatorial theorem and eHdomorphism rings of abelian groups II, pp. 37-86, in Abelian groups and modules, CISM Courses and Lectures, 287, Springer, Wien 1984

Butler Modules and Bext Pat Goeters Department of Mathematics, Auburn University, Auburn, AL 36849-5310

Charles Vinsonhaler Department of Mathematics, University of Connecticut, Storrs, CT 06269-3009

Let R be an integral domain with quotient field Q. An J?-module G is torsionrx = 0 implies ~ = 0 or x = 0 whenever r € R and x € G. The rank of a torsion-free module G is the size of a-maximal linearly independent subset of G, from which it follows that X is torsion-free of rank 1 if and only if X is isomorphic to a submodule of Q. A finite rank completely decomposable module is a finite direct sum of rank 1 modules. In [6], Butler considered the class of torsion-free modules that are quasi-isomorphic to images of finite rank completely decomposable modules. We will call such a module, a Butler module. The definition of quasi-isomorphic modules will be given later, but we remar^here that it follows from Maths' theory of divisible modules [13], that over noetherian domains of Krull dimension 1, the class of images of finite rank completely decomposable modules is closed under quasi-isomorphsm [9], so Butler modules are just the epimorphic images of finite rank completely decomposable modules when R is restricted in this way. In [4], Bican and Sake developed a characterization of Butler modules that has spawned considerable research. A sequence of modules 0 -> A -> B -> C -+ 0

is called balanced, if each rank 1 module is projective relative to the sequence. One then considers the submodule Bext)j(C, A) of Ext]j(C, A) determined by the equivalence classes of balanced sequences. Bican and Salce showed that a Butler module over the integers is a torsion-free module B of finite rank which satisfies Bext^(.B, T) — 0 for all torsion modules T. In this paper we will investigate the relationship between the class of torsion-free modules B of finite rank for which Bextjj(5, T) = 0 for all torsion modules T and the class of Butler modules over noetherian domains of Krull dimension 1. In particular, we show that when R is a countable, noetherian domain of Krull dimension 1 with finite integral closure, G is a Butler module if and only if Bext^(G, T) is 307

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bounded for all torsion modules T (recall that a torsion module U is called bounded if there exists 0 ^ r € R such that rU = 0). Therefore, a module satisfying the criterion of Bican and Salce is necessarily Butler, but we give an example of a subring R of an algebraic number field and a Butler module G over R for which BextJj(G,T) 7^ 0 for some torsion module T.

1. Preliminaries In this paper, R is always assumed to be a noetherian domain of Krull dimension 1 (i.e., every nonzero prime ideal is maximal). The integral closure of R in its quotient field Q will be denoted by R, and from the Krull-Akizuki Theorem [15], R is Dedekind. Definition. Given a torsion-free .R-module G, the largest .R-submoduie of G is denoted by G and coincides with the image of the evaluation map Hom#(.R, G) Cg>£ R-+G. The definition of G works because the torsion-free image of an ^-module is again an ..R-module: to see this, with (j>: M —> G, M an R-module, we note b(x) and because G is.torsion-free, ((a/6)x) = (a/6)(x) whenever a, b e R such that a/b £ R. The price to pay for this notation is that R has two possible meanings, but we always take R to mean the integral closure of R. When M = R/R, the image of HomR(.R, M) <8>£ R —» M is all of M, however M is not an .R-module (unless R = R); so the requirement that G be torsion-free in the definition is necessary. Torsion-free modules G and H are called quasi-isomorphic if G is isomorphic to a submodule G' of H such that H/G' is bounded. * Proposition 1. Assume that R is finitely generated as a module over R and let G be a torsion-free R-module. Then G and G are quasi-isomorphic. Furthermore, G is a Butler module if and only if G is a Butler module over R. Proof. By assumption, there exists 0 ^ r € R such that rR C R. Then, rG C G, and G and G are quasi-isomorphic. If G is a Butler .R-module, then G is a Butler .R-module, and since G is quasi-isomorphic to G, G is a Butler .R-module [9]. Conversely, if G is a Butler .R-module, and is the image of the finite direct sum GX © • • • © Cn of submodules Cj of Q, then G lies between G and the image of G! © • • • © Cn so G is a Butler module over R. D Given a maximal ideal P of R and an element a; in a torsion-free module the P-height of x in M is taken to be the largest positive integer n = n(P) which x 6 PnM when such an integer exists, and is otherwise defined to be symbol oo. In this way we obtain a height vector h(x) = (n(P) P €E max(R)} x in M. Given a torsion-free module G and a maximal ideal P of R, Gp will denote localization of G at P.

M, for the for the

Lemma 2. Let G be a torsion-free module, x & G and let P be a maximal ideal of R. Then the P-height of x in G is equal to the Pp-height of x in Gp.

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Proof. It suffices to show that PnG = PnGP D G for any n > 0. One containment is obvious; for the reverse, let g e PnGp n G. By clearing denominators in the localization, we find there exists an r £ R\P such that rg £ PnG. Since Rr + Pn — R, write 1 = sr + x for some x e P71 and s £ R. Then 5 = srp + xp is in P n G as desired. D We call a submodule H of a module M, relatively divisible, if rG C\H = rH for every r £ R. Lemma 3. Le£ H be a submodule of a torsion-free module G over a Dedekind domain R. Then, H is relatively divisible in G if and only if for all x e H, the P-height of x £ H is equal to the P -height of x measured in G for all maximal ideals P of R. Proof. This lemma follows immediately from the last, since heights and purity localize and the present result is well-known for pid's. D Two height vectors h, h' are said to be equivalent, provided hp| < oo, where hp and hp are the P-components of h and h' respectively, under the proviso that oo ± n = oo and oo — oo = 0. It is easy to show that when X is a rank one module over a noetherian domain of Krull dimension 1, the height vectors of different nonzero elements of X are equivalent [9], so, up to equivalence, there is only one height vector associated with X. In [9] the following two conditions were shown to be equivalent for modules over 1-dimensional noetherian domains: (1) Every Butler module is a relatively divisible submodule of a finite rank completely decomposable module, and (2) Every rank 1 module is determined' up to quasi-isomorphism by the equivalence class of its height vector. In turn, these conditions imply that R is finitely generated over R. This suggests that the proper context in which to consider Butler modules is under the assumption that R is finitely generated over R.

2. Dedekind Domains For abelian groups, Reinhold Baer proved in [6] that Ext^G, T) is torsion free when G is countable torsion-free, and T is torsion. We offer our own proof for Dedekind domains and also show that this condition characterizes Dedekind domains among the noetherian domains of Krull dimension 1. Recall that because R is commutative, the Po-module structure on A and B induces a natural PL-module structure on Ext)j(^4,5) (Theorem 7.16 in [16]). We will assume the natural module structure is used on Ext^(A, B). The next fact is an easy exercise. Lemma 4. // a belongs to the Dedekind domain R, and A and B are R-modules such that multiplication by a is an automorphism of A or B respectively, then multiplication by a on Ext)j(A, B) is an automorphism. We will require a generalization of Baer's result on Ext. The proof, an induction on rank, requires the following.

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Lemma 5. If X is a torsion-free rank 1 module over a Dedekind domain R, and T is a torsion module, then 'Ext1R(X,T) is torsion-free. Proof. Recall, the torsion module T decomposes as T = 0P Tp, where the sum is over all maximal ideals P of R, and Tp = {x e T ann^ x = Pk for some k} (this is proven in more generality in [13]). In turn, ® p Tp < OP^P- Regarding the sequence 0 -» EomR(X, T) ^ HomR(X, [] TP) -» Homfi(X, J] TP/T) P P

we claim that Cokera and Hom#(X, f}p Tp/T) are torsion-free divisible. Once this is done, Im/? is seen to be torsion-free divisible, and to complete the proof it remains to argue that each Ext'R(X,TP) in E-xtR(X,HPTp = HPEyitR(X,Tp) is torsion-free. If / G Homfi(Js:, HP TP), and 0 ^ r € # are given, let Pl5 P2, . . . , Pm be the maximal ideals of R containing r. If r'f £ Ima, and x € X, then since multiplication by r on any Tp where P =£ Pj is an automorphism, if P[, P^, . • • , P'k are the maximal ideals that form the support of rf(x) (i.e., rf(x) € ®j=iTp'.), then f(x) € X)71i -^Pj + Si=i ^P' ' an(^ consequently Cokera is torsion-free. We can write / as / = A + /2 where f i \ X —* ® JLi TP,- and /2 : -X" ~* ILvanyp,^- Then A 6 ImQ . r~lfi 6 Hom fl (X,ripTp), and r^-1^) = f mod Ima. Hence, Cokera is torsion- free divisible. Replacing X with R above quickly shows that fJpTp/T == Hom^(l?, f|p max 7p/T) is torsion-free divisible. It remains to argue that ExtJjpC, Tp) is torsion-free. We use the embedding of X in its localization Xp: For P = {P1 £ MaxR \ P'X ^ X and P' / P}, D is isomorphic to the torsion divisible module ©p,eP Q/Rp>. Since ^? is Dedekind, Ext^(A, B) = 0 for all modules A and B so we obtain the induced sequence; 0 -> HompCp,Tp) -» Hom(X,T P )

^ Ext^(^p,Tp) -> Ext^X.Tp) ^ 0. Since multiplication on Tp by any element of R \ P is an automorphism of T, the map Hom(Xp,Tp) —> Hom(X, Tp) is an epimorphism; that is, given / £ Hom(X,T), define 5 £ Hom(A"p,Tp) by g(x/c) = c-1/^) for any c e E \ P. By Lemma 4, Ext)j(D,Tp) is torsion-free divisible, so Ext^(X, Tp) is torsion-free if and only if Ext}j(Xp,Tp) is torsion- free. But Xp is either -Rp or Q (up to isomorphism); and ExtJj(Q,T) is torsion-free by Lemma 4, while ExtJj(.Rp,Tp) = (RP, Tp) = 0. Therefore, Ext^(X, Tp) is torsion-free. D

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Theorem 6. A Noetherian domain R of Krull dimension 1 is Dedekind if and only if for every torsion-free module G of finite rank, and torsion module T, Ext}j(G, T) is torsion-free. Proof. First assume that R is Dedekind. The previous lemma provides a basis for induction on the rank of G. We may of course assume that T is reduced. Assuming G has rank greater than 1, let X be a rank 1 relatively divisible submodule of G and let H be the quotient G/X. By induction, ExiR(H,T) and Ext^X.T) are torsion-free. Consider the sequence Hom(G,r) 2+ Hom(X,T) - Ext^(/f,T) -» Ext^(G,T) -> Ext^(X,T) -» 0. Since ExtJj(J?, T) is torsion- free, the image of a must contain the torsion submodule of Hom(X, T). We claim that the image of /3 is divisible; this will show that Extfl(G,T) is the extension of a summand of Extp(ff,T) by Ext]}(A",T) and is therefore torsion- free. To verify the claim, we regard X as a submodule of Q containing R. Then for L = X/R, we obtain the sequence 0 -> Hom(£,T) -> Hom(X,T) -> Hom(fl,T). Given a submodule A of a module B such that B/A is torsion, if A modulo its torsion submodule, A/t(A), is divisible, then B modulo its torsion submodule, B/t(B), is divisible (in fact, the natural map A/t(A) —> B/t(B) is an isomorphism). Therefore, to show that Hom(X, T) modulo the divisible submodule is torsion, it suffices to show that Hom(L, T) is divisible modulo its torsion submodule. With Lp and Tf> equal to the respective P-primary components of L and T, Hom(L,T) = npHom(Lp,Tp). Because, TP is reduced, Hom(Lp,Tp) = 0 when Lp is divisible (i.e., when Xp = Q), and Hom(Lp,Tp) is bounded when Xp ^ Q. It follows that the torsion submodule of Hom(L, T) is equal to ®p Hom(Z/p, Tp); a direct sum of bounded primary modules. Therefore, Hom(Z/, T) is divisible modulo its torsion submodule. To show the converse, we will restrict to torsion modules T for which T = Tp and ideals / of Rp for some maximal ideal P, and show that Rp is Dedekind by arguing that / is principal. Observe that Ext^ p (/,T) is an .R-submodule of Extk(7,T), and note that Ext^(/,T) is bounded. By assumption, Ext#(/,T) is torsion-free; from this it follows that ExtlRp (I, T) must be zero. For 0 •£ a e /, I/aRp has finite length as an .Rp-module since RP is Noetherian with Krull dimension 1 [15]. Therefore, to show that I is principal by induction on length, we may assume that J is a principal ideal contained in / such that // J = S is simple. Consider the exact sequence 0 -> Homfip (5, S) -^ Homfip (/, S) '^ Homflp ( J, S) -+ ExtlRp (5, S) -> 0 (the exactness of the last mapping is due to Ext^ p (/, S) = 0 mentioned above). Now, ExtlRp (S, S) / 0 since R/P2 is an extension of P/P2 by P./P which does

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not split, and since J is a principal ideal in Rp, Honiflp(J, S) is simple. Therefore, Ro-mRp(I, S) = H.omRp(S, S) = 5. But then I /PI must be isomorphic to 5", and consequently / is principal by Nakayama's Lemma. D As a consequence of Theorem 6, if R is Dedekind, G is torsion-free of finite rank and T is torsion, then Bextjj(G,T) is torsion-free. 3. Butler Modules and Bext Theorem 7. Assume that R is finitely generated over R. There exists 0 ^ a £ R such that for all Butler modules G and torsion modules T, aBext#(G,T) — 0. Proof. The first step is to satisfy the theorem with a = 1 when R is Dedekind. However this follows mutatis mutandis from the proof for abelian groups; one version [7] will be sketched briefly here. A sequence of modules

is called prebalanced, if for every rank i module X and map g : X —» N, there is a finite rank completely decomposable module C, an embedding h: X —* C, and a map q>: C —* M, such that frfrh = g. The set of equivalence classes of prebalanced exact sequences (*) constitutes the submodule PBext]?(Ar, L) of Ext]j(Af, L). Any relatively divisible submodule H of a finite rank Butler module G over a Dedekind domain is again a Butler module such that 0 _> H -» G -> G/H -> 0

is prebalanced (see [1] for example). Fuchs and Metelli argue that for T torsion, and H a relatively divisible submodule of the Butler group G of finite rank, there is a long exact sequence 0 -> Eom(G/H,T) -> Hom(G,T) -> Hom(ff,T) -> PBext1 (G/#,T) -> PBext1 (G,T) -» PBext1 (H , T) - > • • - . A development of PBext" for n > 2 proceeds along homological lines but is not necessary for our purposes. The existence of this long exact sequence for a torsion module T and a relatively divisible submodule H of & finite rank Butler module G, can be proven by merely changing group to module in the argument from [7]. Furthermore, Theorem 1.5 in [7] showing that Bext(G,T) = PBext(G,T) for all torsion-free G and torsion T carries over to modules over Dedekind domains. To affect this generalization one must change group to module, Z to R, and Q to Q in the proof of Fuchs-Metelli. From the Fuchs-Metelli results discussed above, we can induct on rank G to show that Bextjj(G, T) = 0 when G is Butler and R is Dedekind: If X is a rank 1 relatively divisible submodule of G, then 0 —> X —> G —> H —» 0 is prebalanced and so PBext^tf, T) -+ PBextk(G, T) -* PBext^Z, T) is exact. Since PBext(X, T) = Eext(X,T) = 0, induction is used to obtain Bext^(G,T) = 0.

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For the case R^R, let

(t)

O-»T-+M-^G-^O

be a balanced sequence with T torsion and G Butler. Every torsion-free module is a balanced epimorphic image of a completely decomposable module, so there exists a completely decomposable module G and a balanced epimorphism <j>: C —> G (for example, simply take G = ®{Y Y is a relatively divisible submodule of G of rank 1}). Consider G, G and 4> — 4>\c ^ provided in the definition of Section 1. Since : G —> G is balanced, any map from a rank 1 .R-module X into G C G must lift to a map from X into G which necessarily has image in G. Therefore, the corresponding map : G —> G is an epimorphism that is balanced relative to R. Since TT: M —» G is balanced, there is a map rj from G —> M such that 7T77 = . With M' = Imr?, any splitting of M' ^-* G will be a splitting of TT, where TT' is the restriction of TT to M'. So, in order to show that (f) is split, we may assume that M = M' = C/K where K — Kerr? is a submodule of G. Set M = C/K. Since KCK, the map /3: M -> M given by /3(c + -K") = 5+ AT for c 6 G is well-defined. Call TT = TT/3. It follows that — ff" where 77 is the natural map from C onto C/K. We claim that (f)

0 — » f1 — > A f - ^ > G — > 0

is balanced relative to R, where T = KerTf. Note that TT is an .R-module map since G is torsion-free and M is an .R-module. Suppose g: X —> G for some rank 1 .R-module .?. There is a map /: X —> G such that g = <j>f since : C —> G is balanced relative to .R. Then /' = fjf: X —> M satisfies TT/' = frf?/ = (j>f = g, and (f) is balanced. By Proposition 1, G is a Butler module over R, and so (f) must split. Fix 0 ^ a e R such that a.R C R. Given a splitting map 7: G —> M for ff, define 7': G —> M by 7' = /?7^, where ^i is multiplication by a on G. Then ^7' = /j, = ale, implying that Bext}j(G,T) is bounded by a. D 4. When Bext is Bounded For the converse of Theorem 7 we need to add a countability condition. The maximal divisible submodule of a torsion-free module H will be denoted by div(H). Theorem 8. Assume that R is finitely generated as an R-module, and that G is a torsion-free R-module of finite rank that is the balanced image of a completely decomposable module of countable rank. If Bextt(G. T) is torsion for all torsion modules T, then G is a Butler module. Proof. To show that G is a Butler module, we will argue that G is a Butler Rmodule. I.e., we must argue that G is an epimorphic image of a finite rank completely decomposable .R-module. tlsing our hypotheses we will construct a countable rank completely decomposable module G and a balanced epimorphism /: G —> G. We will then argue that a certain finite rank completely decomposable summand Cn of G actually maps onto G (this is the hard part).

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Let F = ©£_! Rxi be an essential and free submodule of G. By assumption, there exist rank 1 modules Xi,X2,... such that under the definitions (1)

Cn = F®X1®---®Xn

and

C = F ®Q)Xn = \JCn,

we obtain the exact sequences (2)

0 -> K -> G -^ G -» 0

and

0 -»#„-> Cn -» Gn -» 0;

the first of which is balanced. We now consider maximal /J-submodules, G, Cn and K, Kn of G, Cn, and /C, .Kn. respectively. We obtain the schema (1)

Cn = F®Xl®---®Xn

and

G = F ® ffi)Xn = U n G n .

and (2)

0 -» K -> C -A G -> 0

and

0 -> £Tn -> Cn -> Gn -> 0.

Here, / is just the restriction of / to C, and as argued in Section 1, / is an Rmodule map, and analogous to an argument in the proof of Theorem 7, the first sequence in (2) is balanced relative to R. Note that each Gn is a Butler module. Either Gn = G n _i, or there is some maximal ideal Pn of R for which (Gn)pn ^ (Gn-i)pn. It will be convenient to set Pn = R whenever Gn = G n _i. Recall, when Pn ^ R, Rpn is a dvr, and the only .Rpn-submodules of Q are Q and Rpn (up to isomorphism). Hence, over Rpn, the only Butler modules are divisible modules direct sum free modules. Since (G n _i)p ti /div((G n _i)p n ) is a reduced Butler module over the dvr Rpn, (Gn-]}pn/ div((<5 n _i)p n ) is consequently finitely generated and free over Rpn. Hence, there is a least positive integer en such that P^"G n _i C Fpn + div((G n _ 1 )p ti ). Whenever Pn = R, put en = 0. For each n >. 0, define /„ = P^1 • • • P^, and let L = £~=i InKn. With T and M taken to be the modules KfL and C /L respectively, we have a balanced exact sequence (3)

0 -> T -» M ^ G -> 0.

Furthermore, by the balancedness of (2), the induced sequence (3)

0->f->M-^G-»0

of ^-modules, is balanced, where T = K/L, M = C/L, and ff = TTJJ^. By hypothesis, there exists <j>: G —* M and 0 ^ r 6 R such that K(f> = rlc- The image of G in M under (f> satisfies (G) are ^-modules, B must be an ^-submodule of G. Thus B C C, and therefore the map = ^JQ, satisfies 7f0 = ^1(5- But by Theorem 6, the sequence in (3) must split since it represents a torsion element in Ext^,(G, f). To conserve notation, we will use to represent a splitting map for (3); only the fact that (3) is split will be needed below.

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In order to show that G is a Butler module, it suffices to verify that G is a Butler module by Proposition 1. Given y € F C G, if we regard y canonically as an element of M_= [^0(0°^ X i ) ] / L , then (j>(y)-y e KerTf = jS"_/L. Multiplying_by_a nonzero a £ R chosen to lie in some /„, we may assume that acj>(y) - ay = 0 in C/L, since K/L is torsion. Moreover, if the element a is chosen so that a<j>(x) — ax = 0 in C/L for each x belonging to a finite generating set for F, then aij>(y) — ay = 0 in C/L for each y £ F. Because <(> is a splitting map of ff, we obtain (4) the height vector of aif>(y) in C/L = the height vector of ay in G, for all y € F. Henceforth, a will denote a fixed element of R belonging to some /„, satisfying (4). Let j be a fixed integer such that Pj ^ R, and set P = Pj and e = BJ. We will show that aR C Pe. By definition of P, the torsion module Gj/Gj-i has a nonzero P-primary component, and for noetherian domains of Krull dimension 1 (R is Dedekind), it is easy to prove the existence of an x G Gj such that x £ Gj-\ but Px C GJ_I. By definition of e, we have Ph(Px) C Fp + div((Gj_i)p) for some /i, and assuming h is minimal, h < e. Since heights localize by Lemma "2 and any localization of R is a pid, the conditions on x imply there exists y £ F such that (5)

P-height(y) > h + I in G,- while P-height(y) = h in G,-_I.

By (4) and (5), we conclude (6)

P-height(ay) in C/L = P-height(ay) in G > ho + h + I ,

where ho is the P-height of a in R. Thus, there must exist 1 £ L so that P-height(ay +1) > h0 + h + 1 in C (where ay is regarded as an element in C). Write l = £i+ lz with £ e LI = £n=i /„#„ and £2 e L2 = E~=J ^^nNote that P-height(£2) > 2e > e + h in C, by definition of Jn. It follows that P-height(ay + 4) =_P-height(ay + £ - £ 2 ) > min{/i0 +_l,e} + /i in G. Then by (4), ay = 7r(ay + I'l) € G,-_i has P-height /i0 + /i in Gj-_i, so that /i0 + h > min{/io + 1, e} + /i because heights of elements do not decrease under homomorphic images. Therefore, e < h0 and Pe divides a^. In particular, the set {P,- Pj ^ R} is finite, since there are only finitely many maximal ideals of R containing a. Because Pj was chosen arbitrarily as a maximal ideal for which (G,-)^ 2 (Gj_i)p 3 ., it must be that for n large enough, Gp = (G n )p for almost all maximal ideals P. Fix m large enough so that P = {P \ Gp ^ (G m )p) is finite. If P G P: then by our arguments above, aGn C Fp + div(G n )p for all n > m. Because /: C —> G is balanced, for each maximal ideal P, there is an index n such that div(G)p = div(G n )p. Since P is finite, we can choose m large enough so that div(G)p = div(Gm)p for each P € P. Then of course div(G)p = div(G m ) P for all P since_GP = (G m )p for all P <£'P. Note div(G)p = div(G n )p for all P and all n > m because Gm C Gn C G. So, aGn C_F F + div^G n ) P = FP + div(G m ) P for all P_and n >_m. It follows that aG C Fp + div(G m ) P for all P, implying that aG C F + div(G m ). Therefore,

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G/Gm is finitely generated and G is quasi-isoraorphic to the Butler module Gm. By Proposition 1, G is a Butler module as well. D Corollary 9. Let R be a countable, noetherian domain of Krull dimensiion I such that R is finitely generated as a module over R, and let G be a torsion-free R-module of finite rank. The following are equivalent: (1) G is a Butler module. (2) For any torsion module T, Bext}j(G, T) is torsion. (3) There exists 0 ^ r 6 R such that rBextlR(G,T) = 0 for all torsion modules T. Proof. Since R is countable and G has finite rank, G is countable. Therefore, the set S of rank 1 pure submodules is countable, and since the standard map / : C = ®xes X —> G is balanced, the hypothesis of Theorem 8 is met. D When R is Dedekind, conditions (2) and (3) can be replaced by the condition Bext^(G,T) = 0 for all torsion modules T.

5. Example We offer an example to show that Condition (2) of Corollary 9 cannot be replaced by the statement that Bext(G, T) = 0 for all torsion modules T. Example 1. There is a subring R of an algebraic number field that has a finitely generated torsion-free module G and a torsion module T such that Bext}j(G, T) 7^ 0. Proof. Our example is as follows: • S = Z[a] where a is a 5th-root of unity. • p is an integral prime which remains prime in the Dedekind domain S (e.g., • G=R®R+ R(a, a 2 ) C 5 0 S.

• T = S/R. • H = R®R®T + R(a, a 2 , ^ + R) denned inside S ® S ® (p~lS)/R. The projection map TT of H into S © S has image G and kernel T. We claim that the sequence E:Q-*T-*H—>G^Q belonging to Ext^(G, T) is nonsplit and balanced, while pE is split. Indeed, the latter is clear because pT = 0 implies p[E] = pExt^(G,T) = 0, where [E] is the equivalence class of E in Ext^(G,T) ([16], Chapter 7). To see that E is nonsplit, suppose p is a splitting map for TT. Then p(l,0) = (1,0, t) and p(0,l) = (0,1, t') for some t,t' £ T. For any s e S, ps £ R, and psT = ps(S/R) = (psS + R)/R C PT = 0. That is, pST = 0 where the action of pS on T is induced by R . It follows that p(pa,pa2) = (pa,pa2,Q). On the other hand, because p is a splitting map of TT, p(a,a 2 ) — (a,a 2 ,£o+£")i where to = — + R and t" e T. This implies that p(pa,pa2] = (pa,pa2,pto); a contradiction, since pto + 0.

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Finally we show that E is balanced. To this end, let X be a rank-1 submodule of G. We need to construct a map 0: X —> H such that ~n<j) is the identity on X. If X C R ® R our task is simple since R 0 JR C If. On the other hand, in the remainder of the proof we will argue that if X £ R ® R, then X is projective. In this case, can clearly be lifted through H by the definition of a projective module. Assume that X contains an element x = (a, 6) + r(a, a 2 ), with a,b, e R and r e R \ pS. From the definition of R, r = k + ps where k is an integer and s £ S. Absorbing ps(a,a 2 ) into the (0,6) term, we may assume that r, in the expression for x, is an integer prime to p. Write ur + vp = 1, for some integers u, v. Then X contains ux = (ua, ub] + ur(a, a 2 ) = (ua, ub) — (vpa, vpa2) + (a, a 2 ). The first two terms in the last expression can be combined as (a, 6) for a new choice of a, b € R. This discussion shows that our element x can be chosen to have the form x = (a, b) + (a, a 2 ) for some a, b 6 R, which we fix for the rest of the proof. Suppose y = (c, d) +r(a, a 2 ) is another element of X. Since X is rank-1, there are elements m, n 6 S such that mx — ny. Since QR is a field, multiplying by a suitable element of R we may assume that n is an integer. Claim. If m £ S and mx — ny, with n an integer, then (a) p divides n implies m G pS; and (b) mx e G implies m e R. Proof of (a). Suppose n = pn' with n' & TL. Then ny £ pS (BpS, so that mx £ pS ® pS. More specifically, m(a + a) G pS. Since a £ R, a = ao + ps for some ao G Z and s G S, so we can conclude m(ao + a) € p5. Writing m = mo + mice + m2a 2 + msa;3 and noting that a4 = —1 — a — a2 — a3, we obtain the following congruences: m3 = 0 TO

(mod pZ),

m

+ o — a = 0 (mod pZ), + "H — ms = 0 (mod pZ),

o>om3 + m2 —m3 = 0

(mod pZ).

Using linear algebra, we obtain the equation 7713 = (OQ — a2 + a3 ~ ao)m3 (m°d pZ). Thus, either m3 6 pZ, or, — ao is a root of the polynomial equation X4 + X3 + X2 + X + 1 = 0 (mod pZ). The latter is impossible by the choice of p, since a proper factorization of the polynomial equation X4 + X3 + X2 + X + 1over Z/pZ leads to a proper factorization of the ideal pS (Proposition 25, page 27, [11]). Thus 7713 € pZ and it follows easily from the congruences that m; e pZ for each i. Therefore m € pS. Proof of (b). Suppose m € S and nix £ G. Then (m(a + a),m(b + a 2 )) € G. As above, this implies (m(a 0 + a),m(b0 + a 2 )) € G where a0 and &0 are integers congruent to a and b respectively mod pS. Write m = mo + m\Oi + m^a2 + m3a3, as before. Collecting the coefficients of a*, using the first component m(ao + a),

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we obtain as coefficients of a, a 2 , a3 respectively: a:

aoTni + mo —TOS= u\

(mod pZ)

2

(mod pZ)

3

(mod pZ),

a : a0rn2 + m\ — m3 = u2 a : a0m3 + m 2 — m3 = w3

for some integers ui,U2,U3. A similar set of coefficient congruences is obtained from the second component m(b0 + a 2 ): a: fo0mi — m 2 = «!

(mod pZ)

a 2 : £>om2 + mo ~ m2 = ^2 3

a : fr0m3 + mi - m2 = v3

(mod pZ)

(mod pZ)

2

Prom the definition of G = R ® R + R(a, a ) it follows that u2 = «s = vi = vs = 0

and MI = v%

(mod pZ);

(mod pZ).

We will examine cases in order to assume p does not divide OQ, bo, mi, m 2 , ma. If p | 60, then DI = vs = 0 implies m^ = m 2 = 0 (all congruences are mod pZ). Assume p does not divide b^. If p a®, then u2 = ws = 0 implies 7Tii,m 2 ,m3 are mutually congruent mod p, and the congruence froms + mi — m 2 = 1)3 (mod pZ) implies each of mi, m 2 , ms is divisible by p. In this manner we can show that when p divides any of ao,6o) T T l i ) m 2 ,m3, then m 6 R. Likewise, when bo = 1 (mod pZ), m e -R. Therefore, suppose p does not divide a 0 ,60, mj, m 2 , ms, or b0 — 1. The congruences «i = w3 = 0 (mod pZ) produce the following congruences: m2 = 6omi

(mod pZ)

7713 = (1 — bQ1)mi

(mod pZ)

Combining these with the congruence u^ = 0 (mod pZ) yields (1 — 00)7713 = m2 = &omi 1 771

(mod pZ)

(1 — ao)(l — ^o" ) ! = ^omi

(mod pZ)

Since we are assuming p { mi, (1 — ao)(l — b^1) = bo (mod pZ), or, (ao - 1) = b20(l - b^1

(*)

(mod pZ).

Finally, u\ = v% (mod pZ) yields mi0 0 - m3 = m2&o — m 2 = (&o — I)m 2 a

(mod pZ), or,

1 771

mi o — (1 — bo" ) ! = (^0 ~ tybomi (mod pZ). Thus OQ — 1 + fr^1 = fr§ — fro (mod pZ). Substituting (*) into this last expression gives 6g(l -fro)"1+ &o ' =fro- fro (modpZ). Simplifying, we obtain b$ ~ b% + 62 —fro+ 1 = 0 (mod pZ), which again contradicts the irreducibility of %4 + x3 + X2 + X + 1 over th e integers mod p. It follows that m € R.

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It is well-known that the finitely generated module X is projective if and only if for each maximal ideal P of R, the localization Xp is a free .Rp-module. Since X C S © S and 5 is finite rank free as an abelian group, X is finitely generated as an abelian group, and hence as an jR-module. To show that Xp is a free Tipmodule for any given maximal ideal P of R, since P contains an integral prime q it is sufficient to show that the module Xq is cyclic over the ring Rq for every rational prime q. When q ^ p, Rq — Sq. In this case 59 is Dedekind with only finitely many maximal ideals, and so S is a PID (page 86 in [15]). Therefore Xq is a submodule of the free .R9-module G9, so Xq is free. Now consider p. Given y 6 X, write mx = ny with n € Z and m G S. Prom Claim (a), we may assume that p does not divide n, and it follows from Claim (fe) that m S R. Therefore, Rx + pSX = X and so Rpx + pSpXp = Xp. By Nakayama's Lemma, the latter yields Xp = Rpx. D References [1] Ulrich Albrecht, Anthony Giovannitti, and Pat Goeters, A Class of Pure Submodules of Completely Decomposable Torsion Free Modules, Comm. in Algebra (to appear). [2] David M. Arnold, Finite Rank Torsion-Free Abelian Groups and Rings, LNM 931, Springer-Verlag, 1982. [3] Hyman Bass, On the Ubiquity of Gorenstein Rings, Math. Z. 82, (1963), 8-28. [4] Ladislav Bican, and Luigi Salce, Butler groups of infinite rank, Abelian group theory (Honolulu, Hawaii, 1983), 171-189, LNM 1006, Springer-Verlag, 1983. [5] Z. I. Borevich, and I. R. Shafarevich, Number Theory, Academic Press, New York, 1966. [6] M. C. R. Butler, Torsion-Free Modules and Diagrams of Vector Spaces, Proc. London Math. Soc. (3) 15, (1968), 635-652. [7] Laszlo Fuchs, and Claudia Metelli, Countable Butler Groups, Contemporary Mathematics vol. 130, 1992, 133-143. [8] Anthony Giovannitti, A Note on Proper Classes of Exact Sequences, Methods in Module Theory, Lecture Notes in Pure and Applied Mathematics, V. 140, Dekker 1993. [9] H. Pat Goeters, Butler Modules Over 1-Dimensional Noetherian Domains, Abelian Groups and Modules, Birkhauser, 1999. [10] Thomas W. Hungerford, Algebra, GTM 73, Springer-Verlag, 1974. [11] Serge Lang, Algebraic Number Theory, Springer-Verlag Graduate Texts in Mathematics 110 New York, 1986. [12] Eben Matlis, Torsion-Free Modules, University of Chicago Press, 1972. [13] Eben Matlis, 1-Dimensional Cohen-Macaulay Rings, LNM 327, Springer-Verlag, 1973. [14] Daniel Marcus, Number Fields, Springer-Verlag 1977. [15] H. Matsumura, Commutative Ring Theory, Cambridge University Press, 1980. [16] Joseph J. Rotman, An Introduction to Homological Algebra, Academic Press, 1979.

Remarks on the Multiplicative Structure of Certain One-dimensional Integral Domains Franz Halter-Koch Wolfgang Hassler Florian Kainrath Institut fur Mathematik, Karl-Pranzens-Universitat Graz, HeinrichstraBe 36, 8010 Graz, Austria franz.halterkochSuni-graz.at Wolfgang.hasslerQuni-graz.at florian.kainrathSuni-graz.at

Abstract. Finitely primary monoids are models of the multiplicative monoid of certain one-dimensional integral domains. We investigate smoothness criteria for a finitely primary monoid to be v-noetherian and show that these criteria are far from being necessary. We provide several examples of monoids and integral domains to explore the range of validity of those criteria.

1. Introduction In [HK95], the concept of a finitely primary monoid was introduced in order to investigate the elasticity of one-dimensional noetherian integral domains. Subsequently, this concept proved to be useful in all deeper arithmetical investigations of one-dimensional integral domains (see [Ger98]). Nevertheless, the multiplicative monoids of local one-dimensional integral domains have several properties which essentially adhere to their additive structure. This note is addressed to properties of this kind. In [GHK] a new type of monoid, called an AC-monoid, was introduced in order to investigate the arithmetical structure of congruence monoids in Dedekind domains. In [Has], a particular class of these monoids (called Z-monoids) proved to be the appropriate tool in order to investigate the arithmetical structure of a huge class of finitely generated integral domains. In contrast to AC-monoids, finitely primary monoids need not satisfy the ACC for divisorial ideals, i.e., they need not be u-noetherian. In [GHK], Proposition 5.5, necessary and sufficient 321

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conditions were proved for a finitely primary monoid to be an AC-monoid. Finitely primary monoids satisfying these conditions will be called smooth (see Definition 2.3 below). In particular, it follows that a smooth finitely primary monoid is vnoetherian. Smooth finitely primary monoids also play a crucial role in recent arithmetical investigations of monotone chains of factorizations (see [F]). The requirement of smoothness consists of two conditions, one concerning the units and one concerning the exponents (conditions (A) and (B) of Definition 2.3). The purpose of this note is to investigate these conditions more closely. We shall prove that, in the case of integral domains, the condition concerning the exponents is always fulfilled. For noetherian domains, this is a refining of a well-known result concerning the value semigroup. In the noetherian case, we will also provide an ideal-theoretic characterization of smoothness. In section 2 we recall the relevant definitions and results concerning finitely primary monoids and integral domains, and we investigate the criteria concerning smoothness. The main results here are Theorem 2.7. and the characterization of smoothness in the noetherian case (Theorem 2.9). In sections 3 and 4 we consider several examples of monoids and integral domains. In section 3 we use power series to construct examples built finitely primary monoids of rank 1, and in section 4 we obtain examples built with finitely primary monoids of arbitrary rank by means of polynomials. Among others, these examples demonstrate that there exist integral domains R such that R* is finitely primary having the following properties: 1) R is noetherian and R* is not smooth; 2) R is Mori, not noetherian and R* is not smooth; 3) R' is smooth, but R is not noetherian. We close with two examples of finitely primary monoids showing the difference between the theory of rings and the purely multiplicative theory of monoids. 2. Finitely primary monoids and integral domains Let N denote the positive integers, NO = N U {0}, and for a, 6 € Z, let [a, b] be the set of all integers v satisfying a < v < b. For a set X, we denote by \X\ £ NO U {00} its cardinality. By a monoid, we mean a commutative and cancellative semigroup with identity element. If not denoted otherwise, we use multiplicative notation. For a monoid H with unit element 1 = IH, we denote by Hx the group of invertible elements, and we call H reduced if Hx = {!}. Our main reference for the elementary arithmetic and ideal theory of monoids is [HK98]. Note however that in [HK98] all monoids have a zero element. Thus in order to apply results from [HK98] we either have to adjoin a zero element to our monoids or to remove the zero element from the monoids and ideals there. For an integral domain R, we denote by R* = /J\{0} its multiplicative monoid. In our terminology, a local domain need not be noetherian. For a monoid H, we denote by Q(H) a quotient group of H and by H = {x 6 Q(H) | there exists some c € H such that ex" € H for all n £ N} the complete integral closure of H. For an integral domain R with quotient field K, we also denote by R C K its complete integral closure. Observe that R* — R*.

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323

A monoid H is called w-noetherian if it satisfies the ACC on divisorial ideals. An integral domain R is a Mori domain if and only if R° is a i>-noetherian monoid. Recall that a monoid H is v-noetherian if and only if, for every subset X C H, there exists a finite subset E C X such that E~l C X~l (where X~l — {z e Q(H) | zX C H}, see [HK98], Theorem 24.1). Next we recall the definition of a finitely primary monoid (see [HK95] for the definition, and [Ger96] for a thorough investigation). A monoid H is called finitely primary of rank s G N and exponent a e N, if there exists a factorial monoid F = F* x [pi,... ,ps] (with pairwise non-associated prime elements Pi,. • • ,ps) such that H C F is & submonoid satisfying (Pl-...-Ps)aFcH

and H\H* CPl •... -p,F.

In this case, F = H and thus F is uniquely determined by H. If H is finitely primary of exponent a, then it is also finitely primary of exponent (3 for every 0>a. Integral domains R, for which R* is a finitely primary monoid, were characterized in [Ger96], Theorem 2, as follows. Proposition 2.1. For an integral domain R, the following assertions are equivalent: 1. R* is finitely primary. 2. R is a one-dimensional local domain, R is a semilocal principal ideal domain, and there exists some f € R* such that fR C R. In particular, if R is noetherian, then R* is finitely primary if and only if R is one-dimensional, local and analytically unramified. If R is an integral domain such that R* is finitely primary, then R need not be noetherian nor even a Mori domain (see Example 3.1 and Lemma 4.3), although R is a semilocal principal ideal domain and (R : R} ^ {0}. This fact is in contrast to the Theorem of Eakin-Nagata (see [Mat97], Theorem 3.7). However, there is a special case which is worth to notice. Proposition 2.2. Let R be an integral domain such that R is noetherian, f = Aimn(R/R), and suppose that R/^ is a finitely generated R-module. Then R is noetherian. Proof. Since f C R C R, it follows that R/R is a finitely generated ^-module, and the exact sequence 0 -> R -* R -> R/R -» 0 shows that R is a finitely generated .R-module. Thus the assertion follows by the Eakin-Nagata theorem. D In the following definition, we fix the concept of a smooth finitely primary monoid already mentioned in the introduction. Definition 2.3. Let H C F = F* x \pit... ,ps] be a finitely primary monoid of rank s. H is called smooth, if the following two conditions are fulfilled:

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P. Halter-Koch, W. Hassler, and F. Kainrath

(A) There exists a subgroup V C F* of finite index such that V(H \HX) CH. (B) There exists some /? £ N such that, for all j £ [1, s] and x G p^F, we have x £ H if and only if p?x £ J?. Note that the property of being smooth does not depend on the choice of Pi, . . . ,ps. Indeed, suppose that condition (B) holds for pi,. . . ,p3 with some exponent /3 € N, and let qi,. . . ,qa be another system of pairwise non-associated prime elements of F, say qj = £JPJ for all j £ [l,s], where £j £ Fx . We may replace /? by any multiple, and hence we may assume that (Fx : V) | /?. If x £ q?F = P j F , then we obtain q?x = e^p^x £ H if and only if p^x £ H, since E@ £ V . If s — 1, then condition (B) is fulfilled by the very definition of a finitely primary monoid. If If is a finitely primary monoid such that H is reduced, then condition (A) is trivially fulfilled. Proposition 2.4. Every smooth finitely primary monoid is v-noetherian. Proof. By [GHK], Proposition 5.5, a finitely primary monoid is a C-monoid if and only if it is smooth, and by [GHK], Theorem 5.8, every C-monoid is •y-noetheriau. D

Proposition 2.5. Let R be an integral domain such that R* is finitely primary, 1. If R* is smooth, then R is a Mori domain. 2. // .R/f is finite, then R is noetherian and R* is smooth. Proof. 1. By Proposition 2.4, R* is u-noetherian, and thus R is Mori. 2. By Proposition 2.2, R is noetherian. Since R' = {a 6 R* \ a + f £ R/f C .R/f}, it follows that R* is a congruence monoid in R defined modulo f. Therefore R* is smooth by [GHK], Theorem 7.1 and Proposition 5.5. D Next we consider the value semigroup of a finitely primary monoid, which is defined in the same way as for one-dimensional integral domains (see [BDFOO]). Definition 2.6. Let H C F = Fx x [ p i , . . . ,ps] be a finitely primary monoid of rank s. For an element x = ep™1 -. . .-p"s e F (where £ G Fx and n = (ni, . . . ,ns) € Ng), we set v(:r) = n, and we call v(H) C Ng the value semigroup of H. It is easily seen that the value semigroup v(H) does not depend on the choice of Pi, . . . ,ps. The value semigroup v(H) of a finitely primary monoid H of rank s and exponent a is again finitely primary of rank s and exponent a, and v(H) = Ng. If -R is an analytically unramified one-dimensional local noetherian domain, then v(R°) is the usual value semigroup of R which is known to be smooth (see [BDFOO], Lemma 2.4). We prove the following stronger result. Theorem 2.7. Let R be an integral domain such that R° is finitely primary of rank s and exponent a. Then there exist pairwise non-associated prime elements Pi, . . . ,ps of R with the following property: For all j £ [1, s] and x £ p"R, we have £ R if and only if x £ R.

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325

Proof. If s — 1, there is nothing to do. Thus suppose that s > 2, and let q^, . . . , qs be any pairwise non-associated prime elements of R. We must prove that, for every j € [1, s], there exists some e, € Rx with the following property: For all x 6 qfR, we have e~lqjX € -R if and only if x g R. Once this is done, we set PJ = s~lqj to finish the proof. It is sufficient to do the case j' = 1. We set £i = 9i + ( 9 2 - . . . - 9 , ) a e£. Since B\ is not divisible by some prime of R, we have e\ £ R* . If x £ qfR, then x - E^giz = e]"1^ • • • • • qs}ax e #, and therefore we obtain x € R if and only if e^1q±x G /?.

D

Corollary 2.8. Let R be an integral domain such that R* is finitely primary and m its maximal ideal. 1. R* satisfies condition (B), and\/(R*) is smooth. 2. R* is smooth if and only if there exists a subgroup V C Rx of finite index such that Vm C m. Proof. Obvious by Theorem 2.7 (observe that R \ Rx = m).

D

Theorem 2.9. Let R be a noetherian integral domain such that R* is finitely primary, m its maximal ideal and f = AimR(R/R). Then R* is smooth if and only if either R/m is finite, or f — m. Proof. Let first R* be smooth. By Corollary 2.8, there exists a subgroup V C R* such that (Rm : V) < oo and Vm C m. If S = {x e R xm C m}, then S is a ring satisfying R C S C R and V C Sx , whence (R* : 5 X ) < oo. Being a finitely generated ^-module, S is a semilocal noetherian domain. Since J? is a finitely generated S'-module, it follows that every p e Asss(R/S) has a finite residue class field (see [Kai88], Theorem 2.1). If Asss(^/5) = 0, then S = R and thus f = m. If p e A.sss(R/S), then p n R = m, hence R/m C S/p, and thus R/m is finite. Suppose now that R/m is finite. If in is a maximal ideal of R, then j?/ffi is finite over R/m and hence finite. By [N90], Theorem 1.6 it follows that R/f is finite, and therefore R* is smooth by Proposition 2.5.2. If f = m, then mRx C m, and therefore R' is smooth by Corollary 2.8. D For an arbitrary finitely primary monoid H, the valuation monoid v(.ff) need not even be v- noetherian (see Example 4.7). However, it is easily checked that v(H) is smooth whenever H satisfies the condition (B). 3. Examples of rank 1 Example 3.1. Suppose that K C S C '£, where K and L are fields and .S is a .ftT-subspace of L. We consider the ring R = K + Sx + z2L[z] cL[z].

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Since a;2L[x] C R, R\RX C xL[x] and L[x] is a discrete valuation ring with prime element x, it follows that R° is finitely primary of rank s = 1 and exponent a = 2, and R = L[xJ. If L/K is finite, then R is a finitely generated X|o;|-module and thus noetherian. In contrast, if 5 is a ring but not a field, then R is not even a Mori domain (see [BDF86], Example 17). Concerning smoothness of R*, we have the following criterion. Lemma 3.2. Suppose that K C S C L, where K and L are fields, S is a Ksubspace of L, and R = K + Sx + x2I/[x] C L[x]. Then R* is smooth if and only if either S = L or L is finite. Proof. By definition, R* is smooth if and only if there exists a subgroup V C Rx such that (Rx : V) < oo and V(R \RX) C R. Note that R* = Lx x W, where W = l + xLlx]. If S - L, we set V = R*, and if L is finite, we set V = Kx x W to see that R* is smooth. Suppose now that R* is smooth, and let V C Rx be a subgroup such that (Rx : V) < oo and V(R\R*) C R. Let (f>: R* —> L x be the canonical projection. Then U = x ) < (L x : C/) < oo, and by [K88], Theorem 4.3.11 we infer that either L is finite, or L = F and consequently 5 = L. D Example 3.3. Let R = K + Sx + x2L[x] be the ring of Example 3.1, for which R* is finitely primary. If L/K if finite, then R is a noetherian but, by Lemma 3.2, R° is not smooth unless either S = L or L is finite. If S = L, then R* is smooth and hence R is a Mori domain (see also [Bar83], Example 3.8). By [BR76], Theorem 4, R is noetherian if and only if L/K is finite. 4. Examples of arbitrary rank First we show that the phenomena demonstrated by the examples of section 3 can also be achieved for an arbitrary rank. The corresponding constructions are more involved than those in section 3 but follow the same ideas. Finally, we shall give two examples of finitely primary monoids of rank 2 violating condition (B) (a •u-noetherian and a not f-noetherian one). Example 4.1. Suppose that s e N a,nd'K C S C L, where K and L are fields and S is a /f-subspace of L. Let c\ = 0 , C 2 , . . . ,cs e K be distinct. We consider the ring R0 = K + Sx + x2L[x] C L[x]

Remarks on the Multiplicative Structure

327

and, for i € [1, s], the maximal ideals mi = (x- Ci}L[x\ c L[x\

and m; = m. n R0 = {/ 6 R0 /(Q) = 0} C R0-

We obviously have mi = Sx+x2L[x], Ro/mi = K and L[x]/m» = L for all i e [1, s]. ASSERTION 1. For i e [2, s], we have m.i — (x - c;)#0 a™d Ro/™-i — L. Proof of Assertion 1. The map 9i\ RQ —> L, denned by #;(/) = /(c») is a ring homomorphism with kernel m,. If y e L and / = x2(x — c, + c~ 2 y), then f & RQ and /(cj) = y. Hence $» is surjective and I/ = R^/m-i. We obviously have (x — c^Ro C m^ If / 6 m^, then there exists some
R is a local domain with maximal ideal m and residue class field K. To derive its properties, we use the following characterization of the elements of m inside T~lL[x\. ASSERTION 2. Let q e T~~lL[x] be given in the form xni(x-c2)n2 • ,..-(x-cs)n'f 9

vnth (uniquely determined] HI, . . . , ns S NO and relatively prime polynomials f , g £ L[x] such that g(0) — 1, /(O) ^ 0 and f ( a } g ( c i } ^ 0 for all i € [2,s]. Then the following assertions are equivalent: 1. q e m. 2. ni,.. . ,ns £ N, and if ni = 1, then /(O) € S. Proof of Assertion 2. (a) =4- (b). If q € m, then q € T^ttii C (x - ci}T~i L\x] and therefore Ui > 1 for all i 6 [!,«]• Suppose now that n\ = I,/ = flo + £/* and g = 1 + ^5*, where ao 6 L x and f*,g* & L[x]. Since g € m C T~1m1 = T~1(Sx + x2L[x}), we obtain q = ——- , where ai e S and /0, 50 6 £p 1 +xg 0 and consequently 01 + Xfo

1 + a;5o

=

(X~ C2)"2 • . . . • ( J - C,)"' (Op + S/*)

1 + xg*

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At x = 0, this implies ai = (—C2)™ 2 • . . . • (—c s )™ s ao € S, and since c%,... ,cs € Kx, we obtain CQ = /(O) e 5. (b) => (a). Suppose that g = 1 + bx + X2g2, where 6 £ L and g2 € L[x]. If b £ S, we set g\ = g and A = /• If k ^ S1, we set 51 = (1 — bx)g and A = (1 — 6x)/. In any case, we obtain gi e T, (x — c2)"2 • . . . • (x — c3)n'xni A 6 mi n ... nm s , and consequently <7 =

5i

€ m.

D

Now it follows by ASSERTION 2 that x2(x - c 2 ) • . . . • (x - ca)T~lL[x\ C m C x(x - c2) • . . . • (x - c^T"1^], whence R = T

L[x], and R* is finitely primary of rank s and exponent 2.

Lemma 4.2. Suppose that K C S C L, where K and L are fields and S is a K-subspace of L. Let R be the ring of Example 4.1. Then R* is smooth if and only if either S = L or L is finite. Proof. The proof is essentially the same as that for Lemma 3.2, this time using the decomposition Rx = Lx x W, where W = (f- & L(x) (9

f , g € L[x],/(0) = 5(0) - lJ(Ci)g(Cl) + 0 for all i 6 [2, s]

and the projection Lx, given by the evaluation of a rational function at

1 = 0.

n

Lemma 4.3. Suppose that K C S C L, where K and L are fields and S is a ring which is not a field. Then the ring R of Example 4.1 is not a Mori domain. Proof. Suppose that b 6 S* \ Sx and consider the set X = {b~nx2(x - c2) • . . . • (x - cs) \ n € N}. By ASSERTION 2 of Example 4.1, we have X C R, x~[bNX
Ro/mi n . . . n ms ^ JJ Ro/mi ^ K x Ls~l it=l

and

Ri/m £* T-^Ro/T'1^ n ... n m a ) =" T~l(K x Ls~l) =Kx Ls~l.

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Hence Ri/m, is a finitely generated K-sp&ce, say Ri/m = K(gi+m) + . . .+K(gm + m), where g±, . . . ,gm £ RI, and we obtain Ri=Kgi + ...+ Kgm + m = Rgi + ...+ Rgm + R. Thus RI is a finitely generated .R-module. Since RQ = K + Sx + x2L[x] is a finitely generated /f [x]-module, it follows that RQ is noetherian and hence RI = T^^-Ro is also noetherian. By the Eakin-Nagata Theorem, R is also noetherian. D Example 4.5. Suppose that K C S C L, where K and L are fields and 5 is a .ff-subspace of L. Let R be the ring of Example 4. 1 , for which R* is finitely primary. If L/K is finite, then R is noetherian by Lemma 4.4 but, by Lemma 4.2, R* is not smooth unless either S — L or L is finite. If S — L, then R* is smooth and hence a Mori domain, but if L/K is not finite, then R is not noetherian, again by Lemma 4.4. We complete this paper with two purely multiplicative examples which, by Theorem 2.7, cannot be realized in an integral domain. Example 4.6. We consider the additive monoid H = (N x N> 2 ) U {(2",I) | n £ N} U {(0, 0)} C Ng. It is finitely primary of rank 2 and exponent 2, H = N§, and H is obviously not smooth. We assert that H is yet v-noetherian. We must prove that for any subset X C H there exists a finite set E C X such that E~l C X'1. Let X C H be an infinite subset. For m £ N, we set X<m = X n (N x {m})

and X'm = \J X j . j>m

Let k £ N be minimal such that X n ({k} n N) ^ 0, and let n € N be minimal such that (fc, n) £ X. Let E0 be any finite subset of X n ({k} n N) such that \E0\>2 if X n ({/c} n N) | > 2 .

ASSERTION 1. Eg"1 c X'-+\. Proof of Assertion 1. Suppose that (a, 6) e E1^1 C Z2 and ( x , y ) £ X^+2. Then (a + fc, 6 + n) 6 ff C N§, x > k and y > n + 2, whence b + y > b + n + 2 > 2. If |.Eo| = 1, then re > fc + 1 and a + x > a + k + l > 1. If |E0 > 2, then a + fc > 1 and x > a + fc>l. In any case it follows that (a + x, b + y) 6 H . D ASSERTION 2. For every I e [l,n + l], there exists a finite subset EI C Xi such that Ef1 CXf1. Proof of Assertion 2. We may assume that Xi is infinite, and we distinguish two cases. Case 1. There exists some c £ Z suth that Xi = {(c+ 2ni,l) i € N}, where 1 < HI < n2 < . . .. We set EI = {(c + 2™1 , Z), (c + 2"2 , Z)} and obtain X,-1 = {(-c,-Z + l ) } u { ( a , & ) e Z 2 | a > - 2 n i + l , f e > - ^ + 2} =E~l.

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Case 2. Xi = {(nit /) | i € N}, where 1 < HI < n2 < . • ., and for every c € Z, there exists some j e N such that c+ n,- £ {2n n 6 N}. Let h 6 N be maximal such that {(-c + rii i e l , / i ] } c { 2 " n e N}. We set E{ = {(n»,0 | i e [1,fc+ 1]} and obtain

X1 = (a,b) e Z 2 a > - n i + l , 6 > - Z + 2 = E f 1 .

D

By the ASSERTIONS, the set E = E0 U EI U . . . U En+i is finite and satisfies E- CX~1. 1

Example 4.7. For subsets 9 j^T C S C N> 2 , we consider the sets

JT = ({1} x T) U (N> 2 x N> 2 ) c Hs = ({1} x S) U (N> 2 x N> 2 ) U {(0, 0)} c N 2 . Obviously, HS is a finitely primary monoid of rank 2 and exponent 2, and .Hs = N 2 . We shall prove that by an appropriate choice of S and T we can achieve that Jfo 2 ^f l for every finite subset TO C T. Since every finite subset E of JT is contained in some JTO for a finite subset T0 C T, this implies that H is not v-noetherian. ASSERTION 1. For any 0 ^ T C 5 c N> 2 , we /iave J-1=({0}x(S-T))u(NxN0), where S - T = {x 6 N0 | x + T C S}. Proof of Assertion I . We obviously have JT + ({0} x (S - T)) U (N x N0) C HS, and therefore 1 ({0} x (S - T)) U (N x No) C Jfl. If (x, y) e JT 1 C Z 2 , then ({x + 1} x (y + T)) U ((x + N> 2 ) x (y + N> 2 ) C Hs and therefore x > — 1. If a; = —1, then y + T = {0} implies y < —2, which is a contradiction as N x N0 ^ HS. Thus we obtain x e NO, and (o;+N> 2 ) x (y + N> 2 ) c Hs implies y e N0. If x = 0, then y + TcS implies y 6 5 - T. D

ASSERTION 2. // 5 = { 2 ; + 2 3 m f c | m 6 N , Z e [l,m],&eN 0 } ano! T = {I1 \ I e N}, then 0 / T C 5 C N> 2 , (5 - T) = {0} and (5 - T0) ^ {0} /or ever?/ finite subset TO C T. Consequently, by the above, this choice of S yields a monoid HS which is not v-noetherian. Proof of Assertion 2. (Suggested by G. Lettl). Clearly, T c S C N> 2 , and if TO C T is finite and 2* = max(T0), then 23' € (5 - T0). In order to prove that (S — T) = {0}, denote by i>2(o:) the 2-adic exponent of an integer x, and observe that if x G S and v%(x) = /, then x = "2l mod 23m for some m > I. Assume now that there exists some non-zero x £ (S -^ T). Then we have v2(x) = I > 1. Since x + 2!+1 e S, x + 1l+2 € S and v2(x + 2'+1) = v2(x + 2 ;+2 ) = I, there exists some m > I such that x + 21+1 = x + 2l+2 mod 23m. Since 3m > 3/ > I + 2, this is a contradiction. D

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References [Bar83] V. Barucci. On a class of Mori domains. Comm. Algebra 11:1989-2001, 1983. [BDF86] V. Barucci, D. E. Dobbs and M. Fontana. Conductive integral domains as pullbacks. Manuscr. Math. 54:261-277, 1986. [BDFOO] V. Barucci, M. D'Anna and R. Froberg. Analytically unramified one-dimensional semilocal rings and their value semigroups. J. Pure Appl. Algebra 147:215-254, 2000. [BR76] J. W. Brewer and E. A. Rutter. D + M constructions with general overrings. Michigan Math. J. 23:33-42, 1976. [F] A. Foroutan. Monotone Faktorisierungsketten. Thesis, Graz 2002. [Ger96] A. Geroldinger. On the structure and arithmetic of finitely primary monoids. Czech. Math. J., 46:677-695, 1976. [Ger98] A. Geroldinger. A structure theorem for sets of lengths. Colloq. Math., 78:225-259, 1998. [GHK] A. Geroldinger and F. Halter-Koch. Congruence Monoids. Preprint, 2002. [HK95] F. Halter-Koch. Elasticity of factorizations in atomic monoids and integral domains. J. Th. Nomb. Bordeaux, 7:367-385, 1995. [HK98] F. Halter-Koch. Ideal Systems. An Introduction to Multiplicative Ideal Theory. Marcel Dekker, 1998. [Has] W. Hassler. Arithmetic in finitely generated domains. [Kai88] F. Kainrath. A note on quotients formed by unit groups of semilocal rings. Houston J. Math., 24: 613-618, 1998 [K88] G. Karpilowsky. Unit groups of classical rings. Clarendon Press, 1988 [Mat97] H. Matsumura. Commutative ring theory. Cambridge Univ. Press, 1997. [N90] W. Narkiewicz. Elementary and Analytic Theory of Algebraic Numbers. Springer, 1990.

Non-finitely Generated Prime Ideals in Subrings of Power Series Rings William Heinzer Department of Mathematics, Purdue University, West Lafayette, IN 47907-1395

Christel Rotthaus Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027

Sylvia Wiegand Department of Mathematics and Statistics, University of Nebraska, Lincoln, NE 68588-0323

Abstract. Given a power series ring R* over a Noetherian integral domain R and an intermediate field L between R and the total quotient ring of R*, the integral domain A = L n R* often (but not always) inherits nice properties from R" such as being Noetherian. For certain fields L it is possible to approximate A using a localization B of a particular nested union of polynomial rings over R associated to A; if B is Noetherian then B = A. If B is not Noetherian, we can sometimes identify which, prime ideals of B are not finitely generated. We have obtained in this way, for each positive integer n, a. three-dimensional quasilocal unique factorization domain B such that the maximal ideal of B is two-generated, B has precisely n prime ideals of height two, each prime ideal of B of height two is not finitely generated and all the other prime ideals of B are finitely generated. We examine the structure of the map Spec A —> Spec B for this example. We also present a generalization of this example to higher dimensions.

1. Introduction In this paper we analyze the prime ideal structure of particular non-Noetherian integral domains arising from a general construction developed in our earlier papers [4], . . . , [8]. Briefly, with this technique two types of integral domains are The authors would like to thank the National Science Foundation and the National Security Agency for support for this research. In addition they are grateful for the hospitality and cooperation of Michigan State, Nebraska and Purdue, where several work sessions on this research were conducted.

333

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constructed: (1) the intersection of a power series ring with a field yields an integral domain A as in the abstract, and (2) an approximation of the domain A by a nested union B of localized polynomial rings has the second form described in the abstract. Classical examples such as those of Akizuki [1] and Nagata [11, pages 209-211] were created using the second (nested union) description of this construction. As we observe in [7], it is also possible to realize these classical examples as the intersection domains of the first description. In [7] we observe that in certain applications of this technique flatness of a map of associated polynomial rings implies that the constructed domains are Noetherian and that A = B. In [7] and [8], we apply this observation to the construction of examples of both Noetherian and non-Noetherian integral domains. This paper represents a continuation of this project. We present several examples constructed using this technique of building non-Noetherian integral domains inside a power series ring for which the prime ideal structure can be described explicitly. In Section 2 we give the notation for a simplified adaptation of the construction which we use to produce these examples. This adaptation produces "insider" examples A" and B" having the form A and B described above and fitting inside an easier and more straightforward Noetherian domain A' = B' (where the two forms of the construction are equal). We have shown in our previous work that the question of whether or not the insider domains A" and B" that arise in this adaptation are Noetherian or equal is related to flatness of a map of polynomial rings corresponding to the extension B" <—» B' and having the form (f.S:= R[f] *—> T := R\x\, where x := (xi,...,xn) is a tuple of indeterminates over R and / := (/i,.. • , fm) consists of polynomials fa in R[x\ that are algebraically independent over the field of fractions Q(R) of ft. In Sections 3 and 4, we analyze and provide more details about an "insider" example B constructed in [8]. For each positive integer n, this construction produces an example of a 3-dimensional quasilocal unique factorization domain that is a generalized local ring in the sense of Cohen [2] and has as its completion a 2dimensional regular local domain. Moreover, B is not catenary and has precisely n prime ideals of height two. The associated intersection domain A for this construction is a 2-dimensional regular local domain. In Section 4, we examine the prime spectrum map Spec A —> Spec B and describe three types of height-one prime ideals of B by how they relate to primes of A. In Section 5 we prove for each positive integer n > 2 and each positive integer t, the existence of a non-Noetherian integral domain B such that: (a) dim5 = n + l. (b) The maximal ideal of B is generated by n elements. (c) B has exactly t prime ideals of height n and each of these primes is not finitely generated. (d) B is a factorial domain. (e) The completion of B is a regular local domain of dimension n. (f) B is a birational extension of the localized polynomial ring over a field in n + 1 variables.

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2. Background and Notation We begin this section by recalling some details from Section 3 of [8] concerning the insider adaptation of the construction utilized in this article.1 We use the following setting throughout the paper. 2.1 General Setting. Let A; be a field and let x, j / i , . . . , ys be indeterminates over fc; for convenience, we abbreviate the yi by y. (Often we assume k to have characteristic zero. This ensures excellence of the constructed domains in the simplest applications.) Let R := k [ x , y i , . . . ,ya](x,yi,...,y,) = k[x,y]^x
(2.2.1)

For each i e {1,..., n} and r > 0,

pir := ]T

(bijxj)/xr.

j=r+l

Now, for each r > 0, define Ur := R[pir, • • • , Pnr] and set Br to be Ur localized at the multiplicative system 1 + xUr. Then set U := U^L0UT and Bp = U^0Br. Thus U is a nested union of polynomial rings over R and Bp is a nested union of localized polynomial rings over R; clearly U C Bp C AT_. We refer to Bp as the nested union domain corresponding to p. The definition of the UT (and hence also of Br, U and B) is independent of the representation of the pi as power series with coefficients in R [3, Proposition 2.3]. For the two sequences of power series in (2.1), namely T and /, we have the approximation sequences T^ and /. and the nested union domains B^ and Bf, which are localizations of U^ 0 72[Ti r ,..., rnr] and of U£l 0 .R[/i r ,..., fmr] respectively. Clearly Bf C B^ are quasilocal domains with B^ dominating Bf. 1 2

For more detail see [7] and [8]. Again more details may be found in [8].

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2.3 Proposition ([6, Proposition 4.1]). Assume the set-up as in (2.1). Then T —» R*[L/x] is flat, A^_ = B^ and AT_ is a Noetherian regular local ring; moreover, if char k = 0, then AT_ is excellent. 2.4 Insider construction details. Let T = TI, ... ,rn and / = f\,... ,fm be as in (2.1) and (2.2). We assume the constant terms in R = k[x,y] of the fa are zero. Let S := R[f}. The inclusion map S <—> T is an injective .R-algebra homomorphism, and m < n. Let A be the intersection domain for /, i.e. A := Af = Q(S)t~}R*. Let B := Bf_ be the nested union domain associated to the / i , . . . , fm, as in (2.2). We consider the inclusion maps 4>: S t-> T and if): T <^-> R*[i/x] as shown in the following diagram. R*[\/x] (2.4.1)

"'""" ^ RCS:= R[f]

2.5 Theorem ([4, Theorem 2.2], [8, Theorem 3.2]). (1) B is Noetherian and B = A if and only if the map a: S —> R*[L/x] in (2.4.1) is flat. (2) For Q* e Spec(R*[l/x\), the localization aQ.: S -> (^*[I/O;])Q. of the map a in (2.4.1) is flat if and only if the localization p in (2.4.1) is flat. (3) The following are equivalent: (i) B is Noetherian and A = B. (ii) B is Noetherian. (iii) The localized map of (2.4.1) is flat. 2.6 Proposition ([8, Proposition 2.4]). Let R be a Noetherian ring and let X i , ..., xn be indeterminates over R. Assume that / i , . . . , / m £ R[xi,...,xn] are algebraically independent over R. Then (1) if. S := R [ f i , . . . , f m ] <-* T := R[xi,...,xn] is flat if and only if, for each prime ideal P o f T , we have ht(P) > ht(P n S). (2) For Q £ SpecT, TQ is flat if and only if for each prime ideal P C Q o f T , we have ht(P) > ht(P n S). (3) // (fx : S -> T[l/x\ is flat, then B is Noetherian and B = A. 3. Non-Noetherian Examples We review the construction of a series of examples of non-Noetherian integral domains inside power series rings given in [8, Examples 5.1]. 3.1 Specific construction details for the examples of [8]. This construction is a localized version of (2.4), with s = 1. Thus k is a field, R = k[x,y]rxy) is a

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two-dimensional regular local ring and R* = k[y]^[[x]] is the (x)-adic completion of R. Let T = X^liCj'3^' € xfc[[o;]] be algebraically independent over k(x). (It is easy to see that (2.3) still holds; that is, the modified AT := Q(R[r\) n R* — U£o*M(-)=Sr.)

For the insider domains, let p^ £ R \ xR be such that pi-R*, . • . ,pnR* are n distinct prime ideals. For example, we could take pi = y — x1 . Let q — pi • • -pn. We set / := qr (i.e., m = 1). Let B := Bf be the nested union domain associated to / as in (2.4). If rr = X^fLr+i ^~ ig *ne rth endpiece of T, then fr := qrr is the rth endpiece of /. For each r > 0, let BT = R[fr}(x,yjr)- Then each Br is a 3-dimensional regular local ring and B = \J^L0 Br. In this example, the intersection domain Af associated to / is the same as that associated to T; that is, A = AT = Q(R\r}) n R* = Q(R[f]) n R* = Af, since Q(R[T]) = Q(R(f}). 3.2 Proposition ([8]). For each positive integer n, the nested union domain B constructed in (3,1) is a three-dimensional quasilocal unique factorization domain such that (1) B is not catenary. (2) The maximal ideal of B is two generated. (3) B has precisely n prime ideals of height two. (4) Each prime ideal of B of height two is not finitely generated. (5) Each height-one prime ideal of B is principal and each nonzero prime ideal of B is the union of the prime ideals of height one that it contains, so B has infinitely many prime ideals of height one. (6) For every non-maximal prime P of B, the ring Bp is Noetherian. 3.3 Notes. (1) The nested union domain B is not Noetherian by (2.5.1) and (2.6.2); each piR*[\./x\ is a height-one prime of R * [ l / x ] , but piR*[l/x] n S = (p., f}S has height two, thus the map a: S —> R*[l/x] is not flat. (2) The maximal ideal of B is (x,y)B, because B/xB = R/xR. (3) As noted above, BT = AT = A = R* n Q(R(r)) = R* n Q(R(f)) = Af in the notation of (2.1) and (2.2), so that Af is a nested union of three-dimensional regular local domains (although we will see that it is not equal to the nested union domain We consider the inclusion map B t—> A and the map Spec A —» Spec B. The following Proposition is proved in [8]. 3.4 Proposition. With the notation of (3.1) and A = R* n Q(R(f)), we have (1) A is a two-dimensional regular local domain with maximal ideal ni^ = (x,y)A. (2) m.4 is the unique prime of A lying over rriB = (x,y)B, the maximal ideal of B. (3) If P e SpecB is nonmaximal, then ht(PR*) < 1 and ht(PA) < 1. Thus every nonmaximal prime of B is contained in a nonmaximal prime of A. (4) I f P e Spec.B and xq £ P, then htP < 1.

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(5) J / P e SpecB, htP = l andPnR^O, then P = (Pr\R)B. (6) If pA is a height-one prime of A with pA $ {piA,.. .pnA}, then APA = and ht(pA Pi B) = 1. However, PiA ("I B has height two and is not finitely generated. (7) Each piB is prime in B. (8) PiB and Qi := (pi, /i, /2,.. .)B = PiA n B are the only primes of B lying over piR in R. (9) Qi has height two and is not finitely generated. 3.5 Notes. (1) With regard to the birational inclusion B <—> A and the map Spec A —> Spec B, we remark that the following hold: Each Qi contains infinitely many height-one primes of B that are the contraction of primes of A and infinitely many that are not. Among the primes that are not contracted from A are the p^B. In the terminology of [13, page 325], P is not lost in A if PA n B = P. Since piAnB = Qi properly contains ptB, PiB is lost in A. Since ( x , y ) B is the maximal ideal of B and (x, y)A is the maximal ideal of A and B is integrally closed, a version of Zariski's Main Theorem [12] implies that A is not essentially finitely generated as a jO-algebra. (2) The quasilocal domains B constructed in Example 3.1 are generalized local rings in the sense of Cohen [2, page 56], that is, the maximal ideal m of B is finitely generated and the intersection of the powers of the maximal ideal is zero. Cohen proves in [2] that the completion of a generalized local ring is Noetherian. In our situation, the m^-adic completion of B is equal to the m^-adic completion of A and is a 2-dimensional regular local domain. 4. Further analysis of B for n = 1 In this section we consider the ring B = |J^=o ^r °^ *he Previ°us section in the case where n = 1 and q = Pi = y- Thus / = yr, R = fc[x,j/](X)J/), Br = R(yTr\(x,y,yrr) and B = \J™=0R[yTr](X,y,yrr)• This ring B has exactly one prime ideal Q = (y, {yTr}^0)B of height 2. Moreover, Q is not finitely generated and is the only prime ideal of B that is not finitely generated. We also have Q = yA n B, and Q n Br = (y, yrr)Br for each r > 0. If q is a height-one prime of 5, then B/q is Noetherian if and only if q is not contained in Q. This is clear since Q is the unique prime of B that is not finitely generated and a ring is Noetherian if each prime ideal of the ring is finitely generated. The height-one primes q of B may be separated into three types as follows: Type I. The primes q <2 Q, such as xB. As mentioned above, B/q is Noetherian. These primes are contracted from A. To see this, consider q — gB where g £ Q. Then gA is contained in a height one prime P of A. Then g G (P C\ B) \Q so P n B ^ Q. Since na.BA = m^, we have P n B ^ m B . Therefore P n B is a height-one prime containing q, so q = P PI B and Bq = Ap. There are infinitely many primes q of type I, because every element of m^ \ Q is contained in a prime q of type I. Thus m^ C Q U \J{q of type I}. Since

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339

is not the union of finitely many strictly smaller prime ideals, there are infinitely many primes q of type I. If q is a height-one prime of B not of type I, then B = B/q has precisely three prime ideals. These prime ideals form a chain: (0) c Q C (x, y)B = nig. Type II. The primes q C Q, where q has height one and is contracted from a prime p of A = k(x, y) n R*, for example, the prime y(y + r}B. For q of this type, B/q is dominated by the one-dimensional Noetherian local domain A/p. Thus B/q is a non-Noetherian generalized local ring in the sense of Cohen. For q of Type II, the maximal ideal of B/q is not principal. This follows because a quasilocal generalized local domain having a principal maximal ideal is a DVR [11, (31.5)]. There are infinitely many height-one primes of type II, for example, y(y+xnr)B for each n e N. For q of type II, the DVR Bq is birationally dominated by Ap. Hence Bg = Ap and the ideal ^fqA = p n yA. Type III. The primes q C Q, where q has height one and is not contracted from A, for example, the prime yB and the prime (y + xnyr)B for n £ N. Since the elements y and y + xnyr are in m^ and- are not in m^ and since B is a UFD, these elements are necessarily prime. Also the prime ideals (y + xnyr)B and (y+xmyr}B are distinct for n and m distinct positive integers, for if W is a prime ideal of B that contains them both and if m > n, then xm~nyr e W. But x $ Q and Q and ing are the only primes of B containing (y,yr)B. For q of type III, we have If q = yB or q = (y + xnyr)B, then the image m^ of mg in B/q is principal. It follows that the intersection of the powers of m~B is Q/q and B/q is not a generalized local ring. For if P is a principal prime ideal of a ring and P' is a prime ideal properly contained in P, then P' is contained in the intersection of the powers of P [10, page 7, ex. 5]. Another way to examine the height-one primes of B. Observe that B [ l / x ] is a localization of the polynomial ring k [ x , y , f ] while A[l/x] is a localization of k[x,y,T\. Thus the embedding B[l/x] <—> j4[l/a;] is a localization of the map: 4>: k [ x , y , f ] —> k [ x , y , r ] which is defined by (f>(f) = yr. Obviously the nonfiat locus of 0 is defined by the ideal (y) of k[x,y,r}. If q e B is a height one prime ideal which is not contained in Q, then the extension qA is not contained in yA and for every minimal prime p C A over qA the induced map Bq •—> Ap is flat. In particular, q is not lost in A. If q is a heightone prime of B which is contained in Q then q is of type II or III according to the minimal prime divisors of qA: If there is a minimal prime p e Spec A of qA which is different from (y), then the map: Bq Y> Ap is flat, and q is not lost in A. In this case, q is of type II. Note that yA is also a minimal prime divisor of qA. If yA is the only prime divisor of qA, then q is lost in A. In this case the map BQ «—v A( y ) is a localization.

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We remark that B/yB is a rank 2 valuation ring. This can be seen directly or else one may apply [9, Prop. 3.5(iv)]. If we do a similar construction with a prime contained in in2 (instead of y), for example, / = (x2 + y2)? (over Q), then B/(x2 + y 2 ) has a two-generated maximal ideal and cannot be a valuation ring. 5. A More General Construction In Theorem 5.1, we obtain a generalization of the construction that gives the examples considered in Sections 3 and 4. 5.1 Theorem. Let n > 2 be a positive integer. For each positive integer t, there exists a non-Noetherian integral domain B such that: (a) dimB — n + 1. (b) The maximal ideal of B is generated by n elements. (c) B has exactly t prime ideals of height n and each of these primes is not finitely generated. (d) B is a factorial domain. (e) The completion of B is a regular local domain of dimension n. (f ) B is a birational extension of the localized polynomial ring over a field in n + 1 variables. Proof. Let A; be a field and let 1,1/1, . . . ,2/n-i be variables over k. Define: R = k[x,yi, . . . ,yn-i}(x,yi,...,yn_1) and let TI, . . . , Tn-i € ifcffa;]] be algebraically independent over Q(R}- For 1 < i < t, let pi = yi — xl . Set q = pip2 • • -Pt, and consider the element The map:

has as its Jacobian ideal: J = (q,y%, . . . ,yn-i) which is an ideal of height n — 1 that is the intersection of t prime ideals of height n — 1: J = Q-\_r\ • • • n Q t , where Qi = ( p i , y 2 , - - - , 2 / n - i ) . If we construct B as usual as the union of localized polynomial rings of dimension n + 1, then by [8, Theorem 3.9], B is not Noetherian (or use (2.5.1) and an argument similar to (3.3.1)). If n > 2 then J has height > 1 and it follows from [5, Theorem 5.5] or [7, Theorem 6.3] that B = A = Q(R[f}) r\R*. It follows from [5, Theorem 4.5] that B is factorial. Moreover, if we consider our usual set up: R[f\ — > R[Tl, . . . ,!•„_!] [I/a;] — R*[l/x] the preimage in R[f] of JR*[l/x] is the ideal L — (J, f) which has height n. The ideal L corresponds to the intersection of t prime ideals in B of height n, these are the only primes in B of height n and they are not finitely generated. D i 5.2 Remark. It would be interesting to know whether for B as in Theorem 5.1, the prime ideals of B of height n are the only prime ideals of B that are not finitely generated.

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5.3 Remark. With the insider construction given in (2.1)-(2.4), if the dimension of Bf is greater than that of R*, then Bf is not catenary. One way to see this is that in Spec(J3/) there is always a saturated chain of prime ideals that includes (x) and this chain has length equal to dim R*, while if dim Bf > dim R*, then there exists a saturated chain of prime ideals in Bf of length greater than dim R*. References 1. Y. Akizuki, Einige Bemerkungen uber primdre Integritatsbereiche mil Teilerkettensatz, Proc. Phys-Math. Soc. Japan 17 (1935), 327-336. 2. I. S. Cohen, On the structure and ideal theory of complete local rings, Trans. Amer. Math. Soc. 59 (1946), 54-106. 3. W. Heinzer, C. Rotthaus, and S. Wiegand, Noetherian rings between a semilocal domain and its completion, J. Algebra 198 (1997), 627-655. 4. , Building Noetherian domains inside an ideal-adic completion, Abelian Groups, Module Theory and Topology Proceedings in honor of Adalberto Orsatti's 60th Birthday (Dikranjan and Salce, eds.), Marcel Dekker, Inc., New York, 1998. 5. , Intermediate rings between a local domain and its completion, Illinois J. Math. 43 (1999), 19-46. 6. , Noetherian domains inside a homomorphic image of a completion, J. Algebra 215 (1999), 666-681. 7. , Building noetherian and non-noetherian integral domains using power series, Ideal theoretic methods in commutative algebra in honor of James A. Huckaba's retirement (Anderson and Papick, eds.), Marcel Dekker, Inc., New York, 2001, pp. 251-264. 8. , Examples of integral domains inside power series rings, Commutative ring theory and applications (Fontana, Kabbaj, and Wiegand, eds.), Marcel Dekker, Inc., 2002, pp. 233—254. 9. W. Heinzer and J. Sally, Extensions of valuations to the completion of a local domain, J. Pure Appl. Algebra 71 (1991), 175-185. 10. I. Kaplansky, Commutative rings, Allyn Bacon, Boston, 1970. 11. M. Nagata, Local rings, John Wiley, New York, 1962. 12. C. Peskine, Une generalisation du "main theorem" de Zariski, Bull. Sci. Math. 90 (1966), 377-408. 13. O. Zariski and P. Samuel, Commutative algebra volume II, D. Van Nostrand, New York, 1960.

Rings with Finitely Many Orbits under the Regular Action Yasuyuki Hirano Department of Mathematics, Okayama University, Okayama 700-8530, Japan yhiranoSmath.okayama-u.ac.jp Abstract. Let R be a ring and let U(R) denote the group of units of R. Then U(R) acts naturally on R+, the additive group of R, by left multiplication. In this paper, we completely determine the structure of rings R with a finite number of'orbit under the above action of U(R) onR.

1. Introduction Throughout this paper, R denotes a ring with identity and J(R) denotes the Jacobson radical of R. Let U(R) be the group of units of R. We consider a natural group action of U(R) on R, defined by ((u, r) K-> ur) from U(R) x R to R. We call this action the (left) regular action. J. Han [2] proved that if R has only finitely many orbits under the regular action, then the set X of nonzero nonunits of R coincides with the set of all right zero-divisors of R and moreover if X is a nonempty finite set then R is a finite ring. He also proved several results concerning local rings R and U(R) cyclic. In [3], J. Han considered some finiteness properties of a unit-regular ring R under the regular action. In this paper, we study the structure of rings with a finite number of orbits under the regular action. In particular, we prove that a ring R has only finitely many orbits under the regular action if and only if R has only finitely many left ideals, if and only if R is the direct sum of finitely many left uniserial rings and a finite ring. 2. Results Proposition 2.1. If R has only finitely many orbits under the regular action, then R has only finitely many left ideals. In particular, R is left artinian. Key words and phrases, regular actions, uniserial rings, distributive rings, finite rings.

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Proof. We can easily see that every left ideal of R is J7(.R)-invariant. Hence each left ideal is a union of U(R)-oibits. Since R has only finitely many {/(fi)-orbits, the number of left ideals of R is finite. D From this proposition, [2, Theorem 2.2] is clear. Proposition 2.2. If R has only finitely many left ideals, then R/J(K) is the direct sum of finitely many division rings and a finite ring. Proof. By hypothesis, R/J(R) is a semisimple artinian ring, and hence R/J(R) is the direct sum of finitely many simple artinian rings. Let S be one of those simple artinian rings. By Wedderburn structure theorem, S is isomorphic to the full matrix ring Mn(D) with some division ring D and some natural number n. It is easy to see that if D has an infinite number of elements and if n > 1, then Mn(D) has infinitely many left ideals. Therefore, in case n > 1, D must be a finite field. This proves that R/J(R) is a finite direct sum of division rings and finite simple rings. n Proposition 2.3. If R has only finitely many left ideals, then pR has no subfactor which is isomorphic to T © T, where T is a simple module with infinitely many elements. Proof. Suppose, to the contrary, that there exist left ideals L, A and B of R such that R/L has a submodule A/L © B/L, where A/L and B/L are isomorphic simple .R-modules with infinitely many elements. By Proposition 2.2, we may write R/ J(R) = DI © • • • © Dm © F, where Di is an infinite division ring for each i = 1 , . . . , m, and F is a finite semisimple ring. Let Annfi(A/L © B/L) denote the annihilator of A/L ®,B/L in R. Then R/Annfi(A/L © B/L) is naturally isomorphic to one of the D;'s, say D\. Then the lattice of .R-submodules of A/L © B/L is isomorphic to the lattice of subspaces of the 2-dimensional vector space A/L(&B/L over DI. Since the latter lattice has infinitely many elements, this is a contradiction. D A ring R is called a left uniserial ring if its left ideals form a finite chain. An .R-module M is distributive if for any submodules A, B, C of M we have Ar\(B + C)=Ar\B + Ar\C. A ring R is said to be a left distributive ring if R is distributive as a left .R-module. Theorem 2.4. The following statements are equivalent for a ring R: (1) R has only finitely many orbits under the regular action. (2) R has only finitely many left ideals. (3) R is the direct sum of finitely many left uniserial rings and a finite ring. Proof. By Proposition 2.1, we have (1) =>• (2). (2) =» (3). By Propositions 2.1 and 2.2, R is left artinian and R/J(R) = -Di © • • • © -Dfc © PI © • • • © Fm, where each. D i is an infinite division ring and each Fj is a finite simple ring. Assume that k > 1 and m > I. Then there are orthogonal idempotents e:,..., ek, /i, • . . , fm £ R such that e± H \-ek + fi-\ h fm = I , each 6i + J(R) is the identity of Di and each fj + J(R) is the identity of Fj.

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Suppose that epRfq ^ 0 for some p, q, and take a G R such that epafq ^ 0. Then Repafq/J(R)epafq ^ Rep/J(R)ep. Since R/(J(R)ep 0 J(R)epafq) * (Rep/J(R)ep) ® (Rfq/

J(R)epafq)

R/(J(R)ep ® J(R)epafg) has a submodule which is isomorphic to Rep/J(R)ep ® Rep/ J(R)ep. This is a contradiction to Proposition 2.3. Hence etRfj = 0 for all i, j. Suppose next that fqRep ^ 0 for some p, q. Since R is left artinian, rings epRep, fqRfq and (ep + fq)R(ep + fq) are left artinian. Since epRfq = 0, the ring (ep + fq}R(ep + fq)/(epJ(R)ep + fqJ(R)fqRep + fqRepJ(R)ep + fqJ(R)fq) is isomorphic to the generalized matrix ring _ (ePReP/epJ(R)eP

\

M

0

fqRfq/fqJ(R)fq

where M = fqRep/(fqJ(R)f(]Rep + fqRepJ(R)ep). Since epJ(R)ep and fqJ(R)fq are nilpotent, we can easily see that 'M ^ 0. Since M is a right vector space over the infinite division ring epRep/epJ(R)ep, M has infinitely many elements. However, since fqRfq/fqJ(R)fq is a finite ring, the ring A is not left artinian, a contradiction. Hence fiRej = 0 for all i, j. Therefore we have proved that the finite ring (/! + • • • + fm)R(fi + • • • + fm) is a ring direct summand of R. Now consider the ring S = (e\ + ---- h ek)R((ei + • • • + ek). By Proposition 2.3, sS has no subfactor which is isomorphic to T ® T, where T is a simple module. Then $5 is distributive by [1, Theorem 1]. Since S is left artinian, S is the finite direct sum of left uniserial rings by [4, Theorem 2.4]. (3) =>• (1) Suppose that R is the direct sum of finitely many left uniserial rings and a finite ring. To prove that R has only finitely many orbits under the left regular action, we may assume that R is a left uniserial ring. Let R = IQ D /i D • • • D In — 0 be the chain of all left ideals of R and take an arbitrary
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Finally we calculate the number of orbits of some finite rings under the regular action. The first example shows that the assertion in Corollary 2.6 is not true for finite local rings. Example 2.7. Let K be the field of q elements, let V be an m-dimensional vector space over K and consider the ring

w *=((; iv° °. Then R is a finite local ring. Let X be the set of nonzero, nonunits of R. Then the number of orbits in X under the regular action is the number of 1-dimensional subspaces of V, which is equal to the number of elements of the (m — l)-dimensional projective space Pn~1(K). This number is

Hence R has n + 2 orbits under the regular action, where n = (qm — !)/( I. In fact, J(R}2 = 0. In case R is the ring of n x n matrices over a field K, the regular action is familiar in the theory of linear algebra. Example 2.8. Let K be a field of q elements and let n be a positive integer. Consider the ring R = Mn(K) of n x n matrices over K. In this case, U(R) is the general linear group G = GLn(K) over K. Let Er e Mn(K] be the matrix (a^) such that an = • • • = arr = I and a^- = 0 if ( i , j ) ^ (1,1),... (r,r). Let G denote GLn(K] and consider the subgroup Hr = {g £ G | GErg — GEr}. Then u — J I •"• Hr-\\r,

0 n\

A e GLr(K),B e GLn^r(K),C& M (n _ r)xr (^), 1.

Hence \Hr\ = \(qr -l)...(qr- q^1}}^^^^^ - 1) ... (qn~r Hence the number of orbits of R under the regular action is 1+

\G\/\Hr\ = I + r=l

>" - 1) . . . (g" - qr-1)}/[qn-r(qr - 1) . . - (qr - g^1)].

r=l

Finally we give an example of a ring which has only finitely many orbits under the left regular action, but has infinitely many orbits under the right regular action. Example 2.9. Let K be a field and let K(x) denote the field of rational functions in x over K. Let a be a .ftT-endomorphism of K(x) defined by cr(x) = x2. Consider the skew polynomial ring S = K(x)[Y;cr] defined by Y f ( x ) = a(f(x))Y for all f ( x ) e K(x) and construct the factor ring R = S/SY2S. Then # = K(x) + K(x)y = K(x) + yK(x) + xyK(x] whqre y = Y + SY2S € R. Then £ is a left uniserial ring, but is not a right uniserial ring, and hence R has finitely many orbits under the left regular action, but has infinitely many orbits under the right regular action.

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References [1] V. Camillo, Distributive Modules J. Algebra 36 (1975), 16-25. [2] J. Han, Regular action in a ring with a finite number of orbits, Comm. Algebra 25 (1997), 2227-2236. [3] J. Han, Group actions in a unit-regular ring, Comm. Algebra 27 (1999), 3353-3361. [4] W. Stephenson, Modules whose lattice of submodules is distributive, Proc. London Math. Soc. 28 (1974), 291-310.

Projective Covers and Injective Hulls in Abelian Length Categories Otto Kerner Mathematisches Institut, Heinrich-Heine-Universitat, D-40225 Diisseldorf (Germany) kernerScs.uni-duesseldorf.de Dedicated to Roberto Martinez on the occasion of his sixtieth birthday

Abstract. The existence of injective hulls and projective covers of simple objects in abelian length categories is discussed.

An abelian category A is called a length category, if each object M € A has a composition series. We say M has length c(M), if the composition series of M have length c(M). Although a subobject of an object M € A is defined as an equivalence class of iponos i: N —> M, see for example [2], we write (N, i) or even N for a subobject. An object M £ A is called local, if it has a unique maximal subobject (rad M, i). In the induced short exact sequence 0 —> radM -^ M -^ S —> 0, the cokernel S is simple in A and it is called the top of the local object M. Denote by £(S) the class of local objects with top isomorphic to the simple object S. Since S €E £(£"), this class is nonempty and it is closed under nonzero epimorphic images. Dually, an object M € A is called colocal, if it has a unique minimal subobject socM, which then necessarily is simple and is called the socle of M. Denote by £*(5r) the class of colocal objects with socle S. We say that a class X of objects of A has bounded length, if max{c(X) X g X} exists. In this note the existence of projective covers and injective hulls in abelian length categories is studied. Recall that a pair (P, /M) with P projective and /M : -P —> M epimorphic is called a projective cover of M, if for g: X —» P the map p/M only is an epi, provided g is an epi. Injective hulls are defined dually. 2000 Mathematics Subject Classification. 18G05, 16D40, 16D50.

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Motivated by [3] we consider the following problem: When does a simple object S admit a projective cover, respectively an injective hull? Theorem. Let A be a skeletally small abelian length category. (a) A simple object S has a projective cover, if and only if the class £(•?) has bounded length. (b) A simple object S has an injective hull, if and only if the class £*(S) has bounded length. A short proof is presented here, and it goes back to conversations with Henning Krause. Proof. It is enough to show (a). The proof of part (b) is dual. If 5 has a projective cover fs: P(S) -> S, then clearly P(5) 6 £(S). It has maximal length, since each element in £(5) is an epimorphic image of P(S}. Conversely, assume the class £(5) has bounded length. We show that in this case there is a unique element of maximal length P(S) e £(S), up to isomorphisms, and that P(S) is a projective cover of S. Let P(S) be any object of maximal length in £(S). (A) We show that for each M e £(•?) there exists an epi /M : P(S) —> M. Suppose the assertion is false and let J\f(S) be the set of objects in £(5) which are not epimorphic images of P(S). Take jV 6 Af(S) of minimal length. From S £ N(S) we get c(N) > 1. Therefore N has a proper simple subobject (5', i). Let 0-4 S' ^>N ^ N' -> 0

be the induced short exact sequence. Since £(5) is closed under nonzero epimorphic images and c(N') < 'c(N), we get N' e £(S) \ A/"(5). Hence there exists an epi /': P(S) -> N'. Since N e Af(S), the epi /' does not factorise through p. Consider the pullback diagram 0

> 5' —5_ M —?—4 P(S)

> 0

0 > S' ——+ N —£-» N' > 0 Since /' does not factorise through p, the first row is a non-split short exact sequence. But c(M) > c(P(S)) implies M £ £(S). Since M maps nontrivially to S, it cannot be local. Therefore there exists a simple object L and an epi q: M —*• L which does not factorise through g. Application of the functor Hom(—, L) = (—, L) to the first row gives the exact sequence

0 _ (P(S),L) ^ (M,L) -^l (S',L) Since q is not in the image of (g, L), we get 0 ^ (e, L)(q) — eq. Since 5' and L are simple objects, eq is an isomorphism, hence e is a, split mono. Thus the first row splits, a contradiction. Note that the condition (A) already implies that P(S) is the unique element with maximal length in £(5), up to isomorphisms.

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(B) We show that P(S) is projective, that is each epi p: M —> P(S) splits. Let £(M) be the set of subobjects (N, ZAT) of M such that the composition ifjp: N —> P(5) still is epimorphic. Since (M, IM) € £ (M), the set £(M) is not empty. Choose (N0,io) £ £(-W) with c(JVo) minimal. If (N',i) is a proper subobject of NO, the composition Hop: N' —*• P(S) is not epimorphic, hence it has image in the unique maximal subobject radP(S') of P(S). Consequently NO is a local object. Indeed, if MI and Mj are two different maximal subobjects in NO, then (Mj)ip C rad(P(5), hence (No)ip = (Mi +M'2)ip C radP(S), a contradiction. Since NO is a local object and it maps onto P(S), we get NO € -£(5). Since P(S) has maximal length in £(S), the epi ip is an isomorphism. Therefore p is a split epi with section (ip)~ii. (C) Any nonzero map f$ : P(<5) —> 5 is a projective cover of 5, since P(5) is local. D Example. For details on representations of quivers see [1, 5]. Let Q = (Qo, Qi) be some locally finite quiver, with set of vertices Qo, respectively arrows Qi, without oriented cycles, and let fe be some field. Let repfc Q be the category of finite dimensional fc-linear representations of Q, that is those representations M = (M(i),M(a) | i e Qo,a 6 Qi), where dimM = J2i£Q0 dimM(i) < oo. Then repfc Q is an abelian length category. Its simple objects Si are the one-dimensional representations, concentrated at the vertex i. For j S Qo we consider the class C*(Sj) of colocal representations with socle Sj. A representation M £ repfc Q belongs to £*(Sj) if and only if (a) M(j) = k and M(i) = 0, if there is no path from i to j in Q. (b) If i ^ j and o^, . . . , ar are the arrows, starting in i with terminating points ti, . . . ,tr, then P|kerM(o:s) = 0. Consequently (*)

dimM(i) <

holds. Therefore £*(Sj) has bounded length, only if the length of the paths, terminating in j is bounded. Indeed, if it is not bounded, for each natural number N, there exists such a path of length N. The (N + l)-dimensional indecomposable representation, concentrated at this path then is colocal with socle Sj. On the other hand, if the length of paths, terminating in j is bounded, there are only finitely many such paths, since Q is locally finite. If we have dim M(i) = ^dimM(i s ), in (*), for all i ^ j, this representation is of maximal length in C*(Sj). One easily can construct an element Qj of maximal dimension in L*(Sj) hence the injective hull of 5-,-: Let p ( i , j ) denote the number of paths from i to j. If i is any vertex, Qj(i) has dimension p(i,j). Let xl(iti), . . . ,x*(7r p (jj)) be a basis of Qj(i), indexed by the different paths irr from i to j. If a is an arrow, starting in i and ending in i', define M(a)(xl(ns)} = xl (IT'), if TTS = an', and M(a)(xl(T7s)) = 0 else. Normally the indecomposable injective, respectively projective, representations are constructed via the equivalence repfc Q = fcQ-mod, where fcQ-mod is the category of finite dimensional modules over the path-algebra kQ of Q, see for exampie (!}.

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It is well known that every object in an abelian length category has a projective cover, respectively an injective hull, if and only if so have the simple objects. In this case one says, A has projective covers, respectively injective hulls. Therefore we get: Corollary 1. A has projective covers, respectively injective hulls, if and only if for each simple object S 6 A the class £(S), respectively £*(S), has bounded length. Assume A has only finitely many isomorphism classes of simple objects, and let {Si, . . . , Sn} be a complete set of representatives of simple objects. If P(Si) is the projective cover of Si, then P = ©™=1 P(Si) is a small projective generator in A, and each object of A is finitely generated by P. Hence A = (End^(P))op-mod, by Morita's theorem. Since A is a length category, the ring (End_4(P)) op has to be left artinian. Therefore we have the following corollary: Corollary 2. An abelian length category with finitely many isomorphism classes of simple objects is equivalent to A.-mod for some left-artinian ring A if and only if for each simple object S the class £(S) of local objects with top S has bounded Remark. For a left artinian ring A the abelian length category A-mod contains projective covers. In [4] an example for a left artinian ring A is constructed, where no indecomposable injective left A-module has finite length. References [1] M. AUSLANDER, I. REITEN AND S. SMAL0. Representation theory of artin algebras. Cambridge Studies in Advanced Mathematics, 1994. [2] J. P. FREYD. Abelian Categories. Harper & Row, New York, 1964. [3] I. REITEN AND M. VAN DEN BERGH. Noetherian hereditary abelian categories satisfying Serre duality. Preprint Mathematics 10/2000, Trondheim. [4] A. ROSENBERG AND D. ZELINSKY. Finiteness of the injective hull. Math. Z. 70 (1959), 372-380. [5] C. M. RlNGEL. Tame Algebras and Integral Quadratic Forms. Springer Lecture Notes in Math. 1099 (1984).

Semigroup Rings that are Inside Factorial and their Cale Representation Ulrich Krause Department of Mathematics, University of Bremen, Bibliothekstr. 1, Bremen (German}') krauseSmath.uni-bremen.de

1. Introduction: Not factorial but still nice? Concerning the ubiquitous property of factoriality, Robert Gilmer and Tom Parker in a beautiful paper proved that a semigroup ring D[S] is factorial precisely if both the integral domain D and the semigroup 5 are factorial (and 5 satisfies some additional property which is described in [10, Theorem 7.17] and [11, Theorem 14.16]). Even for a field D, a non-factorial but otherwise nice semigroup S leads, according to the Gilmer-Parker Theorem, to a non-factorial semigroup ring D[S]— which, nevertheless, can be quite nice. In this paper, we address the question of whether the semigroup ring D[S\ still enjoys some nice factorization properties— albeit neither D nor S is assumed to be factorial. The answer is given by Theorem 3.2 in section 3 where we show that D[S] is inside factorial if both D and 5 are inside factorial and D[S] is solid. Before proving this result, we define and examine in Section 2 the concepts of inside factoriality and Cale representation, respectively. But before doing so, we discuss the meaning of "nice" as used above and recall some fundamental notions. Let D denote an integral domain with identity (or just a domain for short). Let S denote an additively written abelian semigroup with 0 which is cancellative (i.e., x + z = y + z implies x — y where x,y,z 6 S). Furthermore, assume S to be torsion free (i.e., nx = ny implies x = y for x,y £ 5 and n > 1) and that S contains only trivial invertible elements (i.e., x, — x e S implies that x = 0). For an integral domain D and a semigroup 5, the semigroup ring D[S] is the set of all mappings /: 5 —> D with /(s) ^ 0 only for finitely many s 6 S equipped with an addition defined by (/ + g)(s) = f ( s ) + g(s) and a multiplication defined by (/ • g)(s) = Et+u=s /(*)$(«)• An element / of D[S] we write as £ s€S f ( s ) X s (Cf. [11]). Considering examples, by the GilmerLParker Theorem for D = K a field and S the factorial (or free) semigroup N" (N = {0,1,2,...}), the semigroup ring K[S\ — 353

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K[Xi,...,Xn] (i.e., the ring of all polynomials in n indeterminates over K) is factorial. As soon, however, as the semigroup S is not isomorphic to some N", the semigroup ring K[S] is not factorial. A first interesting case, in one dimension, is a numerical semigroup S (i.e., S ^ N and S — S = Z). Whatever the integral domain D, the semigroup ring D[S] is never factorial and, in particular, for D = R and S = N \ {1} it shows a very bad factorization behavior. In higher dimensions, for Diophantine monoids S = {x £ N™ Ax = 0} where A € Zmxn, a variety of factorization behaviors can be observed (see Section 3). In general, the semigroup ring K[S] for a field K is not factorial, but factorization behavior can be nice in that it is inside factorial as, for example when S — {x € N3 Ixi + 5z2 — ^x3}. Considering the above mentioned case of bad factorization behavior for S = N\{1} and D = R, one has that D[S] = D[X2, X3} is isomorphic to R[X, Y] modulo the ideal generated by Y2 — X3. For any number of indeterminates one has that the polynomial ring R[Xi,... ,Xn] modulo the ideal generated by X2+X^-Ji \-X2 — l is factorial for n > 3 (see [9]) and not factorial but with a nice factorization property for n = 2 (see [16]). In the final section 4 we apply the general results of section 3 to subsemigroups S of N". If S is a Diophantine monoid, given by equations or by inequalities, the inside factoriality of the corresponding semigroup ring is characterized in terms of equations (Theorem 4.1). For S finitely generated, the notion of a Gale matrix is introduced and employed to describe toric ideals (Proposition 4.2). The author wishes to thank an anonymous referee for many helpful suggestions. 2. Inside factoriality and Gale representations for monoids and domains Let H denote a monoid, which means a multiplicatively written abelian semigroup with 1 which is cancellative, and denote by Hx the group of units of H. The following definition presents the main notions for all that follows. Definition 2.1. The monoid H is inside factorial with base Q c H \ Hx if for each x & H \ H* there exists a positive integer rn(x) and nonnegative integers {x(q)}qeQ such that 1.

where m(x] is minimally chosen, u g Hx and only finitely many of the x( e ffx and each s(q) is a nonnegative, then v = u and s(q) = x(q) for each q. The base Q is called a tame base if'for every q 6 Q there exists a (minimally chosen) e(g) e N* such that ^ffi S N for all a; G jff \ F x . We will call an inside factorial monoid with tame base a Gale monoid where each nonunit x e H has a Gale representation (*).

Semigroup Rings that are Inside Factorial and their Cale Representation 355

A domain R is called inside factorial if its multiplicative monoid H = R* = R \ {0} has the properties outlined above. Remark 2.2. A monoid H is inside factorial if the submonoid F generated by Q and Hx is the "biggest" factorial monoid sitting inside of H in such a way that every element of the quotient H/F has torsion. For inside factorial monoids and domains see [3], [13], [14] and in particular, [4]. The concept of inside factoriality for domains is closely related to the "fastfaktorielle Ringe" analyzed by U. Storch (see [19]) and to "almost factoriality" as studied by M. Zafrullah (see [21]). Definition 2.3. For two monoids with H C S and with the same identity the set Tfs = {s e S sn <= H for some n £ N*} is called the root-closure of H in S. H C S is called a root extension or H is root-closed in S if H s = S. In the particular case where S is the quotient group q(H) of H, HS is simply called the root-closure of H denoted by H and H is simply root-closed if H — H. (Note that in [11, p. 151] the term integral closure is used for root-closure.) For two domains the notions of root-closure, root extension and root-closed are defined as above with respect to the corresponding multiplicative monoids. (For the root-closure in commutative rings see [1], for the root-closure in algebraic orders see [15].) A domain D is called solid if the root-closure of its multiplicative monoid D* is additively closed or, equivalently, if D = D* U {0} is a subdomain of q(D). The main result we will need about inside factoriality is the following characterization of Cale monoids and Cale domains ([4, Theorems 4 and 7]). Theorem 2.4. (i) The monoid H is a Cale monoid if and only if H is a Krull monoid with class group Cl(H) a torsion group. (ii) The domain D is a Cale domain if and only if its integral closure D is a Krull domain with class group Cl(D) a torsion group and if D C D is a root extension. 3. Cale semigroup rings This section presents the main result of this paper. The following Lemma will be needed in its proof. Lemma 3.1. Let D be a Cale domain and let S be an additively written monoid which is torsion free. (i) D is solid, (ii) the following statements are equivalent: (a) D[S] is solid, (b) D[S] C D[S] is a root extension,

(c) D[S]=D[S}.

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Proof, (i) By Theorem 2.4 (ii), D c D is a root extension, i.e., D* C D*. Obviously, D* C D* and, hence, D = D. Since D is a subdomain of q(D] the same holds for 15. _ (ii) First we show that (a) implies (b). By assumption D\S\ is a subdomain of q(D[S\). Obviously, ~D C D[S]. Consider Xt for t e 5, i.e., * = si - s2 with nt G 5, where si,s 2 6 S and n € N*. It follows that X* = X^~^ e g(-D[S]) and (JiT*)" e D[5]. Therefore, A"* e P[5]. Since S is a domain by (i), it follows for any / = £tes /(*)^* e ~D\S} that / 6 D[5]. Thus Z5[5] C D^]. Furthermore, by [11, Corollary 12.11(1)] Z^SJ = D[~S}. Since D is Gale one has as injtep (i) that D=~D. Altogether, D\S] = Z)[S] C D[S], which means that D[S] C D[S] is a root extension. _ ^___ __ Similarly, from (b) it follows that D[S] = D[S} = D[~S], that is (c). __ Finally, since D is Gale it follows from (i) that D is a domain and, hence, D[S\ is a domain. This shows that (c) implies (a). D Now we are ready to extend the beautiful result of Gilmer and Parker concerning factoriality to the weaker but still "nice" property of inside factoriality. Theorem 3.2. Let D be a domain and let S be an additive abelian semigroup with 0 which is cancellative, torsionfree with trivial invertible elements. Then the semigroup ring D[S] is a Gale domain if and only if D is a Cole domain, S is a Gale monoid, and the semigroup ring is solid. Proof. (=>•) Suppose first that D[S] is a Gale domain and let Q be a Gale base of D[S}. We shall show' that QD = Q n D is a Gale base of D and that Qs = {s € 5 usXs E Q for some unit us of D} is a Gale base of S. Note that units in D[S] must be units in D because invertible elements of S are trivial. Also, for a product in D[S] which is monomial both factors must be monomial (see [11, Theorem 11.1]). With respect to QD, let / £ D c D[S] be a non-unit in D[S] with Gale representation fmW = w]jLeQ < 7'^ 9 ' ) ! u um* m &[&]• As mentioned u must be a unit in D and for f(q) > 0 one has that q 6 D. This shows that QD is a Gale base of Z?. Concerning Qs, let s 6 S, s ^ 0 and consider the Gale representation of Xs in D[S\, namely for some m 6 N* (Xs)m = uY[q(-Q qn(-i\ u a unit in D[S\. Again, u must be a unit in D and for q e Q with n(q) > 0 one has that q = u(q)Xs^\ where u(q) is a unit in D and s(q) 6 S \ {0}. This implies that 'Mn(q\

u' a u n i t i n g

?eQ

and, hence, ms —

s(q)n(q),

where s(q) 6 Qs.

Semigroup Rings that are Inside Factorial and their Cale Representation 357

For the uniqueness of such a representation assume that ms = Y^ tn(t) = Y^ tp(t)

with n(t),p(t) € N.

It follows that

Xms = and, because of utXt = qt € Q with a unit ut of .D, wHt?" = ^Ilt?? with units w, v of I?. Since Q is a Cale base of D[S] it follows that u — v and n(t) = p(t) for all t. This shows that Qs is a Cale base for S. (<=) Suppose now that D is a Cale domain, S is a Cale monoid and D[S] is solid. From Theorem 2.4 we have that D is a Krull domain with a torsion class group Cl(D) and that 5 is a Krull monoid with a torsion class group Cl(S}. By [11, Corollary 12.11] it holds that E>[S} = Dfi] and, by [11, Theorem 15.6], D[S] is a Krull domain. Furthermore, from [11, Corollary 16.8]we have that Cl(D\S\) — Cl(D) ® Cl(S). Since Cl(D] and Cl(S) are torsion groups, Gl(D\S}) is a torsion group, too. Thus, D[S] is "a Krull domain with a torsion class group. By assumption D[S\ is solid and I? is a Cale domain and, hence, by Lemma 3.1 one has that D[S] C D[S] is a root extension. Theorem 2.4 (ii) yields that D[S] is a Cale domain. D Because of the relation D[S] — D[S\, one obtains easily from Theorem 3.2 the following consequence. Corollary 3.3. Let the domain D and the monoid S be as in Theorem 3.2 and assume in addition that D is integrally closed and that S is root-closed. Then the semigroup ring D[S] is Cale if and only if D and S are Cale. Remark 3.4. Obviously, for D a Cale domain Theorem 3.2 means that D[S] is a Cale domain iff S is a Cale monoid and D[S] is solid. As has been suggested by an anonymous referee, the latter property means that for D[S] root-closure and integral closure coincide. In what follows, we discuss the importance of this requirement. (In [15] one finds examples of algebraic orders R which are root-closed but not integrally closed; such an R is solid but R C .R.) (i) Consider D = R and S = N\{1}. It is easy to see that S, which is not factorial, is a Cale monoid with, for instance, Q = {2} as base. The semigroup ring D[S] = D[X2,X3], however, is not a Cale domain—the bad factorization behavior as mentioned in Section 1. The reason is that R = D[S] is not solid, because one has that 1, X £ R, (1 + X) £ q(R) but no power of 1 + X is in R. This example also shows that the assumption of root-closedness for S in Corollary 3.3 cannot simply be omitted. (ii) If, however, D is a finite field, the situation is different. If, for instance, D = F 2 , then a, b e F2 implies that (a+bX)2 = a2+b2X2+2abX € F2[JsT2, X3} and, hence, W2[X] = W2[X2,X3}. In particular, R = D[S] is solid, and, by

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Theorem 3.2, R = ¥2[X2,XZ} is a Gale domain. It is easily verified that the squares of all irreducible polynomials in F2[X] form a Gale base of F2[X 2 , X3}. The same argument can be applied to show that for any arbitrary finite field D and any numerical semigroup S the semigroup ring D[S\ is a Gale domain. (iii) Considering, more general, any S which is not root-closed, the following nice characterization has been communicated by F. Halter-Koch [12] to the author: For a Gale domain D and a not root-closed Gale monoid S, root-closure and integral closure for D [S] coincide if and only if D has characteristic p > 0 and for each s e S there exists k £ N such that pks & S. Furthermore, Corollary 3.3 shows for an arbitrary field K and S root-closed that K[S] is always solid and that K[S] is Gale precisely if S is Gale. Therefore, domains K[S] with S root-closed but not Gale exhibit solid domains which are not Gale. Examples are provided in the next section. 4. Diophantine monoids, Gale matrices, and toric ideals In the following, we will draw some conclusions from the general theory developed in the first three sections. For A £ Z m x n the set S = {x 6 N" | Ax = 0} is a monoid for the (componentwise) addition in N" which is torsion free and has only trivial invertible elements. Similarly, the set S = {x e Nn Ax ^ c} U {0} with A € N m x n and c e Mm with components C{ > 0 is a monoid which also possesses these properties. Monoids of this kind are called Diophantine monoids given by equations or inequalities, respectively. Whereas Diophantine monoids when defined by equations are Krull monoids, this is not the case when defined by inequalities (for Diophantine monoids given by equations see [5], which also contains a characterization of these monoids as those (reduced) monoids that are root-closed and finitely generated). Finitely generated Krull monoids have been a topic of interest in the recent literature ([8]). For semigroup rings over Diophantine monoids, one obtains the following result. Theorem 4.1. Let D be a factorial domain. (i) For a Diophantine monoid S given by equations the following statements are equivalent: (a) D[S] is a Gale domain. (b) S is a Gale monoid. (c) S can be described by a matrix A for which each row has just one positive entry or just one negative entry. (ii) For a Diophantine monoid S ^ {0} given by inequalities with D[S] solid the following statements are equivalent: (a) D[S] is a Gale domain. (b) S is a Gale monoid. (c) // Ci > 0 then the i-th row of A is positive. Proof, (i) D is integrally closed and Gale and S is root-closed. Therefore, the equivalence of (a) and (b) is implied by Corollary 3.3. By Theorem 2.4 (i) the

Semigroup Rings that are Inside Factorial and their Cale Representation 359

monoid S, which is a Krull monoid, is a Cale monoid if and only if Cl(S) is a torsion group. From [5, Theorem 3.1] it follows that Cl(S) is a torsion group iff (c) holds. This proves (i). (ii) Since D[S] is assumed to be solid, the equivalence of (a) and (b) is implied by Theorem 3.2. For the equivalence of (b) and (c) assume first that S is Cale. Suppose that for some i one has that Cj > 0 but a,ij = 0 for some j. If n = 1 then this contradicts the assumption that 5 ^ {0}. Therefore, let n > 1. Then x — ej e N™ and A(kx) ^ c is impossible for every k > 1. That is, BJ £ S. On the other hand, there exists x e 5 with Ax ^ c and, hence, A(x + kej) ^ c for all k > 1. Thus, x + kej e S for all fc > 1 and, in particular, ej € q(S). By Theorem 3.2, the root-closure S is a Krull monoid and for all valuations v defining S in q(S) one has that v(x + kej) > 0 for all fc > 1. This implies V(BJ) > — k~lv(x) for all fc > 1 and, hence, v(ej) > 0. That is BJ € S, which is a contradiction. Therefore, statement (c) must hold. Conversely, assume that (c). We shall show that 5 = Nn which, by Theorem 3.2, implies that (b) must hold. From (c) it follows in particular that for some r > 1 one has that r = (r,..., r) G 5 and, hence, for any x 6 1n and k > 1 big enough x + kr G S and fcf £ S. This shows x = x + kr — fcr G S — S and, hence, Zn = S - S. In particular, S C N". Furthermore, let x 6 Nn c 5 - S, x ^ 0. From (c) it follows that there exists fc > 1 such that for all 1 < i < m it holds that k^2j dijXj > a. Thus, kx £ 5 and, hence, N" C 5. This shows 5 — Nn which proves (b). D We illustrate Theorem 4.1 by some examples. Examples. 1. Let 5,= {x £ N3 | 2rci + 5o;2 = 3xs} be a Diophantine monoid given by just one equation. Then for any field K one has that K[S] is Cale by Theorem 4.1 (i) because the matrix A — [2 5 —3] has just one negative entry. The monoid 5" is generated by the irreducible elements gi = (3,0, 2), gi = (0, 3, 5), 53 = (1, 2,4) and 54 = (2,1,3). Therefore, the semigroup ring K[S] is equal to the monomial subring

K[X3Z2, Y3Z5,XY2Z\ X2YZ3} of the ring of polynomials K[X, Y, Z] (for monomial subrings see [20, Chapter 7]). 2. Let 5 = {x 6 N4 | xi + x3 = x 2 + x4} be another Diophantine monoid given also by just one equation. Then one has for an arbitrary field K that K[S] is not Cale by Theorem 4.1 (i) because A = [1 1 —1 — 1] has two positive and two negative entries. (5 cannot be represented otherwise, actually Cl(S) = Z.) The monoid S is generated by the irreducible elements §i = (1,0,1,0), g% = (1,0,0,1), 53 = (0,1,1,0) and 54 = (0,1,0,1). Therefore, the semigroup ring K[S\ is equal to the monomial subring

K[XU, XV, YU, YV] of the ring of polynomials K[X,Y, C/, V]., Let S = {x e N2 | x ^ (1,1)}(J{0} be a Diophantine monoid defined by inequalities, where A is the 2 x 2 identity matrix and c = (1,1). From Theorem 4.1 (ii) it follows that K[S] cannot be Cale whatever the field K. Obviously, K[S] = K+XYK[X, Y}.

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The example S = N \ {1} discussed earlier is, for A = [1], c = 2 also an example of a Diophantine monoid denned by inequalities. As already seen, in this case K[S] is not Gale for K = R but it is Gale for K = W^. In this case 5 satisfies property (c) of Theorem 4.1 (ii), but R[S] is not solid whereas ¥2\S} is solid. For a Gale monoid which is finitely generated, such as a Diophantine monoid defined by equations, the essence of the Gale representation can be captured by a matrix of natural numbers, the Gale matrix. More precisely, let 5 be an additively written Gale monoid which is finitely generated, torsionfree and with trivial invertible elements. Let gi, . . . ,gi be a generating set of atoms (irreducible elements) of S and let pi, . . . , pfc for fc < Z be a Gale base Q of S. The Gale representation then yields

for all 1 < j < I, where rn,- G N* is minimally chosen and the Cij 6 N are uniquely determined. Let m = lcm{mi, . . . , m/} and m'- = mm"1. The Cale matrix of S with respect to Q is defined as

Obviously, C is of the form [mlk • • •} where Ik is the k x k identity matrix, and S is factorial if and only if k = I and C = IkWe illustrate the concept by two simple examples: Examples. 1. Let S be an arbitrary numerical semigroup, i.e., 5 — g± . . . +p;N, with positive integers i = (3,0,2), p2 = (0,3,5), p3 = (1,2,4),
The Cale matrix is connected also'with a presentation of a semigroup ring D[S] by the generators of S which might be called the Redei presentation because it is reminiscent of Redei's theorem that every finitely generated monoid is finitely presented (cf. [18, Theorem 5.12]).

Semigroup Rings that are Inside Factorial and their Cale Representation 361

More precisely, consider any additive submonoid S C N™ that is finitely generated by atoms S defined by $(o) = 2^i=i ai9i ^s a surjective hornomorphism of additive monoids. For a domain D the mappings $ induces a surjective ring homomorphism $*: D\Kl] -> D[5] by

where for u = (HI, . . . , ut) € N* the monomial X^X^2 . . . X^ is denoted by Xu. The ideal Is = ker$* of the polynomial ring D[N!] is generated by the binomials Xa - Xb, where a, b 6 Nz with $(a) = $(6), and it is called the tone ideal of S with respect to {gi, . . . ,gi}. (Cf. [2, Section 5.1], [20, Chapter 7] where it is called the presentation ideal). Called the kernel ideal, this ideal was already analyzed in [11, §7] for more general semigroup rings. Obviously, D[S] ~ Z3[N;]//s and under certain conditions on the semigroup ring, D[S] is the coordinate ring of the affine toric variety belonging to 1$. (For D a field see [6], [7], [20]). In case the monoid S is Cale, its toric ideal can be described by the Cale matrix as follows. Proposition 4.2. Let D be a domain and let S be an additive, finitely generated submonoid ofNn which is a Cale monoid with Cale matrix C e N f c x '. (i) The toric ideal of S is generated by the binomials Xa — Xb where a, b 6 N' with min(aj, 6j) = 0 for all i and Ca = Cb. (ii) Particular binomials given by the Cale representation are X™j - X?> X°22i . . . Xckkj

forl<j
Proof, (i) The toric ideal of 5 is generated by Xa - Xb for a, b e N' with 4>(a) = $(&). Hence 53 7=1 aj9j = ^2j=i^j9} f°r a fi-xed system of generators g\,...,giIf a = min( ai , 6i) > 0 then Xa - Xb = Xc(Xa' - Xb') for c = (GI, . . . , c{) and a',b' 6 N' with min(a^,6^) = min(aj — Cj,6j — c») = 0 for all i. Also, 2j=i a'j9j = !Cj=i b'j9j- Therefore, without loss we may assume that min(ai, fej) = 0 for all i. Since g\, • • • ,gk form a Cale base we have that k

k

m

j9j = ^2cij9i and, hence, mgj = i=l

t=l

This implies that ;

i

k

m$(a) = m

j / n

Therefore, by the uniqueness of the Cal^ representation, $(a) = $(6) is equivalent to Ca = Cb. (ii) By the Cale representation for fixed 1 < j < I rrijgj = J^i=1 c^pj and, hence, for a = (0, . . . , nij, . . . , 0) 6 N' and for b — (cy, . . . , cjy , 0, . . . , 0) e N' one

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has that $(a) = m,jQj = ^3t=i cy'#« = $(&)• Thus, the binomials Xa — Xb with a and 6 as above are special elements of the toric ideal of 5. D Of course, by the definition of the homomorphism $ as $(a) = Sj=i aj9jt one has that the toric ideal of S is always generated by the binomials Xa — Xb where a,6, 6 N' with min(aj.&i) = 0 for all i and Ma = Mb where M is the so called associated matrix given by M = [5152 • • • <7i]- (See [20, p. 202/203].) The point of Proposition 4.2 is that for a Gale monoid one may use in such a description a possibly smaller matrix, namely the Gale matrix. In contrast to the associated matrix, the Gale matrix makes sense also for finitely generated monoids that are not submonoids of some N™ . As an illustration consider the following examples, discussed already for the Gale matrix. Examples. 1. Let S be an arbitrary numerical semigroup S = where g\ < . . . < gi are relatively prime. It turned out that the Gale matrix of 5 with respect to Q = {gi} is C = \g\gz • • •
^

(with respect to the base chosen).

1

The associated matrix is given by " 3 0 1 2 0 3 2 1

(with respect to the generators chosen).

2 5 4 3 Thus, Gale matrix and associated matrix differ. According to Proposition 4.2, to determine the toric ideal of D[S] one has to solve the system of linear Diophantine equations Cu = 0 for u — (MI,U2, «s,u±) € Z 4 . One obtains uy, = —3ui — 2it4, u^ = 2ttj + 144 and choosing parameters r = ui,s = u4 one has that u = ru1 + sv? for u1 = (1,2,-3,0), u2 = (0,1,-2,1). The solutions u1 and u2 correspond to the binomials X^X?, — X% and ^2X4 — X2. It is easily verified that the toric ideal Is is generated by these two binomials and, hence, the affine toric variety defined by Is is given by V = V(Ig) = {x € D4 | x\x\ = x\ and x%X4 = x2}.

Semigroup Rings that are Inside Factorial and their Gale Representation 363

References [I] D. F. Anderson. Root closure in commutative rings: a survey. In: Advances in Commutative Ring Theory. Lecture Notes in Pure and Appl. Math. 205:55—71, Marcel Dekker Inc., New York, 1999. [2] W. Bruns, J. Gubeladze, and N. V. Trung. Problems and algorithms for affine semigroups. Osnabriicker Schriften zur Mathematik, Heft 230, preprint, 2001. [3] S. T. Chapman and U. Krause. On the Gale property in integral domains and monoids. Unpublished manuscript, 2000. http://www.trinity.edu/schapman/cale.pdf [4] S. T. Chapman, F. Halter-Koch, and U. Krause. Inside factorial monoids and integral domains. J. Algebra, 252:350-375, 2002. [5] S. T. Chapman, U. Krause, and E. Oeljeklaus. On diophantine monoids and their class groups. Pac. J. Math., 207:125-147, 2002. [6] D. Cox, J. Little, and D. O'Shea. Ideals, Varieties, and Algorithms. Springer Verlag, New York etc., 1992. [7] D. Eisenbud and B. Sturmfels. Binomial ideals. Duke Math. J., 84:1-45, 1996. [8] A. Facchini. Direct sum decompositions of modules, semilocal endomorphism rings, and Krull monoids. J. Algebra, 256:280-307, 2002. [9] R. M. Fossum. The Divisor Class Group of a Krull Domain. Springer Verlag, Berlin etc., 1973. [10] R. Giimer and T. Parker. Divisibility properties in semigroup rings. Mich. Math. J., 21:65—86, 1974. [11] R. Gilmer. Commutative Semigroup Rings. The University of Chicago Press, Chicago, 1984. [12] F. Halter-Koch. Private communication, October 2002. [13] U. Krause. Eindeutige Faktorisierung ohne ideale Elemente. Abh. Braunschweigische Wiss. Ges., 33:169-177, 1982. [14] U. Krause. Semigroups that are factorial from inside or from outside. In J. Almeida et al., editor, Lattices, Semigroups and Universal Algebra, pages 147—161. Plenum Press, New York, 1990. [15] M. Picavet-L'Hermitte. Root closure in algebraic orders. Arch. Math., 78:435-444, 2002. [16] M. Picavet-L'Hermitte. Cale bases in algebraic orders. Annales Mathematiques Blaise Pascal. to appear. [17] G. Picavet and M. Picavet-L'Hermitte. Trigonometric polynomial rings. In: Commutative Ring Theory and Applications. Lecture Notes in Pure and Appl. Math., 231:419—433. Marcel Dekker Inc., New York, 2003. [18] J. C. Rosales and Garci'a-Sanchez. Finitely Generated Commutative Monoids. Nova Science Publishers, New York, 1999. [19] U. Storch. Fastfaktorielle Ringe. Schriftenreihe Math. Inst. Univ. Miinster, Heft 36, 1967. [20] R. H. Villarreal. Monomial Algebras. Marcel Dekker Inc., New York, 2001. [21] M. Zafrullah. A general theory of almost factoriality. Manuscripta Math., 51:29—62, 1985.

Reduced Modules Tsiu-Kwen Lee Department of Mathematics, National Taiwan University, Taipei 106, Taiwan tkleeSmath.ntu.edu.tw

Yiqiang Zhou Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, Newfoundland A1C 5S7, Canada zhouSmath.mun.ca

Abstract. Extending the notion of a reduced ring, we call a right module M over a ring R a reduced module if, for any m g M and a 6 R, ma = 0 implies mR D Ma = 0. Various results of reduced rings are extended to reduced modules. Moreover, these modules are used to obtain results on certain annihilator conditions of the polynomial extension and the power series extension of a module.

All rings R are associative and have identity, and modules are unitary right modules. A ring R is called a reduced ring if a2 = 0 in R always implies a = 0. The notion of reduced rings has been studied by many authors. Some of the known results on reduced rings can be recalled as follows: R is reduced iff R[x] is reduced iff -R[[a;]] is reduced; R is reduced iff R is a subdirect product of domains by Andrunakievic and Rjabuhin [2]; recently it was proved in Anderson and Camillo [1] that, for n > 2, R is reduced iff R [ x ] / ( x n ) is an Armendariz ring where an Armendariz ring is any ring 5 such that if (53iLoa»a;I)(I3j=o^'a;J') = 0 in S[x] then a,ibj — 0 for all i and j. For an endomorphism a of R, Krempa [12] obtained that the skew polynomial ring R[x; a] is reduced iff R is a-rigid, that is, for any a € R, aa(a) = 0 implies a = 0. The concept of a reduced ring is very useful in the investigation of certain annihilator conditions of R[x] and .R[[#]]. A ring R is called a Baer (resp. right p.p.-) ring if the right annihilator of any non-empty subset (resp. any element) of R is generated by an idempotent of R. A well-known result of Armendariz [3] states that, for a reduced ring R, R is Baer (resp. right p.p.) iff so is R[x], and there exist non-reduced Baer rings whose polynomial ring is not 1991 Mathematics Subject Classification. Primary 16D80, 16S36.

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Baer. Recently, this result has been extended in several directions by BirkenmeierKim-Park [5], Han-Hirano-Kim [6], Hirano [7], Hong-Kim-Kwak [8], and Kim-Lee [11]. In this paper, the notion of a reduced module is introduced as a generalization of reduced rings to modules. Reduced modules are rich of supply and they behave as well as their analogues of rings. Various results of reduced rings are extended to reduced modules. These modules are also used to obtain results on certain annihilator conditions of the (Laurent) polynomial extension and the (Laurent) power series extension of a module. We write MR to mean that M is a right Rmodule. Throughout, a is an endomorphism of R (unless specified otherwise), that is, a: R —> R is a ring homomorphism with a(l) = 1. The set of all endomorphisms (resp. automorphisms) of R is denoted by End(.R) (resp. Aut(JZ)). 1. Reduced modules We start with the following: Definition 1.1. A module MR is called a-reduced if. for any m G M and any aeR, 1. ma = 0 implies mR n Ma = 0. 2. ma = 0 iff ma(a) = 0. The module MR is called reduced if MR is 1-reduced. Lemma 1.2. The following are equivalent for a module MR: 1. MR is a-reduced. 2. The following three conditions hold: For any m 6 M and a £ R, (a) ma = 0 implies mRa = mRa(a) = 0. (b) maa(a) = 0 implies ma = 0. (c) ma 2 = 0 implies ma = 0. Proof. The proof is straightforward.

D

Examples 1.3. Several easy examples of reduced modules can lie given: 1. R is a reduced ring iff RR is a reduced module. 2. Every submodule of a reduced module is reduced. In particular, if I is a right ideal of a reduced ring R, then IR is a reduced module. 3. Letp be a prime number and n > 1. T7i.enpn~1Zpn is a reduced module over 1pn. 4. Every direct product of reduced. R-modules is a reduced R-module. 5. If Mt is a reduced Rt-module for each t € F, then Y[t Mt is a reduced JTt Rfmodule. 6. A module M over TL is reduced iff, for any m 6 M, either m is torsion-free or the order of m is square-free.' In particular, for n > 1, Zn is a reduced module over Z iff n is square-free. Remark 1.4. Let R be a subring of a ring S with lg € R, a & End(S') such that Cf(R) C R and MR C LS- If LS is a-reduced then MR is also a-reduced.

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367

We use R[x], R[[x}], R[x, x x] and R[[x, x 1]] for the polynomial ring, the power series ring, the Laurent polynomial ring and the Laurent power series ring over R, respectively. For a module MR, we consider oo

M[[x;a]l := {Y^ mixi : mi 6 M}, i=0 s

M[x; a] := { Y~] m,x* : s > 0, m,i 6 M}, i=0 oo

M[[x, x"1; a]] := { Y^ mixi : s > 0, rm 6 M} and

M[x, x^-a] := { E "^ : s > 0, t > 0, m; e M}. i=—s

Each of these is an Abelian group under an obvious addition operation. Moreover, M[[x;a]] becomes a module over #[[3;; a]] under the following scalar product operation: For m(x) = £^ mi^ G M[[ox; a}} and f ( x ) = ^ a^x* € R[[x; a}}, (1.1)

m(x)f(x) = k

i+j=k

Similarly, M[x; a] is a module over R[x; a]. The modules M[x; a] and M[[x; a]] are called the skew polynomial extension and the skew power series extension of M respectively. If a G Aut(.R), then with a scalar product similar to (1.1), M[[x, x"1; a]] (resp. M[x,x~i;a]) becomes a module over ^[[rrjX" 1 ;^] (resp. R[x,x~1;a]). The modules M[x,x~1;a] and M[[x, x -1 ;a]] are called the skew Laurent polynomial extension and the skew Laurent power series extension of M, respectively. Next we investigate when the skew (Laurent) polynomial extension and the skew (Laurent) power series extension of M are reduced. Lemma 1.5. Let M be a module such that, for any m € M and a £ R, ma = 0 implies mRa(a) = 0 and ma? = 0 implies ma = 0. If m ( x ) f ( x ) = 0 where m(x) = Efco mixi e M\[x' a}] and f ( x ) = ESo aixi e R\[x'' «]]> then miO'i(ai) = ° f°r al1 i and j . Proof. From m(x)f(x) = 0 we have Ei+j=fc rn « Q;t ( a j) = 0 for fc = 0, 1, . . . . So mooto = 0. Assume that s > 0 and mjO: z (aj) = 0 for all i, j with i + j < s. Note that we have (1.2)

m0as+i + mia(as} -i ----- h msas(ai) + ms+ias+1(a0) = 0.

Multiplying (1.2) by as+l(a0) from the right, one obtains (1.3)

m 0 a, +1 a s+1 (a 0 ) + • • • + msa'(a1Ja'+1(a0) + ms+las+l(a0)as+1(a0) = 0.

By assumption, m^o^ao) = 0 for i = 0, . . . , s, and hence miRas+l(ao) = 0 (for all 0 < i < s). Thus, moas+ias+l(ao) = ••• — msas(ai)as+1(ao) = 0, and so (1.3)

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becomes m 5+ i(a s+1 (a 0 )) 2 — 0. So ms+1as+l(a0) = 0 by assumption. Thus, (1.2) becomes (1.4)

m 0 a s+ i + mia(a s ) H ----- h m s a s (ai) = 0.

Multiplying (1.4) by as(ai) from the right and using the assumption, we have m s (o; s (ai)) 2 = 0, and thus maas(ai) — 0. Continuing this process, we have ms+ias+1 (do) = msas(a-\_) = • • • = moas+i = 0, so we have proved that mial(aj) = 0 for all i, j with i + j < s + 1. By the induction principle,TO,a* (a.,-) = 0 for all i and j. D Theorem 1.6. The following are equivalent for a module MR: 1. MR is a -reduced. 2. M[x; a].R[x;a] is reduced. 3. M [[x;a}]f^[x.a^ is reduced. If a e Aut(.R), then the conditions (l)-(3) are equivalent to each of (4) and (5): 4. M[x,x~^\oi\p^x^x-i.t0^ is reduced. 5. M[[x,x~l\a]}R^x-i.a^ is reduced.

Proof. "(5) => (3) => (2)" and "(5) => (4) =» (2)" are clear by Remark 1.4. "(2) =>• (1)". Clearly, MR is reduced. If ma = 0 where m e M and a £ R, then mJ?[x; a] a = 0 by Lemma 1.2. Thus ma(a)a; = mxa = 0 and so ma(a) = 0. If ma(a) = 0 where m £ M and a £ R, then Tnaa(a) = 0. Thus, max(aa;) = maa(a)a;2 = 0. By (2), max = 0 and so ma = 0. Thus M is ct-reduced. "(1) =>• (3)". Let m(x)f(x) = 0 and m(x)g(x) = l(x)f(x) where m(x) = ESo ^^l- K^) = E~o 10* e M[[a;; a]], and /(i) = E~o ^'. 5(^) l= ESo fci^ e ^2[[o;;a]]. It suffices to show that l(x}f(x) = 0. By Lemma 1.5, mia (aj} = 0 for all i and j, so mi/?a*+s(a.,-) = 0 for all i, j, s by Lemma 1.2. Then

i=0

i=0

i=0

Thus, [l(x)f(x)]f(x) = m(x)g(x)f(x) = 0. By Lemma 1.5, i(a:)[i:~0 0*^(0^2;'] = l(x)(f(x)a,k) = (l(x) f (x))a,k = 0 for fc = 0 , 1 , . . . , and again by Lemma 1.5, li<x*(ajaj(ak)) = 0 for all i,j,k. Thus, lial(ak)al+k(ak) = liai(akak(ak)) — 0. By Lemma 1.2(2)(b), lial(ak) = 0 for a^l i and fc, showing that l(x)f(x) = 0. "(3) ^. (5)". Suppose that m(x)f(x) = 0 and m(x)g(x) = l(x)f(x) where m(x},l(x] e Mf^jX" 1 ;^] and f(x),g(x) 6 ^[[Zja:" 1 ;^]. There exists fc > 0 such that m(a:)a:fc,/(z):r'i: e M[[o;;a]] and f ( x ) x k , g ( x ) x k e E[[i;a]]. Write f(x) =

Reduced Modules

^oo

j

i

/ \

369

\ >oo

7

•»

mi

,i=_fc a » x and SW — E»=_fc OiX . ihen oo

[TO(X)Xk/!

/ >

- fc

= rn(x)/(x)x 2fc = 0,

i=-fc

[ m ( x ) x f c ] ; a-^ L\=_k and both (ES-* Q ~ fc ( a *) a;i )^ fc and (ES-fca"*^)*')** are in #[[*;<*]. By (3), Mz)z fc ][(ES-fca~ fc (&i)z i y c ] =°- Hence m(x)g(x)x2k = 0, and thus m(x)g(x) = 0. D Corollary 1.7. M# is reduced iff M[[x,x~l]]R[[x,x-1]]

is reduced.

Following [8], the ring R is called a-rigid if aa(a) = 0 in R implies a = 0. It can be verified that R is a-rigid iff RR is a-reduced (see [8, Lemma 4]). Corollary 1.8 ([12]). R is a-rigid iff R[x; a] is reduced iff R\[x\ a}} is reduced. We write Mn(R) for the n x n matrix ring over R. For a module MR and A = (a y ) e M n (fl), let MA = {(may) : m £ M}. For n > 2, let V = E^ ^i,i+i where {Eij : 1 < i, j < n} are the matrix units, and set Vn(R) = RIn + RV + +RVn~l and Vn(M) = MIn + MV + • • • + MVn~l. Then vJ^R) is a ring and Vn(M) becomes a right module over Vn(R) under usual addition and multiplication of matrices. There is a ring isomorphism 9 : Vn(R) —> R[x]/(xn) given by 0(r0In + r\V + • • • + rn_iVn-1) = (r0 +nx + ••• + ^.ix""1) + (x™), and an Abelian group isomorphism (f>: Vn(M) —* M\x]/(M\x\(xn}) given by (moln + m\V-\ \mn_iVn-1) = (TOO + mix + • • • + mn^ixn~i) + M[x](xn) such that (f>(WA) = (W)9(A) for all W e Vn(M) and A e Vn(R). Following Anderson and Camillo [1], a module M is called Armendariz if, whenever m(x]f(x) — 0 where m(x) — Ei=o rn » x ' ^ M[x] and /(x) = Ei=o a«:rl ^ J2[x], we have m^a., = 0 for all i and j. By Lemma 1.5, every reduced module is Armendariz. Theorem 1.9. Letn>1 be an integer. Then MR is reduced iff M[x\/M[x](xn) an Armendariz right module over R [ x ] / ( x n ) .

is

Proof. By the remark above, it suffices to show that MR is reduced iff Vn(M)vn(R) is Armendariz. "<=". Let ma = 0, but mr = m'a ^ 0 where m,m' e M and a,r 6 R. Then (m/ n ) - (m'£i,n)a: 6 Vn(M)[x], (aln) + (rEi,n)x e Vn(R)[x], [(m/n) (m'E ljn )x][(a/ n )+(r£; lin )x] = 0 but (m/ n )(r£ lin ) = (mr)Ei,n ^ 0. So ^(Af)^^) is not Armendariz. "=»". Suppose that W(x)A(x) = 0 where W(x) = Ei=0 ^^ € Vn(M)[x] and ^(^) = ELo A*xi e ^n(^)[x]. Write Wi = m^In + m^V +••• + m^V^1 and Ai = a^ In + a{i}V + ••• + a^V71'1 for i = 0 , 1 , . . . , s. It follows from

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T.-K. Lee and Y. Zhou

W(x)A(x) = 0 that [(m0(i)/n + rm(x)V + ••• + mn^(x}Vn-1} • ((a0(x)In + ai(x)V + ••• + an^(x)Vn-1} = 0 where rm(x) = m< 0) + m^x + •••+ mf V andOi(x) = a(°} + a^x + •••+ a^x* for i = 0 , 1 , . . . , n — 1, and hence Ysi+j=k m-i(x)aj(x) = 0 for k = 0 , 1 , . . . , n — 1. Assume that 0 < I < n — 1 and rrii(x)aj(x) = 0 for all i, j with 0 < i + j < I. Note that we have (1.5)

m0(x)ai+i(x) + mi(x)o;(x) H

h mi(x)a1(x) + mi +1 (x)a 0 (x) = 0.

l

Since MR is reduced, M[x,x~ }R\XtX--L} is reduced by Corollary 1.7. Thus, M[x]B[x] is reduced by Remark 1.4 and so (1.6)

mi(x).R[x]a.,-(i) = 0 for all i, j with 0 < i + j < I.

Multiplying (1.5) by aQ(x) from the right, by (1.6), we have m;+1(x)a0(a:)2 — 0. So m; + i(x)a 0 (x) = 0 since Mfa;]^^] is reduced. Thus, (1.5) becomes (1.7)

m0(x)ai+i(x) + mi(x)aj(x) H

h m/(x)ai(x) = 0.

Multiplying (1.7) by ai(x) from the right and using (1.6), we have rn;(:r)ai(a:)2 = 0, and thus mi(x)a\(x) = 0. Continuing this process, we have mijri(x)ao(x) = mi(x)ai(x) = • • • = mQ(x)ai+i(x} = 0. So we have proved that m,i(x)aj(x) = 0 for all i, j with 0 R given by f ( x ) i—> f ( c ) is a ring homomorphism, and hence M becomes a module over R[x] via m o f ( x ) = m/(c). 2. The map 9^f: M[x] —> M given by m(x] \—> m(c) is an JZ[x]-module homomorphism, Proof. The verification is straightforward.

D

For m(x] = m0+ m^x H h msxs e M[x] with ms ^ 0, s is called the degree of m(x) and is denoted by 9(rn,(o;)); if m(x) = 0 we define d(m(x)) = —oo. Remark 1.11. Let .R be a subring of a, ring 5 with lg e Jf? and MR C Lg. If LS is Armendariz then M^j is also Armendariz. Theorem 1.12. MR is Armendariz iff M[x,x~1]mXiX-i]

is Armendariz.

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Proof. If M[x, x I}R[X,X-I] is Armendariz then MR is Armendariz by Remark 1.11. Suppose that MR is Armendariz. Let [m 0 (x)+mi(x)y-| hms(x)ys][f0(x) + l h(x)y + ••• + / t (x)y*] = 0 where m;(x) £ M(x,x~ ] (i = 0,1,... ,s) and fj(x) £ R[x, x"1} (j = 0 , 1 , . . . , t). There exist u > 0 and v > 0 such that m,i(x)xu £ M[x] (for all 0 < i < s) and fj(x)xv £ R[x] (for all 0 < j < t). Then [m0(x)xu + u u s v + fi(x)xvy + ••• + /t(x)xV] = [mo(x) + mi(x)x y + •••+ ms(x)x y ][f0(x)x s + A(x)y + • • - + /t(x)2/t]xu+" = 0. mi(x)y + •••+ m s ( x ) y ] [ f 0 ( x ) u Let n = •m.ax.{d(mi(x)x ),d(fj(x)xv') : i = 0 , . . . , s; j = 0 , . . . ,t}. Since x™ is a central element of R[x], we have by Lemma 1.10(2) that [TOO(X)X U + mi(x]xuxn + ••• + ms(x)xuxsn}[f0(x)xv + fi(x)xvxn -\ 1- ft(x)xvxtn] = 0. Since MR is Aru v mendariz, [mi(x)x ][fj(x)x ] = 0, and hence mj(x)/j(x) = 0 for i = 0 , . . . , s and j = 0 , . . . ,t. So M[x,x~1]R[XtX-i] is Armendariz. D Letting MR = RR yields the following: Corollary 1.13. Let n > 2 be an integer. 1. R is reduced iff R [ [ x , x ~ 1 } } is reduced. 1. [Ij R is reduced iff R[x]/(xn) is Armendariz. 3. [1] R is Armendariz iff R[x,x~*} is Armendariz. It is well-known that every reduced ring is a subdirect product of (not necessarily commutative) domains (see [2]). We conclude this section by proving the following theorem. For a submodule N of MR, the set {a £ R : Ma C N} is denoted by (N : M)R or simply by (N : M). Recall that MR is said to be torsion-free if for every non zero-divisor a £ R and 0 / m G M we have ma j^ 0. Theorem 1.14. Let MR be a non-singular reduced module. Then MR is a subdirect product of modules {Mi : i £ 1} where, for each i £ I, R/(Q : Mi) is a domain and Mi is a torsion-free module over R/(0 : Mi). Proof. First we assume that R is a reduced ring. Let {Pt : t G T} be the set of all minimal prime ideals of R. Then it is well-known that all R/Pt are domains and f| ter P t = 0. For t £ T, let Nt = {m e M : ma = 0 for some a £ R\ Pt}. Since R\Pt is closed under multiplication, Nt is clearly a submodule of M by Lemma 1.2. For any 0 ^ b £ Pt, there exists a £ R \ Pt such that ba = 0 by [2] (or see [12, Lemma 1.3]). So Mba = 0, showing that Mb £ Nt. Thus, Pt C (Nt : M) and so M/Nt is a module over R/Pt. If mf = 0 where m £ M/Nt and f £ R/Pt, then mr £ Nt and so mrat = 0 for some at £ R \ Pt. If r £ R \ Pt then rat £ R\Pt and thus m £ Nt. So M/Nt is a torsion-free module over R/Pt. We now prove that HteT ^* = 0- In f act > if m € flter -^ then, for each t £ T, there exists at £ R\ Pt such that mat = 0. By Lemma 1.2, mRat = 0. Thus, mK = 0 where K = Ystzr RatR- It Kr\aR = 0 where 0 ^ o € R, then a.fiT = 0 and a £ R\Pt for some t 6 T since f\eT-^ = 0- -^ f°ll°ws that 0 = aat £ R\Pt, a contradiction. This shows that K is an essential right, ideal of R. Hence m = 0 since MR is non-singular. So we proved flteT-^t = °> an<^ hence f]{Nt :t£T,Ntj^ M} = 0. Without loss of generality, we can assume that Nt ^ M for all i € T. We prove next that (Nt : M) = Pt for every t £ T. If a e (Nt : M) \ Pt then, for any m£ M,

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ma e Nt and so maat = 0 for some at £ R \ Pt- Thus, m e Nt. This shows that M = Nt, a contradiction. So (Wt : M) C Pt. Since Pt C (Nt : M) as above, (Nt : M) = Pt. Therefore, M is a subdirect product of {M/Nt : t <E T] where, for each t, (0 : M/Nt) = Pt, R/Pt is a domain and M/Nt is torsion-free over R/Pt-_ Let MR be a non-singular reduced module and R = R/(0 : M). Then R is a reduced ring and MR is a non-singular reduced module. As above, M^ is a subdirect product of ^-modules {Mt : < e T} where each R/(0 '. Mt)R is a domain and Mt is torsion-free over R/(0 : Mt)R. Therefore, MR is a subdirect product of fl-modules {Mt : t £ T} where each Mt is torsion-free over R/(0 : Mt)R, and R/(Q : Mt}R ^ R/(0 : Mt)R is a domain. D 2. Baer modules, p.p.-rnodules and quasi-Baer modules The notion of a reduced ring has been used to obtain results on various annihilator conditions, such as the Baer, p.p. and quasi-Baer properties, of the (Laurent) polynomial extensions and the (Laurent) power series extensions of rings (see [3, 4, 5, 7, 8, 9, 10, 11]). In this section, some of the results will be extended from reduced rings to reduced modules. For a subset A* of a module MR, let TR(X) = {r £ R : Xr = 0}. Extending the notions of Baer, quasi-Baer, and p.p.-rings, we define the following: Definition 2.1. Let MR be a module. 1. M is called Baer if, for any subset X of M, rR(X) = eR where e2 = e e R. 2. M is called quasi-Baer if, for any submodule X C M, TR(X) = eR where e2 = e e R. 3. M is called p.p. if, for any m e M, rfl(m) = eR where e2 = e £ R. Clearly, R is a Baer (resp. p.p. or quasi-Baer) ring iff RR is a Baer (resp. p.p. or quasi-Baer) module; if R is a Baer (resp. p.p. or quasi-Baer) ring then, for any right ideal / of R, IR is a Baer (resp. p.p. or quasi-Baer) module. For a subset U of MR and a € End(-R), the set of all skew polynomials (resp. skew power series, skew Laurent polynomials, or skew Laurent power series) with coefficients in U is denoted by U[x; a] (resp. C/[[x;a]], U[x,x~1; a], or Z/[[:E, a:"1;**]]). Lemma 2.2. Let MR be a module and a e End(.R). 1. The following are equivalent: (a) For any V C M[x; a], (TR[x.a](V) n R)[x; a] = rR[l.a](V). (b) For any m(x) = Y.l=omixl £ M [ x ; a ] , f ( x ) - Y
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373

.R)[[x;a]]. Then all ctj & rfl[[x;a]](m(a;)). It follows that midl(aj) = 0 for all z and 3-

"(2) (6) => (2) (a)". It is clear that, for V C M[[x;a]], (r R[[x.a]](V) n R)[[x; a}} C rfl M (n Let f ( x ) = Eto^i e r fl[[i;a]] (K). By (2) (6), ^ = 0 for all j and so a, e rfl[[:r;Q]](V) n fl. Thus, f ( x ) & (rR[[x-a]](V) n #)[[x;a]]. So (rfi[[:E.Q]](F) n D

Lemma 2.3. Lei MR be a module and a £ Aut(JJ). 1. JTie following are equivalent: (a) For any V C M[x,x~l;a], rfl[ X)X -i ;a ](F) = I[x,x~1;a] where I = (b) For any m(x) = £);=_„ TO;!; e M^x"1; a], f(x) = £•=_„ a»Xi 6 .R[x, re"1; a], m(x]f(x] — 0 implies mja l (aj) = 0 /or aZZ i anrf j. 2. The following are equivalent: (a) For any V C M[[x, x~l;a}}, TR([x.x-i;a]](V) = /[[a;,^-1; ajj w/iere I = rR[\x,x-i;a]](V)T\ R. (b) For any m(x) = ^'JI_ u m i x i e M^.a;-1; aj], f(x) = }J*L_V aiXi £ R[[x,x~1;a\], m(x)f(x] = 0 implies miOl(aj) = 0 for all i and j. Proof. Similar to the proof of Lemma 2.2.

D

Definition 2.4. A module MR is called a-Armendariz if the following conditions (1) and (2) are satisfied, and the module MR is called a-Armendariz of power series type if the following conditions (1) and (3) are satisfied: 1. For m € M and a e R, ma = 0 iff ma(a) = 0. 2. For any m(x) = X^=o m i xt e Af[x;a] and /(x) = X^o^1' e -R[x;a], m(x)f(x] = 0 implies mjO: l (aj) = 0 for all i and j. 3. For any m(x) = £^0™^ e M[[x;a]] and /(x) = E^o0^' e ^[[I;Q]], m(x)/(x) = 0 implies mjal(a.,-) = 0 for all i and j. The module M/j is Armendariz iff M# is 1-Armendariz; we call MR Armendariz of power series type if MR is 1-Armendariz of power series type. If MR is areduced then MR is a-Armendariz of power series type (by Lemma 1.5); if MR is a-Armendariz of power series type then MR is a-Armendariz. In general, an a-Armendariz module of power series type is not a-reduced. (For example, for a field F, let R = {(So) : a,° G F}. Then RR is not reduced, but is Armendariz of power series type.) Theorem 2.5. The following statements hold for a module MR:

1. Leta£~End(R). (a) If M[x] a] m.j..Qj is Baer then so is MR', the converse holds if MR is a-Armendariz. , (b) If M[[x; a]]fl[[x;a]] is Baer then so is MR; the converse holds if MR is a-Armendariz of power series type. 2. Let a 6 Aut(fl).

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(a) If M[x,x 1;d\R[XtX-i;a] *s Baer then so is MR; the converse holds if MR is a-Armendariz. (b) If M[[x,x~1;a]\R^XtX-i.a]] is Baer then so is MR; the converse holds if MR is a-Armendariz of power series type. Proof. (l)(a). Let M\x\ a]R[x.a] be Baer. For U C M, rR(U) C rR(U)[x;a] = R[z;a](U) = f ( x ) • R[x;a] where f ( x ) is an idempotent of R[x;a}. Write f ( x ) = Z^*=oa»xi where all a, 6 rR(U). Then, for any a € rR(U), a — f ( x ) g ( x ) for some g(x) G J?[z;a], and so f ( x ) a = f(x)f(x)g(x] = f(x)g(x] = a. It follows that a = a 0 a for all a G r fl (J7). Thus, TR(U) = a0R with 0% = a 0 . So MR is Baer. Let MR be Baer and a-Armendariz. For any V C M[x;a], by Lemma 2.2(1), T R[x;a](V) = r\R = rR(V). Since MR is a-Armendariz, I = TR(CV) where Cy is the set of all coefficients of polynomials in V. Since MR is Baer, rR(Cv) = eR where e2 = e G R. So rR[x.a](V) = I[x;a] = (eR)[x;a] = e • R[x; a], and thus M[x; a]fl[x;a] ig Baer. (!)(&). Similar to the proof of (l)(a). (2). Similar to the proof of (1). D r

Corollary 2.6. Let RR be a-Armendariz of power series type. 1. R is Baer iff R[x\a] is Baer iff R[[x;a}] is Baer. 2. If a £ Aut(J?), then R is Baer iff R[x;a] is Baer iff R[[x;a}} is Baer iff R[x,x~l;a] is Baer iff R[[x, a;"1; a]] is Baer. Corollary 2.7. 1. Let MR be an Armendariz module. Then MR is Baer iff M[x]R[x] is Baer iff M[X,X~I]R[X!X--L} is Baer. 2. Let MR be at} Armendariz module of power series type (in particular, MR is a reduced module). Then MR is Baer iff M[x]R[x] is Baer iff M[[x]]R[[x]] is Baer iff M[x,x~^]R^x^-i^ is Baer iff M[[x,x~ 1 ]] fi [[ a . ?a .-i]] is Baer. The next corollary is obtained in [5] when R is reduced. Corollary 2.8. Let RR be Armendariz of power series type. Then R is Baer iff R[x] is Baer iff R[[x]} is Baer iff R[x,x~l] is Baer iff R[[x, x~1]] is Baer. Lemma 2.9. Let MR be a module such that, for any m e M and a £ R, ma = 0 implies ma(a) — 0. Then ma(e) = me for any m € M and any e2 = e € R. Proof. From me(e— 1) = 0, we have 0 = mea(e-l) = me(a(e) — l ) = mea(e)—me, so mea(e) = me. From m(e — l)e = 0, we have 0 = m(e— l)a(e) = mea(e) — ma(e), so ma(e) = mea(e) = me. D Lemma 2.10. Let MR be an a-Armendariz module. If me = 0 where e2 — e G R and m e M, then mfe = 0 for any /2 = / e R. Proof. Suppose that mfe ^ 0 where / 2 '= / € R. Let l(x) = m(e — 1) — [m(e — l)fe]x 6 M\x\ a] g(x) =e + [(e- l)fe}x G R[x\ a].

and

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Then, by Lemma 2.9, l(x)g(x) = m(e — l)fex — m(e — \)fea(e)x — m(e — l)fea((e — l)fe)x2 = m(e — l)fex — m(e - l)fex - m(e — l ) f e ( e — l)/ex 2 = 0, but m(e — 1) • (e — l)fex = m(e — l)fex = -mfex ^ 0. This shows that MR is not a-Armendariz, a contradiction. D Theorem 2.11. The following statements hold for a module MR: 1. Leta^End(R). (a) If M[x; a]fl[x;a] is p.p. then so is MR; the converse holds if MR is a-Armendariz. (b) If M[[x; a]\R[[x;a]} is p-p. then so is MR. (c) Let MR be a-Armendariz of power series type. Then M[[x; a]]fl[[x;a]] is p.p. iff, for any countable subset X of M, TR(X) = eR where e2 = e e R. 2. Leta^Aut(R). (a) If M[x,x~i;a]p^XtX-i]a] is p.p. then so is MR; the converse holds if MR is a-Armendariz. (b) IfM[[x,x~'L;a]}R[[x.x-i.afl is p.p. then so is MR. (c) Let MR be a-Armendariz of power series type. Then the module M[[x,x~1;a]}R[[x_x-i.a^ is p.p. iff, for any countable subset X of M, rR(X) = eR where e2 — e e R. Proof. (1)(6) and (l)(c). Similar to the proofs of Theorem 2.5(1). Just note that the set of coefficients of a power series of M[[x; a]] is a countable set, and conversely. (l)(a). The proof of first part is similar to that of Theorem 2.5(1). Suppose that MR is p.p. and a-Armendariz. For m(x) & M[x; a], by Lemma 2.2, TR[x-ia](m(x)) = 7[x;a] where I — r^[ a; . a ](m(x)) n R — rR(m(x}). Since MR is a-Armendariz, I = r/j(C m ) where Cm = {rrii : i = 0 , . . . , s} is the set of all coefficients of m(x). Since MR is p.p., rR(Cm) = rii=o r fl( TO i) = C\i=oeiR where e? = ei e R and rR(m,i) = BiR. We prove next that Di=o e^ = £R wnere e2 = e e R. By Lemma 2.10, Ta\e^e\ = 0, and so e$e\ € r^(m,i) = e\R. Thus, e\e,§e\ = e$e\. Let /i = eoei- Then /j2 = fa and e0RneiR = f\R. Again by Lemma 2.10, m 2 /ie 2 = 0. This shows fie2 £ e^R, and hence e2/!e2 = /ie2. Let /2 = fie2. Then /| = /2 and /iR n e^R = f^R. Continuing this process, we obtain f% = fs G R such that nU 6 ^ = f»R- Thus' TR(x;a](m(x)) = I[x;a] = (f,R)[x;a] = f, • R[x\a]. So M[x; a]jj[x;Q] is a p.p.-module. (2). Similar to the proof of (1). D Corollary 2.12. 1. Let MR be an Armendariz module. Then MR is p.p. iff M[x]R[x] is p.p. iff M[x,x-l]R[x,x-^ is p.p. 2. Let MR be an Armendariz module of power series type (in particular, MR is a reduced module). Then M[[x]]R^x^ is p.p. iff M[[x,x~l]}R\\(X^-i^ is p.p. iff, for any countable subset X OM, TR(X) = eR where e2 = e G R. Theorem 2.13. Let MR be a module such that, for any m £ M and a € R, ma = 0 iff ma(a) = 0. Then the following statements hold:

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1. MR is quasi-Baer iff M[x; a]R[x.t0l] is quasi-Baer iff M[[x; a}]R^X.!C^ is quasiBaer. 2. If in addition a 6 Aut(.R), then MR is quasi-Baer iff M[x,x~l; ci\R\XtX-i.a] is quasi-Baer iff M[[x,x~l; a]]fi[[i,x-i;Q]] is quasi-Baer. Proof. (I). We prove that MR is quasi-Baer iff M[[x;a]]#[[x;a]] is quasi-Baer, and it is similar to prove that MR is quasi-Baer iff M[x;a]/j[x;Q] is quasi-Baer. Suppose M[[o;; £*]]#[[.,..£,]] is quasi-Baer. For a submodule U of MR, U[[x;a]} is clearly a submodule of M[[x;a]]fi[[x;a]], so r fl[ [ x;a ]](C/[[x;a]]) = f (x) • R[[x; a]] where /(x)2 = /(x) e R[[x; a}} and thus /(x). R[[x; a}} C rR[[x.a]](U) = rR(U)[[x; a}}. For 9(x} = Ejlo&j'^' 6 rR(U)[[x;a]], Ubj = 0 for all j > 0 and hence Uak(bj) = 0 for all j > 0 and all k > 0 by assumption. For any u(x) = Y^Lo uixl ^ U[[x;a]], u(x)g(x) = EiEj^^+i = 0. So g(x) £ rR[[x.ia]](U[[x;a]}). Thus, rfl(t/)[[x;a]l = /(x) • fl[[z;a]]. Write /(x) = £~oa*xi where all a; e rfi(C7). As in the proof of Theorem 2.5(la), a — a0a for all a 6 r#([/). Thus, r#(J7) = a0R with an = a0. So MR is quasi-Baer. Suppose MR is quasi-Baer. Let V be a submodule of M[[x; a]]jj[[x;a]]. Let U = {u & M : u ^= 0 and there exists some Y^>kaiX* ^ ^ with a^ = w}. Since MR is quasi-Baer, rR(UR) = eR where e2 — e e J?. We claim that r fl[[z;a]](^0 = e • -^[[^iQ1]]- First let m(x) = Y^LomixZ ^ V- ^ m(x}e ¥" 0 then, by Lemma 2.9, m(x)e = Xli^o771'0^6)2-* = ^3i^omJe:rl = 53i>A; T7l«ea;I with 771^6 ^ 0 for some k. Since m(x)e e V, mke € Z7 and so mke = (rn^e)e = 0, a contradiction. Thus, m(x)e = 0 and hence e • R[[x; a]} C rfl[[x;Q]](Vr). Next let /(x) = ^^fcfliX 1 e rR[[ x;Q ]](V) where afc ^ 0. For any I £ C/ and any r e -R, there exists h(x) — ~^i>s hx* 6 V with ^ = L Then h(x)r & V and so 0 = \h(x)r]f(x) = Ei>s ^at(r):r;Z]/(:E) = las('r)as(ak}xs+k + [las(r)as(ak+i) + ls+ias+i(r')as+l(ak)]xs+k+1 + • • •,showing that las(r)as(ak) = 0. Thus, lrak = 0 by assumption. So ak € TR(UR] = eR, implying that ak — eak. Now f(x)—akxk = Si>fc+i aix* e *R[[x;a]](V)- Arguing as above with /(x) - akxk replacing /(x), we have Ofc+i = ea^+i. A simple induction shows that Oj = eaj for all i > k. Therefore, /(x) = e/(x) € /(x) • fl[[x;a]]. So rR[[x.a]](V) = e • R[[x;a}}. (2). Similar to the proof of part (1). D Corollary 2.14. MR is quasi-Baer iff M[x]R[x] is quasi-Baer iff M[[x]]R^x^ is quasi-Baer iff M [ x , x ~ 1 ] R [ X i X - i ] is quasi-Baer iff M[[x,x~l]\R^XjX-i^ is quasi-Baer. Corollary 2.15. Let a g End(B) such that ab = 0 in R iff aa(b] = 0. Then R is quasi-Baer iff R[x; a] is quasi-Baer iff R[[x; a]} is quasi-Baer. If in addition a £ Aut(.R), then R is quasi-Baer iff R[x,x~l;a] is quasi-Baer iff R[\x,x~l; a}] is quasi-Baer. Corollary 2.16 ([5, Theorem 1.8]). R is quasi-Baer iff R[x] is quasi-Baer iff R[[x]] is quasi-Baer iff ^[XjX" 1 ] is quasi-Baer iff J?[[x,x -1 ]] is quasi-Baer .

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Acknowledgements The second author is grateful to the kind hospitality of National Taiwan University when part of the work was done during his visit there. The research is supported by NSERC of Canada and NSC of R.O.C. (Taiwan). References [1] D. D. Anderson and V. Camillo, Armendariz rings and Gaussian rings, Comm. Algebra 26(7)(1998), 2265-2272. [2] V. A. Andrunakievic and Yu. M. Rjabuhin, Rings without nilpotent elements and completely prime ideals, Dokl Akad. Nauk SSSR 180(1968), 9-11. [3] E. P. Armendariz, A note on extensions of Baer and p.p.-rings, J. Austral. Math. Soc. 18(1974), 470-473. [4] G. F. Birkenmeier, J. Y. Kim and J. K. Park, On polynomial extensions of principally quasiBaer rings, Kyungpook Math. J. 40(2000), 247-253. [5] G. F. Birkenmeier, J. Y. Kim and J. K. Park, Polynomial extensions of Baer and quasi-Baer rings, J. Pure Appl Algebra 159(2001), 25-42. [6] J. Han, Y. Hirano and H. Kim, Some results on skew polynomial rings over a reduced rings, International Symposium on Ring Theory (Kyongiu, 1999), 123-129, Trends Math., Birkhauser Boston, Boston, MA, 2001. [7] Y. Hirano, On annihilator ideals of a polynomial ring over a noncommutative ring, J. Pure Appl. Algebra 168(2002), 45-52. [8] C. Y. Hong, N. K. Kim and T. K. Kwak, Ore extensions of Baer and p.p.-rings. J. Pure Appl. Algebra.-L5-L(20QQ), 215-226. [9] S. J0ndrup, p.p. rings and finitely generated flat ideals, Proc. Amer. Math. Soc. 28(1971), 431-435. [10] I. Kaplansky, Rings of Operators, Math. Lecture Note Series, Benjamin, New York, 1965. [11] N. K. Kim and Y. Lee, Armendariz rings and reduced rings, J. Algebra 223(2000), 477-488. [12] J. Krempa, Some examples of reduced rings, Algebra Collog. 4(4)(1996), 289-300. [13] T. K. Lee and T. L. Wong, On Armendariz rings, Houston J. Math., to appear. [14] M. B. Rege and S. Chhawchharia, Armendariz rings, Proc. Japan Acad. Ser. A Math. Sci. 73(1997), 14-17.

The Mori Property in Rings with Zero Divisors Thomas G. Lucas Department of Mathematics, University of North Carolina Charlotte, Charlotte, NC 28223 tglucasSemail.uncc.edu

Abstract. A Mori domain is an integral domain which satisfies the ascending chain condition on divisorial ideals. In this paper a study is begun to extend this property to commutative rings with zero divisors. A Mori ring is defined to be a ring which satisfies the ascending chain condition on regular divisorial ideals. As with Mori domains, this property is shown to be equivalent to having the ring satisfy the descending chain condition on those chains of regular divisorial ideals whose intersection is a regular ideal. A Qo-Mori ring is defined to be a ring which satisfies the ascending chain condition on semiregular divisorial ideals and satisfies the descending chain condition on those chains of semiregular divisorial ideals whose intersection is semiregular. Examples are provided to show that having only one of the two chain conditions is not enough to guarantee the other for semiregular ideals. Throughout the paper R will denote a commutative ring with nonzero identity.

1. Introduction In the last thirty years or so, a great deal has been learned about Mori domains, those integral domains that satisfy the Ascending Chain Condition on divisorial ideals. (A particularly good source for information is [2].) For one it is known that such domains can also be characterized as those that satisfy the Descending Chain Condition when the chains are restricted to being descending chains of divisorial ideals which have nonzero intersection [14, Theoreme I.I]. We shall consider both ACC and DCC on certain types of divisorial ideals in commutative rings with zero divisors. Of the two, ACC is the easier to state, we simply ask that the ring satisfy ACC on a particular type of divisorial ideal. For DCC we will always have some additional restriction placed on the intersection of the ideals in the descending 2000 Mathematics Subject Classification. Primdry 13F05, 13E99. Key words and phrases. Mori rings, Qo-Mori rings, ring of finite fractions.

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chain. When we say that a ring satisfies the restricted DCC on a certain type "X" of divisorial ideal, we will mean that the intersection of the ideals in the chain also is of type "X". For a Mori domain, we would simply say that the domain in question satisfies the restricted DCC on nonzero divisorial ideals, meaning that we only care what happens to a descending chain of divisorial ideals when the intersection of the ideals in the chain is known to be nonzero. We deal with two different types of divisorial ideals in this paper. Namely, regular divisorial ideals and semiregular divisorial ideals where "regular" means the ideal contains an element that is not a zero divisor (i.e., a regular element) and "semiregular" means the ideal contains a finitely generated ideal which has no nonzero annihilators. For a ring R we let T(K) denote the total quotient ring of R and let Qo(R) denote the ring of finite fractions over R. The ring of finite fractions over R can be viewed either as the subring of T(R[x]) (the total quotient ring, of the polynomial ring R[x]) consisting of those elements b(x)/a(x) for which a(x) = ^o^x', b(x) — ^biX1 ^ R[x] with a(x) regular and a,ibj = ajb,- for all i and j, or as the equivalence classes of .R-module homomorphisms on semiregular ideals with two such homomorphisms declared to be equivalent if they agree on some semiregular ideal. The latter method of construction is modeled after the construction Lambek presents for the complete ring of quotients [7, section 2.3]. There are several sources where one can see both constructions presented. In particular, see any of [8], [9] or [10]. Gilmer's book [5] is a good source for basic information about fractional ideals and *-operations on an integral domain. Much as Huckaba extends these notions to R and its total quotient ring [6, Section 20], we will do the same for R and its corresponding ring of finite fractions. First, we say that an .R-module B C Qo(R) is a fractional ideal of R if there is a semiregular ideal I C R such that IB C R. If, in addition, B contains an element that is not a zero divisor and there is a regular element r G R such that rB C R, then it is a regular fractional ideal of R in the usual sense of being an .R-submodule of T(R) which contains an element which is not a zero divisor (see [5, page 25]). For B to be considered a semiregular fractional ideal, B must contain a finitely generated fractional ideal that has no nonzero annihilator and there must be a semiregular ideal I of R such that IB C R. Note that each regular fractional ideal of R is also a semiregular fractional ideal of R and every ideal of R is by default a fractional ideal as well. We let f(R) denote the set of finitely generated semiregular ideals of R. For a semiregular fractional ideal J, let (R : J) = {t e Qo(R) \ tJ C R}. By default, (R : J) is a semiregular fractional ideal of R as it contains a finitely generated semiregular ideal and this finitely generated subideal times any finitely generated semiregular fractional ideal contained in J will be a finitely generated semiregular ideal of R that multiplies (R : J) into R. Moreover, if J is a regular fractional ideal of R, then (R : J) contains a regular element of R and it is easy to show that (R : J) is a regular fractional i'deal as well (Lemma 2.1) so it is contained in T(R). We say that (R : (R : J)) is the divisorial closure of J and that J is divisorial if (R : (R : J)) = J. We employ the standard condensed form Jv to denote (R : (R : J)). Clearly, if J C A with both semiregular fractional ideals,

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then (R : A) C (R : J) and hence Jv C Av. It follows that (R : J) = (R : J)v for each semiregular fractional ideal J. Thus (R : J) is divisorial no matter whether J is divisorial or not. We say that a semiregular fractional ideal I is v-invertible if (I(R '• /))„ — R- A related operation on ideals is the t operation. For a fractional ideal / of an integral domain, its "i" is the ideal It = \J Jv with the union taken over the finitely generated fractional ideals that are contained in I. We shall have need of the operation only for semiregular fractional ideals. As the v operation preserves order and we are taking a union, we can set It = [_) Jv and take the union over those finitely generated semiregular fractional ideals that are contained in the semiregular fractional ideal /. If / = It, then I is said to be a t-ideal. As mentioned at the beginning, an integral domain is said to be a Mori domain if it satisfies ACC on divisorial ideals. We say that a ring R is a Mori ring if it satisfies ACC on regular divisorial ideals. In Theorem 2.22, we show that J? is a Mori ring if and only if it satisfies the restricted DCC on chains of regular divisorial ideals (so only on those chains with regular intersections). On the other hand, the rings in Examples 3.4, 3.5 and 3.6 show that when we deal with semiregular divisorial ideals, it is possible for a ring to satisfy ACC on semiregular divisorial ideals yet have an infinite descending chain of semiregular ideals whose intersection is semiregular, and it is possible for a ring to satisfy the restricted DCC on chains of semiregular divisorial ideals without satisfying ACC on semiregular divisorial ideals. Since both Mori domains and Mori rings have both ACC and some form of DCC, we shall require a Q0-Mori ring to satisfy both ACC on semiregular divisorial ideals and the restricted DCC on semiregular divisorial ideals.

2. Mori Rings and Qo-Mori Rings A useful fact when dealing with the v and t operations in integral domains is that an ideal of the form (a :rj b) (:= {d 6 D db E aD}) is always a divisorial ideal of D. A simple manipulation shows that (a \D b] = (D : (1,6/a)). We will use a similar result in our work. Lemma 2.1. Let J be a semiregular fractional ideal of a ring R. Then (a) (R : J) is a semiregular divisorial fractional ideal of R. In particular, for each nonzero t G Qo(R), the ideal (R : (l,t)) is a semiregular divisorial ideal of R. Moreover, t € R if and only if (R : (!,£)) = R. (b) If J is regular, then (R : J) C T(R) and (R : J) is a regular fractional ideal ofR. Proof. We gave the basic outline of a proof for the first statement in (a) in the introduction. As tl C R for some finitely generated fractional ideal / of R, R + tR is a semiregular fractional ideal that contains 1. Thus (R : (l,t)) is contained in R and it is semiregular and divisorial by the argument given above in the introduction. The third statement is obvious. Assume J is regular. Then some regular element c of R multiplies J into R and there is an element a of J that is not a zero divisor. The element c is in (R : J), this takes care of half of the requirements for (R : J) to be regular. The product of

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a and c is a regular element b 6 J n R that multiplies (R : J) into R. Hence (R : J) is a regular fractional ideal of R. Let t € (R : J) and consider the product tb — r G R. Since 6 is a regular element of .R we have t = r/b 6 T(.R). Thus (R : J) C T(R). This also forces J C T(.R) since J C Jv. D Lemma 2.2. Let I be a semiregular ideal of a ring R. If It = R, then I contains a finitely generated semiregular ideal J for which J~l — R. Proof. Assume It = R. Then we have R = \J{JV J C I and J e F(R)}. Which implies there is a finitely generated semiregular ideal J C. I such that 1 e Jv. The inverse of such an ideal must be R. D For a prime ideal P, we let R(P) = {t e T(.R) | rt £ R for some r £ R\P} denote the so-called large quotient ring of R with respect to P (see, for example, [5, Exercise 16, page 49] or [6, page 28]). For an ideal / of R we let <J)(p) = {t & R(P) | rt G / for some r € R \ P}. In general, (-O(p) contains IR^p), the simple ideal extension of / to R(p), but the two ideals need not be equal. In a similar fashion we may form a quotient ring based on P inside the ring of finite fractions, we let R(p) = {q & Qo(R) rq e R for some r € R \ P} and refer to this ring as the QQ- quotient ring of R with respect to P. For an ideal / we again have a "large" extension {/) ^ — {q € R(P) \ rq £ I for some r G R \ P} and the simple extension IR(M)- Note that for an element t e R \ P, both large extensions (tR)^ and (tPt)(p) blow up to R(P) and R^p), respectively, since t • 1 6 tR puts 1 in each large extension. Hence (/)(p) = -R(P) and (-O(P) = R(P) for each ideal / not contained in P. On the other hand, one or both simple extensions of / may survive. Survival can occur even if / properly contains P. In the next lemma we show that if R has a proper t-ideal, then maximal tideals exist and each such ideal is prime. We also show that each proper t-ideal is contained in a maximal t-ideal. Several other t related properties are established as well. All are extensions of general facts about ideals in integral domains. Lemma 2.3. Let R be a ring which contains at least one proper t-ideal. Then: (a) Maximal t-ideals exist and each of these is prime. (b) Each proper t-ideal is contained in a maximal t-ideal. (c) // A and B are semiregular fractional ideals, then (AB)V — (ABV)V = (AVBV)V. (d) /// and J are semiregular fractional ideals, then (U)t = (Ut)t = (ItJt)t(e) If J is a proper t-ideal and P is a prime minimal over J, then P is a t-prime.

(f) R= n

PgtMax(fl)

R(P)-

Proof. To prove (a) and (b), we will first show that Zorn's Lemma applies to the set of proper i-ideals of R. To this end',, let {JQ} be a chain of proper ^-ideals of R. We must show that J = |J Ja is a proper t-ideal. Let / be a finitely generated semiregular ideal contained in J. Then some Ja contains / and, since Ja is a tideal, it also contains Iv. Hence Iv C J and J is a proper t-ideal. Now apply Zorn's

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Lemma to get maximal t-ideals. We also have that each proper t-ideal is contained in a maximal t-ideal. It remains to show that each maximal t-ideal is prime. Let P be a maximal t-ideal and let r £ R \ P. We will show that there is a finitely generated semiregular ideal / C P such that (r.7?+/)~1 = R. If no such ideal exists, then for each finitely generated semiregular I C P, we have (rR + I)v ^ R. Set Q = \J{(rR + /)„ | / C P and / e F(R)}. Since (rR + /)„ contains /„, Q properly contains P and Q is a proper t-ideal of R, a contradiction. Hence there is a finitely generated semiregular ideal I C. P such that (rR +1)"1 = R. Now let ab e P. By way of contradiction, assume neither a nor 6 is in P. By the above, there must be a finitely generated semiregular ideal I Q P such that both (aR+I}~[ and (bR+I)'1 are equal to R. But for each finitely generated semiregular ideal J C P, (abR + J)'1 ^ R. Thus there is an element s e (dbR + I)~l \R. Since (aR + I)~l = (bR + I)'1 = R, we have a contradiction: sa 6 (bR + I)~l = R and sb e (aR + I)'1 = R implies s 6 (bR + I)-1 = (aR + I)*1 = R. Proofs of statements (c) and (d) follow the same outline as in the related statements for integral domain. The only difference is we must use semiregular ideals instead of simply nonzero ones. For those unfamiliar with the proofs for integral domains we provide a brief sketch. First we show that for semiregular fractional ideals A and B, (AB)V = (ABV)V = (AVBV)V. For this it suffices to show (R : AB) = (R : ABV). Since AB C ABV, we have (R : ABV) C (R : AB). For the reverse containment, consider t & (R : AB). Then tAB C PL. Hence tAC(R:B) = (R: Bv). Thus tABv C R and the equalities follow. For (d), let / and J be a pair of semiregular fractional ideals of R and let C be a finitely generated semiregular fractional ideal contained in ItJt- Then there are finitely generated semiregular fractional ideals A C It and B C Jt such that C C AB. Since the v operation preserves containment, there are finitely generated semiregular fractional ideals A' C. I and B' C. J such that A C A'v and B C B'v. Hence Cv C (AB)V C (A'VB'V)V = (A'B')V C (IJ)t. The conclusion follows from the fact that the t operation preserves order. The statement in (e) is a direct consequence of (d). Let J be a proper t-ideal of R and let P be a prime minimal over J. Let / be a finitely generated semiregular ideal which is contained in P. Since P is minimal over J, there is a positive integer n and an element r € R \ P such that rln C J. Thus (rR + J) is a semiregular ideal that is not contained in P but does multiply In into J. By (d), taking the "t" of both sides of (rR + J)In C J yields (rR + J)t(In)t C Jt = J C P. Since rR + J is not contained in P, (In)t must be contained in P. The result now follows from (d) since we have (/„)» = (/ t ) n C ((7t)")t = (In)t. For (f), let t€Q0(R)\R and let J = (R : ( I , t ) ) . By Lemma 2.1, J is a proper semiregular divisorial ideal of R. Thus J = Jt and some maximal t-ideal must contain J. If M is such an ideal, then it is easy to see that t cannot be a member of R(M). It follows that R = f| P.(P>. D P6tMax(.R)

Later we extend the statement in (f) to semiregular divisorial ideals. We also prove a similar statement about regular divisorial ideals.

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Lemma 2.4. Let {Jn} be a nonempty collection of semiregular divisorial fractional ideals of a ring R. If f) Jn contains a semiregular fractional ideal of R, then |~| Jn is a semiregular divisorial fractional ideal. Proof. Set / = P| Jn. That / is semiregular follows from the fact that it contains a semiregular fractional ideal. As it is contained in each Jn, it is a fractional ideal as well. Note that (R : Jn) C (R : 7) for each n. Hence Iv C (J n ) v = Jn and therefore / = /„. D Theorem 2.5 (cf. [12, Theoreme 1]). The following are equivalent for a ring R. (1) R satisfies ACC on semiregular divisorial ideals. (2) For each ascending chain of finitely generated semiregular ideals {Im}, there is an ideal In in the chain such that (/„)„ = (/&)„ for each k > n. (3) For each semiregular ideal I there is a finitely generated semiregular ideal J C / such thai Jv = Iv. Proof, The equivalence of (1), (2) and (3) is fairly straightforward, and the standard proof for the same equivalences for integral domains carries over with appropriate addition of the semiregularity of the ideals under discussion. That (1) implies (2) is clear. Assume the statement in (3) is false for some semiregular ideal / and let Ji be a finitely generated semiregular ideal that is contained in I. Then we have that (Ji),, is properly contained in Iv. Hence there is an element 03 € / such that •^2 = J\ + a^R is a finitely generated semiregular ideal whose v properly contains (J\}v. As with Ji, (Jz)v is properly contained in /„. As we have assumed Iv is not the v of a finitely generated semiregular ideal, we may continue this process and construct an ascending chain of finitely generated ideals which violates the statement in (2). Hence (2) implies (3). Now assume the statement in (3) holds and let {/«} be an ascending chain of semiregular divisorial ideals. Let / = |J /„. Then by (3) there is a finitely generated semiregular ideal A C. I such that Av = Iv. As I is the union of the chain {In}, there is an integer m such that A C Im. But as Im is divisorial, we must have Iv = Av C Im C /. Hence the chain {In} stabilizes. D Corollary 2.6. Let R be a ring which satisfies ACC on semiregular divisorial ideals. Then It = Iv for each semiregular ideal. In particular, each semiregular maximal t-ideal is divisorial. Proof. Let / be a semiregular ideal. Then there is a finitely generated semireguiar ideal A C / for which Av = Iv. It follows that It = Iv. Hence each semiregular t-ideal is divisorial, no matter whether it is maximal or not. D We can make a somewhat similar characterization of the restricted DCC on semiregular divisorial ideals. Theorem 2.7 (cf. [14, Theoreme I.I]). The following are equivalent for a ring R.

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(1) R satisfies the restricted DCC on semiregular divisorial ideals. (2) If J is a semiregular fractional ideal that contains R, then there is a finitely generated semiregular fractional ideal B contained in J such that Bv = Jv. Proof. Assume R satisfies the restricted DCC on semiregular divisorial ideals. Let J 2 R be a semiregular fractional ideal. Recall that this means there is a semiregular ideal I of R such that IJ C R. Let R = Jo C J^ c J% C • • • be a chain of finitely generated semiregular fractional ideals with each Jj contained in J. Now consider the semiregular ideals (R : J) C • • • (R : J2) C (R : J:) C (R : J0) = R. Each of these ideals is divisorial. Since R satisfies the restricted DCC on semiregular divisorial ideals, there is an integer n such that (R : J/.) = (R : Jn) for each k > n. It follows that (Jk)v = (Jn)v for each k > n. Thus there must be a finitely generated semiregular fractional ideal B C J such that Bv — Jv. Now assume that each semiregular fractional ideal J that contains R also contains a finitely generated semiregular fractional ideal B such that Bv = Jv. Let /i 3 /2 2 ^s 2 • • • be a descending chain of semiregular divisorial ideals such that P| Ik = I is semiregular. Consider the semiregular fractional divisorial ideals (R : In). Recursively select finitely generated semiregular fractional ideals BnC(R: In) such that (£„)„ = (R : /„) and Bn D B n _i. Set B = \JBn. Since (R : /„) C (R : I) for each n, B C (R : /). Thus B is a semiregular fractional ideal. By our assumption, there is a finitely generated fractional ideal C C B such that Cv = Bv. But by the construction of B as the union of an ascending chain of finitely generated semiregular ideals, C must be contained in some Bn. Thus Bv — (Bn}v = (R : In}. Since each of the ideals 1^ is divisorial, the descending chain I\ D J2 2 • • • must stabilize at /„. Therefore, R satisfies the restricted DCC on semiregular divisorial ideals. D Remark. With regard to chains of regular divisorial, it is quite easy to show that the three statements in Theorem 2.5 are equivalent when restricted to regular ideals as are the two statements in Theorem 2.7. For each proof, simply take the corresponding proof given above and replace "semiregular" with "regular". In fact all five statements are equivalent when one deals exclusively with regular ideals and regular fractional ideals (Theorem 2.22). It bears repeating that Examples 3.4, 3.5 and 3.6 below show that Theorems 2.5 and 2.7 are separate results, a ring may satisfy the equivalent statements of one and not the other. Lemma 2.8. Let J C. B be semiregular fractional ideals of a ring R and let I be a fractional v-invertible ideal of R. (a) I f ( I J ) v = (IB)v, thenJv = Bv. (b) // (I(R : J))v = (I(R : B))v, then Jv = Bv. Proof. By Lemma 2.3, (IJ)V = (IJV)V = (IBV)V(IB)V. Multiply by (R : I) to obtain (IJV)V(R : I) = (IBV)(R : /) arid then calculate the divisorial closure. Again making use of Lemma 2.3, we obtain [(IJV)(R : /)]„ = [(I(R : I))VJV]V = [(I(R : I))VBV}V = \(IBV)(R : /)]„. But since J is v-invertible, (I(R : I)}v = R and therefore the two middle expressions reduce to simply Jv = Bv.

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Both (R : J) and (R : B] are semiregular divisorial fractional ideals, so by part (a) we have (R : J) = (R : B). The conclusion follows as Jv = (R : (R : J)) = (R : (R : 5)) = Bv. H If D is a Mori domain, then each nonzero element is contained in at most finitely many maximal t-ideals and, for each maximal i-ideal M, DM is a Mori domain. The converse holds as well (see [3, Theoreme 2.3] or [4, Proposition 3.4]). In some sense, what helps make this equivalence is the fact that any localization of D at a prime ideal yields a ring with the same quotient field as D. The equivalent statements involving either the total quotient ring or the ring of finite fractions is rare for a ring with zero divisors. Instead we rely on the large quotient rings and Qo-quotient rings defined earlier. A more general quotient ring can be obtained using a multiplicatively closed set of ideals. Let Ti. = {Ia} be a nonempty collection of semiregular ideals of R. As in [1] (or many other places) , we say that H is a multiplicative system of ideals if for each pair Ia, If) in H, Ialp is in H. We may further assume that H is complete in the sense that if an ideal / contains a member of H, then / is in H, but this is not necessary. We allow Ji to be the trivial set {R}. For such a system of ideals, we let RH = {t € Qo(R) \ tla C R for some Ia e H}. It is clear that RH = \J (R : Ia}- For an ideal J of R, we let Ia€H

JH = {q € RH | qla Q J for some Ia € Ji} and refer to JH as the W-extension of J. And for an ideal B of RH, we let B° = B f| RWe may form two other extensions of R based on "H. We let Ti.f = {Ia £ "H Ia contains a finitely generated ideal Ip £ T~i} and li.T = {Jp \ Jp is regular and contains some Ia € 7^}. Both sets are closed to finite products, and if H. is complete, then both Hf and Hr are complete. In general, we have R-^r C RH} C RH. Note that Rnr is always contained in the total quotient ring of R. We first show that both R(p) and R(p) are just special types of rings of the form Rnr and RH, respectively. Note that while the rings are the same, the large extension of an ideal as defined for R(p) and R(p) need not be the same as the Hrextension and Ti-extension, respectively. For example as we saw above, if we choose an element t £ R\P, then its large extensions (t R) (p) and (tR) ^ blow up to R(p) and R(p), respectively. If t is a regular element, then 1/i is in both R(p) and R(p) so that the simple extensions of t blow up as well. But if t is a zero divisor, no semiregular ideal can multiply 1 into tR. Thus neither [tR]-nr nor [tR]n contains 1. On the other hand, the large extension in R(P) and the Hr-extension will coincide for each regular ideal and the large extension in R^p) and the W-extension will coincide for each semiregular ideal. Lemma 2.9. Let P be a prime ideal of a ring R. Then R{p) = RH and -^(P) = RHT where Ji is the multiplicative system of semiregular ideals of R that are not contained in P. Moreover for each semiregular ideal I and regular ideal J, (I)(p) = IH Proof. Since P is prime, a finite product of ideals is contained in P if and only if at least one of the ideals is contained in P. Thus Ti. is closed to finite products and

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therefore it forms a multiplicative system of semiregular ideals. Both it and Hr are nonempty since each includes R. Since no ideal of H is contained in P, tla C R implies t e R^ when Ia e H is semiregular and it implies t £ R(P) when Ia e Hr is regular, the latter since RH. c TOR). Let t 6 R(p). Then t e Qo(-R) so there is a semiregular ideal J such that t J C /?. We also have an element r £ R\ P such that rt G .R. Hence the ideal rR + J is in Ji and we have t e /?-^. Thus R(P) = RT-CNow let t £ R(p). Then t & T(R) so there is a regular element r e R and an element s £ R such that t = s/r. We also have an element b £ R\ P such that 6i G .R. Thus t(r, b) C R for the regular ideal (r.6) which is not contained in P. So -R(P) = -R«,.Let / be a semiregular ideal and let t e .R
^.

n

Lemma 2.10. Let I be a finitely generated semiregular ideal of a ring R and let "H be a multiplicative system of semiregular ideals of R. Then (!H)V — ((Iv}n}vProof. Obviously (!-H)V C ((Iv)u)v So it suffices to show (Ru '• /W)(^) / H Q RT-CAssume / = (61, b^,..., bn) and let t € (Ru • IH) and a 6 (IV)~H- Thus, tbi € RK for each i and there is an ideal Ja e Ti. such that aJa C /„. Since / is finitely generated and H is closed to finite products, there is an ideal J^ € Ji such that i6i Jp C /J for each i. Thus t J/3 C (R: I) and we have taJaJp 6 /J. But Ja J/3 £ 7i, so ta € .R-ft as desired. D It is now time to consider the Mori property in rings of the form R-^. Querre proved that if D is a Mori domain, then DH is a Mori domain for each multiplicative system of nonzero ideals [12, Theoreme 2.2]. The next three results give ways to extend Querre's result to rings with zero divisors. In the first, we consider only ACC on semiregular ideals. Next we do Qo-Mori rings, and in the third we do Mori rings. Theorem 2.11. Let R be a ring which satisfies ACC on semiregular divisorial ideals and let H. be a multiplicative system of semiregular ideals. Then (1) There is a multiplicative system y = Qf such that Rg = RK and each ideal ofH contains a finitely generated semiregular ideal that is contained in Q. (2) RT-I satisfies A CC on semiregular divisorial ideals.

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A crucial step in this proof is that each semiregular ideal of R contains a finitely generated semiregular ideal with the same divisorial closure. We have not been able to prove the related statement about DCC without assuming the ring R is (Q Proof. It does no harm to assume Ji is complete and we shall do so. The first statement is a consequence of R satisfying ACC on semiregular divisorial ideals. For each Ia G Ti., let Ja be a finitely generated semiregular ideal contained in /„ such that (JQ)v = (Ia)v Let J denote the set consisting of all finite products of Jas. Now let Q = {/ | / D J some J € J}. Obviously, Q — Qf and each ideal in "H contains a finitely generated one in Q. Moreover, by our construction of Q and the fact that (AB)V = (AVBV)V, we have (J (R : Bp) = (J (R : /Q); i.e.,

Rgf = Rg = RHFor the rest of the proof we may assume H = Q . To show that R-H satisfies ACC on semiregular divisorial ideals, it suffices to show that each semiregular divisorial ideal of R-H is the extension of a semiregular divisorial ideal of R. Let J be a semiregular divisorial ideal of RH and let A — J \~] R. We first show that A is semiregular. Let B = (&i, b^ . . . , bn) be a finitely generated semiregular ideal of R-H that is contained in J. Then for each i there is a semiregular ideal Iai G % sucn that bila. C R. Since H is a multiplicative system of semiregular ideals, / = H Iai is semiregular and in "H. Thus BI C _ R p ) J = A i s a semiregular ideal. Next we show that A is a divisorial ideal of R. Since R is Q0-Mori there is a finitely generated semiregular ideal of (GI, c2, . . . , Cn) = C C A such that Cv = Av. Let u e (R-H : J). T,hen uci e R-H for each i. Since H is closed to finite products and C is finitely generated, there is an ideal Jp e H such that uCi Jp C R for each i. Thus uJf} C (R : C) = (R : Av}. But this implies uAv C RH. Hence Av C J and we have A = A^ is a divisorial ideal of J?. To complete the proof we show that J — A-H. Let j e J. Then there is an ideal Ia € T~i such that j/Q C R. So j/ Q C A and we have J C AH- Now assume r 6 A-J-I. Then there is an ideal Ip £ H. such that rip C A. Thus we at least have rip C J. Let v € (.R^ : J). Then vrlp C RH. Therefore vr £ ^?H and we have r 6 Jv = J as desired. D Theorem 2.12. Let R be a Qo-Mori ring and let H be a multiplicative system of semiregular ideals. Then RT-I is a Qo-Mori ring. Proof. By the proof above, each semiregular divisorial ideal of R^ contracts to and is the extension of some semiregular divisorial ideal of R. Thus if {Jn} is a descending chain of semiregular divisorial ideals of R-H with f| Jn semiregular, then { Jn n R} is a descending chain of semiregular divisorial ideals of R with f](Jn n -R) semiregular. Thus { Jn n R} stabilizes1 and this implies that the original chain stabilizes as well. Hence R-H is a Qo-Mori ring. D We obtain a similar result for systems of regular ideals and Mori rings.

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Theorem 2.13. Let "H be a multiplicative system of regular ideals of a ring R. If R is a Mori ring, then RK is a Mori ring. Proof. Let J be a regular divisorial ideal of R. It suffices to show that (Jc)n — J • Let t £ J. Then there is a regular ideal Ia € H such that tla C R. So tla C Jc and we have that J° is regular. This also shows that J — (JC)'H.. Since R is Mori, there is a finitely generated regular ideal C = (GI, 02, . . . , cn) of R such that Cv = (J°)v (see the remark following the proof of Theorem 2.7). Let u € (RH '• J}- Then uci £ RU for each i. It follows that there is regular ideal Ip &H such that ucjp C R. Hence ulp C (R : C} = (R : ( J c ) v ) ) . Thus we also have u( J°)v C R-^. As J is divisorial, we have (J°)v Q J • Therefore Jc is a divisorial ideal of R with J = (J°)-Hd An important corollary to these last three theorems is the following. Corollary 2.14. Let P be a prime ideal of a ring R. (a) If R satisfies ACC on semiregular divisorial ideals, then R(p) also satisfies ACC on semiregular divisorial ideals. (b) If R is a Qo-Mori ring, then R(p) is a Q^-Mori ring. (c) If R is a Mori ring, then R^ is a Mori ring. Here is the result promised above extending statement (f) of Lemma 2.3 to divisorial ideals. Lemma 2.15. Let I be semiregular ideal of a ring R. (a) If I is divisorial, then I = r\{((I)(M)}v \ M £ tMax(jR)}. (b) /// is finitely generated, then Iv = [}{((!) (M))v \ M £ tMax(.R)}. Proof. To prove (a), assume / is divisorial and let Iv = r\{((I)(M)}v | M £ tMax(.R)}. Let b £ /" and let s £ (R : I). It suffices to show that 6s £ R. Since R = (~]{R(M) I M £ tMax(.R)}, all we need show is that s € (R(M) '• (-O(M)) for each M € tMax(.R). Let M be a maximal i-ideal and let r £ (I)(M)- Then there is an element t £ R \ M such that tr £ /. Hence str £ R and we have sr £ -R(M) as desired. For (b), assume J is finitely generated (and no longer assume that it is divisorial). By (a), we have Iv = f \ { ( ( I v ) (M))v M £ tMax(fi)}. But by Lemmas 2.9 and 2.10, we have ({/){M))« — ({-M(M))u ^or eac" maximal t-ideal M. Hence ~ M£tMax(/?)}. D Next is a technical lemma that helps when dealing with Mori properties and rings of quotients. Lemma 2.16. Let M be a maximal t-ideal of a ring R and let I be a semiregular ideal of R that is contained in M. If there is a finitely generated semiregular ideal B C (/)(M) such that Bv = ((-f)(M))uj then there is a finitely generated semiregular ideal AC I such that ((A)^M-))V = Proof. Let B = (61, 62, . . . , bm) C (I)(M) be a semiregular ideal of R^M) with the property that Bv = ({I)(M))v- F°r each b, there is finitely generated semiregular

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ideal J; such that 6jJj C R and Jj is not contained in M. Since B C (/){M}> fejJi G / and from the fact that B is finitely generated we have that the ideal J = n J, is finitely generated, semiregular and not contained in M. The ideal A = hi J+&2 J+' • • bm J is also finitely generated and semiregular but it is contained in/. Since (A)(M) — {t e R(M) \tr C A for some r e R \ M}, (A}(M) contains B. n Itfollowsthat ({^(M)),, = ((I)(M))v Both Raillard and Querre proved that if D is an integral domain which can be realized as an intersection of Mori domains |"| Dj such that each nonzero element of D is & unit in all but finitely many of the DiS, then D is & Mori domain [14, Theoreme 1.2] and [13, Theoreme 2.1]. Such an intersection is said to have finite character. Obviously, a regular element of a ring R can be a unit in some overring, but a zero divisor cannot. For Mori rings, restricting our concerns to only the regular elements seems reasonable. On the other hand, for a Qo-Mori ring it seems likely that we will have to develop some notion of a finite character property for semiregular ideals. What we shall use with regard to Qo-quotient rings is to ask that each semiregular ideal be contained in at most finitely many maximal t-ideals. Theorem 2.17. Let R be a ring for which each semiregular ideal is contained in at most finitely many maximal t-ideals. Then: (a) If for each maximal t-ideal M, R(M) satisfies ACC for semiregular divisorial ideals, then R satisfies ACC for semiregular divisorial ideals. (b) If for each maximal t-ideal M, R(M) satisfies the restricted DCC for semiregular divisorial ideals, then R satisfies the restricted DCC for semiregular divisorial ideals. Proof. For (a), note that Theorem 2.5 allows us to simply show that for each semiregular ideal / there is a finitely generated semiregular ideal A C. I such that Av = lv. It does no harm to assume It ^= R for otherwise there is a finitely generated semiregular ideal A C. I such that Av — R. In this case we have Av = Iv = R. We will not assume Iv ^ R. Let MI, MZ, . . . , Mn be the maximal t-ideals that contain /. By Lemma 2.16, there is for each i a finitely generated semiregular ideal Ai C I such that ((Ai)(Mi))v = ((I)(Mi)}v As long as we stay in /, it does no harm to make Ai larger, so we set A = £) At. Now we have ( ( A ) ( M . ) ) V = ((I)(Mi))v for each i. There are at most finitely many other maximal t-ideals that contain A. For each such maximal t-ideal N j , we may select an element Cj e / \ Nj. Set C = A + ^CjR. Then A C C C /. so we have ({C)(Mt))v = ((I)(Mi))v- Furthermore, as with /, MI, M.2, . . • , Mn are the only maximal t-ideals that contain C. Hence for M € tMax(#) \ {Mi,M2, . . . ,Mn}, R(M} = ((C)(M})V = ( ( I ) ( M } ) V . Since a semiregular divisorial ideal is completely determined "by the divisorial closures of its extensions to the rings R(M) (Lemma 2.15), we have Cv = Iv as desired. For (b) we assume {/„} is a descending chain of semiregular divisorial ideals such that f}In is semiregular. By Lemma 2.4, we have that J = P| /„ is divisorial.

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Thus / is contained in only finitely many maximal t-ideals. As above let MI, M%, ..., Mn denote these ideals. In each ring R(M,) there is an integer rrij such that ((Imi)(M))v = ((Ik)(M))v for each k > m*. Let m = max{mi,m2,... ,mn}. For all other maximal i-ideals N we have ((Im)(N))v = ((I)(N))v = R(N)- Thus since / and each Ik are divisorial, Im — I by Lemma 2.15. D Theorem 2.18. If R is a Qo-Mori ring, then each semiregular divisorial ideal is contained in only finitely many divisorial prime ideals. In particular, each semiregular ideal is contained in at most finitely many maximal t-ideals. Proof. Let / be a semiregular divisorial ideal of R and let {Pa} denote the prime divisorial ideals that contain /. By Lemma 2.4, the ideal (~}Pa is divisorial. Assume the set is infinite and select a countably infinite subset {Pn}- Since R is Qo-Mori, this subset contains no infinite chains. Thus we may assume the Pns were selected in such a way that there are no containment relations among them. But by Lemma 2.4 and prime avoidance, we would then have that the ideals Im = Pit=i Pi are a^ distinct divisorial ideals. Thus the chain \Im} would be an infinite descending chain of semiregular divisorial ideals whose intersection is semiregular and divisorial, contradicting the assumption that R is Qo-Mori. The second statement follows from the fact that ACC on semiregular divisorial ideals implies It = Iv for each semiregular ideal /. D Theorem 2.19. The following are equivalent for a ring R. (1) R is a Qo-Mori ring. (2) Each semiregular ideal is contained in at most finitely many maximal t-ideals and for each maximal t-ideal M, R^M) *s a Qo-Mori ring. (3) Each semiregular divisorial ideal is contained in at most finitely many maximal t-ideals and for each maximal t-ideal M, R(M) is a Qo-Mori ring. (4) Each semiregular fractional ideal that is comparable with R contains a finitely generated semiregular ideal with the same divisorial closure. Proof. Obviously, (2) implies (3) and the previous theorem shows that (3) implies (1). The equivalence of (1) and (4) follows from simply combining Theorems 2.5 and 2.7. We will finish the proof by showing (1) implies (2). Assume R is a Qo-Mori ring. Then by Corollary 2.14, R(M) is a Qo-Mori ring for each maximal t-ideal M. We also have that each t-ideal is divisorial since each semireguiar ideal contains a finitely generated semiregular ideal with the same divisorial closure. In particular, each maximal t-ideal is divisorial. Let / be a semiregular ideal of R and let -M(J) denote the set of maximal t-ideals that contain I. Each M in M(I) is divisorial and the intersection of all of them contains /. Thus we may construct a descending chain MI D MI n My D MinM^My, 2 ••• of divisorial ideals whose intersection is semiregular. This chain must stabilize since R satisfies the restricted DCC on semiregular divisorial ideals. Hence M(I) must be a finite set. This establishes (1) implies (2) and (3). D

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With the addition of an assumption that each semiregular divisorial ideal contain a u-invertible ideal, we can separate the statements about ACC and the restricted DCC. Also we can add that each semiregular fractional ideal contains a finitely generated semiregular fractional ideal with the same divisorial closure, no matter whether the ideal compares with R or not. Theorem 2.20. Let R be a ring for which each semiregular divisorial ideal contains a v-invertible ideal. Then the following are equivalent for R. (1) R is a Qo-Mori ring. (2) R satisfies ACC on semiregular divisorial ideals. (3) R satisfies the restricted DCC on semiregular divisorial ideals. (4) For each semiregular fractional ideal J there is a finitely generated semiregular fractional ideal B C J such that Bv — Jv. (5) For each semiregular ideal I, there is a finitely generated semiregular ideal A C I such that Av = Iv. (6) Each semiregular ideal is contained in at most finitely many maximal tideals and for each maximal t-ideal M, R(M) satisfies ACC on semiregular divisorial ideals. (7) Each semiregular ideal is contained in at most finitely many maximal tideals and for each maximal t-ideal M, R(M) satisfies the restricted DCC on semiregular divisorial ideals. Proof. We have already established the equivalence of (2) and (5) in Theorem 2.5. Theorem 2.7 can be used to show (4) implies (3). Thus Theorem 2.19, is enough to declare that (1) implies all of the other statements except (4). As (4) implies (5), we have that (4) implies (1). By Theorem 2.17, we have that (6) implies (2) and that (7) implies (3). Note that none of these implications requires us to invoke the hypothesis that each semiregular ideals contains a w-invertible ideal. To complete the proof it suffices to show that (5) implies (4) and that (3) implies (2). We start with the proof that (5) implies (4). Here we will use the assumption that each semiregular divisorial ideal contains a v-invertible ideal. We also use this assumption in the proof of (3) implies (2). Let J be a semiregular fractional ideal of R and assume that each semiregular ideal R contains a finitely generated semiregular ideal with the same divisorial closure. Then there is a semiregular ideal I of R such that IJ C R and we may I is u-invertible, since any ideal contained in / would also multiply J into R. Now construct a chain of finitely generated semiregular fractional ideals {Bn} based on the following recursive condition: First choose some finitely generated semiregular fractional ideal B± C J. Then for n > 1, if (Bn}v is properly contained in Jv, choose a 6n+i G J \ (Bn)v and set Bn+i = Bn + bn+iR. Otherwise, stop at n as we have (Bn)v — Jv as desired. Let B = (J Bn and consider the ideal IB. This is a semiregular ideal of R. Thus there is a finitely generated ideal A C IB such that Av = (IB)V. Since A is finitely generated and B is a union of comparable ideals, there is an integer m such that A C IBm. Hence we have Av = (IBm)v — (IB)V. Since / is v-invertible, we have (Bm)v = Bv and therefore our recursive construction stopped in finitely many steps so (Bm)v = Jv.

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Next we show (3) implies (2). So assume R satisfies the restricted DCC on semiregular divisorial ideals and let {In} be an ascending chain of semiregular divisorial ideals of R. Let / be a v-invertible ideal that is contained in /i. Then the set of inverses {(R : /„)} forms a descending chain of fractional semiregular ideals and multiplying all of them by / gives a descending chain of semiregular ideals {I(R : In)}- To complete the proof, first apply the u-operation to the ideals in this chain. This yields a descending chain of semiregular divisorial ideals {(I(R '• In))v}Note that as each (R : In) contains R, I is contained in each of the members of this chain. Thus this chain stabilizes and we have that there is an integer n such that (I(R : h))v = (I(R '• In))v for each k > n. By Lemma 2.8, we have (R : Ik) = (R : In) for each k > n. As each of the ideals Im is divisorial we have that the chain stabilizes at /„. D Corollary 2.21. Let R be a Qo-Mori ring. If each semiregular ideal contains a v-invertible ideal, then Jt = Jv for each semiregular fractional ideal J. Proof. This is simply a consequence of the existence of a finitely generated semireg,,!„». r~a,\^ij±unaji ;j,,«i j_/ D r~ (n~ T,rV,;^l, D — fTy< nnt-;^r,n\ ±\_iodii. m.o/i i.j. v— t/T j.\jj. vv 14.1*^11 x-»u i n' All seven statements in Theorem 2.21 hold for Mori rings with the appropriate substitution of "regular" where ever "semiregular" appears (or is understood). Note that every regular ideal contains a regular element and thus a regular principal (invertible) ideal. So there is no need to add the hypothesis that each regular ideal contain a v-invertible ideal, it always will. Theorem 2.22. The following are equivalent for a ring R. (1) R is a Mori ring. (2) R satisfies the restricted DCC on regular divisorial ideals. (3) For each regular fractional ideal J there is a finitely generated regular fractional ideal B C J such that Bv = Jv. (4) For each regular ideal /, there is a finitely generated regular ideal A C I such that Av = Iv. (5) Each regular ideal is contained in at most finitely many maximal t-ideals and for each regular maximal t-ideal M, R(M) satisfies ACC on regular divisorial ideals. (6) Each regular ideal is contained in at most finitely many maximal t-ideals and for each regular maximal t-ideal M, R(M) satisfies the restricted DCC on regular divisorial ideals. 3. Examples Let D be an integral domain and let T3 be a nonempty set of prime ideals of D. For each Pa e P, let Ka be the quotient field of Da = D/Pa. Let X = {(a, n) n £ N, Pa 6 P}. For each i = (a, n) <= I, let Kt = Ka and let B = £ie2- Ki. Construct the ring R = D + B from the direct sum D ® B by defining multiplication by (r, a) • (s, b) = (rs, rb + sa). The ring R is said to be "the ring of the form A + B corresponding to D and P" (see [6, section 26] and [10, page 411]). The following

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theorem collects the relevant properties we need to verify the claims made in our examples. Theorem 3.1 (cf. [10, Theorem 19]). Let D be an integral domain and let P be a nonempty subset o/Spec(D). Then the following statements hold for the ring R of the form A + B corresponding to D and P. (a) T(R) can be identified with the ring DS + B where S = {r e R no prime in P contains r}. (b) An ideal J of R is semiregular if and only if J = IR = I + B for some ideal I of D which contains a finitely generated ideal that is contained in no prime from the set P. (c) Qo(R) can be identified with the ring E + B where E = (JT(I) where I ranges over those ideals of D which generate semiregular ideals of R and T(I) = \J(D : /"). (d) Each semiregular divisorial ideal ofR is of the form IR where I is a divisorial ideal of D which contains a finitely generated ideal that is contained in no prime from the set P. Moreover, if I is a divisorial ideal of D such that IR is semiregular, then IR is divisorial. Proof. Statements (a), (b) and (c) are directly from Theorem 19 of [10]. For (d), statement (b) lets us restrict our efforts to those ideals I of D that contain a finitely generated ideal that is contained in no prime from the set P. First assume IR is divisorial. By (c), (D : I) + B is contained in Qo(R) = E + B and it is clear that each element of (R : I) must be contained in (D : 7) + B. It follows that (IR)V = IvR where /„ is the divisorial closure of / in D. Hence (d) holds. D i Example 3.2. Let R be the A + B corresponding to D = K + (Y, w}K(z)[Y, w\ and P = Max(£>) \ {(Y, w)K(z)(Y, w}}. Then (a) R = T(R) and Q0(R) = K(Z)[Y, w] + B. (b) (R : (Y, w)R) = (R : MR) = Q0(R) where M = (Y, w)K(z)[Y, w}. (c) MR is the only proper semiregular divisorial ideal of R, so R is trivially a QcrMori ring. But no proper semiregular ideal is w-invertible. (d) RMR — DM is a Mori domain. In Corollary 2.14. we showed that if R is Qo-Mori ring, then so is R(p\ for each semiregular prime ideal P. In contrast, the ring in our next example shows that in general, a localization of a Qo-Mori ring need not be a Qo-Mori ring. In this particular case, we have a reduced Qo-Mori ring R with a single semiregular maximal t-ideal M which is also a regular maximal ideal and RM is an integral domain that is not a Mori domain. Example 3.3. Let R be the A + B ring corresponding to D = K{z, {ymzn 1 < m,n& Z}] and P = Max(.D) \ {zD}. Then: (a) T(R) = QQ(R)=D[l/z\ + B. (b) M = zD+B is the only semiregular/regular maximal ideal and it is divisorial with inverse (l/z)D + B.

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(c) R is a Qo-Mori ring, the divisorial semiregular ideals are the ideals of the form znD + B. (d) RM is isomorphic to DZD which is not a Mori domain. In our next example we construct a reduced ring R which satisfies ACC on semiregular divisorial ideals but does not satisfy the restricted DCC on semiregular divisorial ideals. Example 3.4. Let E be a Dedekind domain with a maximal ideal M = (a, 6) where no power of M is principal and let D = E + YEa[v] + zEb[z] + (y, z)2K[Y,z} where K is the quotient field of E and Ex represents E localized at the powers of the element x e E. Let R be the A + B ring corresponding to D and P the set of nonzero prime ideals of D that do not contain both Y and z. Then: (a) Q0(R) = (T(M) + (Y, z)K\Y, z}) + B. (b) P — YEa[Y]+zEb[z] + (Y,z)2K[Y.z] is a prime of D and P.Ris a semiregular divisorial ideal of R. Moreover, (R : PR) = Q0(R). (c) R satisfies ACC on semiregular divisorial ideals but {MnR} is an infinite descending chain of semiregular divisorial ideals whose intersection is a semiregular divisorial ideal. (d) Qo(R) is a semiregular fractional ideal of R but is not the v of a finitely generated semiregular fractional ideal. In our next example, we construct a reduced ring with ACC on semiregular divisorial ideals with a semiregular divisorial ideal that is contained in infinitely many maximal t-ideals. Before presenting the example, we need to set a little notation. Let K be a field and let X = {xn}^=l be a countably infinite set of (algebraically independent) indeterminates over K indexed over the positive integers. Let J- denote the set of finite products of elements from X and for each positive integer m, let AfH denote the set [x™}y=i. Example 3.5. Let D = K [ Y , Z , X$\ XW,YF,zF] and let P denote the set of primes of D which do not contain both Y and z. Then the following hold for the ring R = D + B associated with D and P. (a) For each positive integer n, the ideal (Y, Z, vf , Z? , x%, X^)R is a semiregular maximal i-ideal that is divisorial and each semiregular maximal t-ideal ideal has this form. (b) The ideal (Y, z)v is contained in infinitely many semiregular maximal divisorial ideals and is the unique minimal semiregular divisorial ideal of R. (c) R = T(R) satisfies ACC on semiregular divisorial ideals, but does not satisfy the restricted DCC on semiregular divisorial ideals. Proof. Let P be the prime of D generated by the set {y, z, YF, zF}. The choice of primes for the set P guarantees that each semiregular ideal of R will contain a power of (Y, z)R. As D is contained in S = K[Y, Z , X ] , a Krull domain, and shares the ideal ( Y , z ) K [ Y , z , X ] = PK[Y,z,X] with S, each semiregular ideal of R will

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have its inverse contained in S + B. Thus 5 + B = Qo(R)- From this we get that (y, z)v = P + B = PR is the minimal semiregular divisorial ideal of R. For each positive integer n the ideal Pn = (y, z, YF, zT, x%, X^)R is a semiregular prime ideal. Obviously, xn multiplies Pn into R so (Pn)v is not R. In fact, (Pn)v = Pn since no polynomial outside of Pn can multiply xn into D. Thus we have that (y, z)v is contained in infinitely many divisorial prime ideals. Now suppose g G D is a nonzero polynomial with no terms involving either y or z. Let J = gR + PR and consider J~l = (D : I) + B where / = gD + P. Since g is a polynomial, it involves only finitely many Xns from the set X, say xni, xn2, . . . , xnk. It is clear that (D : I) C K[Ytz,xni,...,xnk,YF,zF,XW,X®]. Let Oj denote the smallest power of Xni that appears in g. By the definition of D, di is either 0 or an integer strictly greater than 1. If dj > 2 for each i, then each term of g is divisible by (x ni x n2 • • • xnk)2. Thus each element of (xni,... ,xnk)K[xni,... ,xnk] will multiply g into D. Thus (D:I)= K\Y,z,xn.,...,xnk,Y?,z?',X^,x^} and it follows that Jv = Iv + B where Iv = (y, z, vf, zT, {f[ x%. \ b> > 2})D. Now suppose some a,; are 0. As we have assumed no particular order for the integers ni, 7 1 2 , . . . , n^, we assume aj = 0,2 = • • • = a.,- = 0 and all other OjS are at least 2. We may concentrate on those elements of (D : I) that are in (xni,...,Xnk)K[xni,...,xnk}. Let s £ (D : I)C\(xni,...,xnk)K[xni,...,xnk] and let m = dY[x%. be a nonzero monomial term of s which is not in D, so at least one 6, is 1. For i > j, fcj can be any nonnegative integer as X^ is a factor of each term of g. But for i < j, no 6j can be 1 (but &j could be zero). Thus (D : I) = K[Y,z,xn,+1 ,...,xnk,YF,zF,X^,#[3]] and Jv = Iv + B where /„ = (y,z,y.F,.z.F, {TIx£. * > J + Mi > 2})-°- It: follows that the ideals designated Pn above are the only maximal t-ideals of R and each is divisorial. Also by the form for the ideals /„ it is clear that R satisfies ACC on divisorial ideals. D As for the previous example, we have several sets to construct to build the ring in our next example. The ring we construct is a reduced ring which satisfies the restricted DCC on semiregular divisorial ideals but does not satisfy ACC on p- i ^ semiregular divisorial ideals. Let Q := {qi = -^ | pi < pz < •• • the odd ^Pi

primes} and T : = { i e < Q > | 0 < £ = m + a\qi + • • • + anqn with m and each a; nonnegative integers} and let {Wj} be a set of indeterminates indexed over the positive integers. For each integer k > 0, let V\4 := {Jl^fc ^7* I ^ —ri e ^> some n > k}. We assume that 1 G Wfc for each k (as the product where all exponents are zero). The last two sets are U := {u¥rzs U G Wo, r, s G T with r + s > 1} and V := {VY<*,VZ<>< | v € Wi, i > 1}. Finally let D = K[Y, z,U, V] where K is a field. Example 3.6. Let P be the primes of the domain D — K\Y,z,U,V] that do not contain both y and z and let R be the A + B ring corresponding to D and P. Then: (a) R = T(R). (b) If J is a semiregular ideal of R, then J = IR where I is an ideal of D that contains a power of the ideal (y, z)D. (c) Qo(,R) = (R : QR) = D[W0] + B where Q = (Y, z,U)D and QR = (QR)V.

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(d) M = (Y, Z, U, V)D is a maximal ideal of D and MR is a semiregular maximal divisorial ideal of R. (e) Recursively define ideals Qt of D by setting Ql - Q+(y«1Wi)I?+(z'J1W1)D and then Q{ = Q^ + (YqiW,).D + (zqiWi)D. Each Qt generates a semiregular divisorial ideal of R. (f) The only semiregular divisorial ideals other than MR and QR are the ideals QiR. Thus R satisfies the restricted DCC on semiregular divisorial ideals. (g) The chain QR c Q\R C Q^R C • • • is an infinite ascending chain of semiregular divisorial ideals. Thus R does not satisfy ACC on semiregular divisorial ideals. Proof. The radical of the ideal (Y, z) in D contains the sets U and V, but no principal ideal contains a positive power of the ideal (y, z). Thus (d, b) is a zerodivisor of R when d is a nonunit of D and therefore R = T(R). By Theorem 3.1, we also have that each semiregular ideal of R has the form IR where / is an ideal of D whose radical contains (y, z) [10, Theorem 19]. But this occurs only when Let E = tf [{y'WoW, {^W0}ter] = ^[{^}t€r, {zf}t€T, {^}i>i] = D[WQ] &ndF = ^[{ytWo}teT\{o},{^tW0}teT\{o}]. Consider the ideal (Y,z)R = (y,z)D+ B. Since D C E, (D : (y, z)D) C (E : (y, z)E) = E. Let J be a proper semiregular ideal of R. Then from the above we have that J = IR = I + B where / is an ideal of D that contains a power of (v,z)D. It follows that (R : J) C E + B. Also note that each element of the maximal ideal M is multiplies each element of F into D. Thus we also have (R : J) D F. Each nonzero element of / has the form / = ^diYqi + J2bjZqi + ^CkYrkzSk where r^, Sk £ T with r> + s^ > 1 and each Oj, bj and c^ some polynomial in the elements from the appropriate set Wm-Wi for Q.J, Wj for bj and Wo for c^. Note that each element of the form CkYrkZSk multiplies each element of E into D, so to know what elements are in (D : /) we need only consider which elements of E will multiply ^2a,iYqi + ^bjZqj into D. There is nothing to do if this part of / is 0, so assume it is not. Then let t = inf ({qi q^ > • • • is strictly decreasing, so t ^ 1/2 implies t = qi for some i. In this case, each element from the set Wi will be multiplied into D by each element of/, but no element from Wi_i \ Wi will be. Also no polynomial with at least one term from the set Wj_i \ Wj will be multiplied into D by each element of J. Thus (D:I)= £>[Wj] = (D : Qi). tyloreover, if g € D \ Qi, then the infimum for the ideal A = gD + Qi is less than qi. It follows that Wj is not contained in (D : A). Thus Qi is divisorial. Therefore the only semiregular divisorial ideals of R are the ideals QR C QiR C Q2R C • • • C MR, with MR = [J QiR. It follows that

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R satisfies the restricted DCC on semiregular divisorial ideals but does not satisfy ACC on semiregular divisorial ideals. D 4. The Mori Property and Idealization Remark. In this section I provide an answer to a question that Professor Paolo Zanardo raised at the end of my talk at the 2002 Algebra Conference held in Venice the first week of June, 2002. Essentially, his question was, "If D is a Mori domain and B is a D-module, does the idealization of B yield a Mori ring?" I did not know the answer at the time, but not long after the conference, I was able to construct an example to show that idealization of a module over a Mori domain need not yield a Mori ring. First a reminder about the basic notion of idealization of a module. Let D be a ring and let B be a nonzero D-module. Then the idealization of B is the ring R = D(+)B formed from the sum D®B with multiplication defined as (r, b)(s, c) = (rs, re + sb). In this way B is transformed into an ideal of a ring that is similar to D [11, page 2]. The product of two elements of B is trivial, thus all such rings are nonreduced. Our first result of this section gives a sufficient condition for the idealized ring D(+)B to be a Mori ring when D is a, Mori domain. Theorem 4.1. Let D be a Mori domain, B be a D-module and Z(B) = {d £ D db = 0 for some nonzero b € B}. If B = Bs where 5 = D\Z(B), then R = D(+}B is a Mori ring. Proof. It is known that T(R) = DS(+)BS (see for example [7, Corollary 25.5]). Also having B = BS guarantees that each regular ideal of R is of the form IR = I(+)B for some ideal / of D that is not contained in Z(B); i.e., / n 5 ^ 0 [7, Theorem 25.9]. From here it is easy to see that for such an ideal IR, the inverse is the regular fractional ideal (D : I)R = (D : I)(+)B. Thus (IR)V = IV(+)B. As D is a Mori domain, R is a Mori ring. D To build the example we take two infinite sets of algebraically independent indeterminates over a field K, y := {vn \ n > 1} and 2 :— {zn \ n > 1} and an additional set Q := {yn/ Y[ zTm ] rm > 0,m > n > I } . Example 4.2. Let D = K[y,Z] and let B = K[y,Z,Q}. Then (a) For each n > 1, the ideal In = ((YI, 0), ( v 2 , 0 ) , . . . , (vn, 0)) is a regular ideal. (b) For each n>l, (0, z~1} € (R : /„) \ (R : In+1). (c) R is not a Mori ring even though D is a Mori domain (in fact D is & UFD). Proof. The module B has no zero divisors in D, thus S = D \ {0}. It follows that each of the ideals /„ is regular. Since z^ is not allowed in the denominator of Yn/Ylz^T when k is less than n, it is eas,y to see that (0, Z ~ l ) is not in (R : In+i)On the other hand, it is in (R : /„). Thus the chain (/].)„ C (I%}v • • • is an infinite chain of regular divisorial ideals. Hence R is not a Mori ring even though D is a Mori domain. D

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The ring of finite fractions over an idealized ring D(+)B is generally more complicated to describe than the total quotient ring, but its form is known [8, Theorem 3]. Let D be a domain, C a D-module and R = D(+}C the idealization of C. Let A(C) := {AnnD(B) B C C a submodule of C} and let S(C) denote the set of finitely generated ideals of D which are contained in no ideal of the set A(C). Let E = \J{(D : I) \ I e S(C)}. Consider a pair of ideals / and J in the set S(C) and homomorphisms / £ Hom(I,C) and g e Hom(J,C). Declare / and g to be equivalent if /(a) = g(a) for each a 6 I(~}J- Then, as with the ring of finite fractions, we obtain a D-module Qo(C) made up of equivalence classes of homomorphisms. It turns out that Qo(C) is also an ^-module. Moreover, we may identify the ring of finite fractions over R with the idealized ring E(+)Qo(C) [8, Theorem 3]. Theorem 4.3. Let D be an integral domain and let C be a D-module such that C = Qo(C). If J is a semiregular divisorial ideal of R = D(+)C, then J = I(+)C for some divisorial ideal I of D that contains a finitely generated ideal which is contained in S(C). The converse holds as well. Proof. Since C = Q0(C), (0)(+)C is a common ideal of R and Q0(R). Thus if / is a divisorial ideal of D which contains a finitely generated ideal from the set S(C), then I(+)C will be a divisorial ideal of R with inverse (D : 7)(+)C. Let J be a semiregular divisorial ideal of D and let / = {d € D \ (d, b) e J for some b € C}. Then for each element (/, g) 6 (R : J) we have (f,g)(d,b) = (fd,dg + f b ) . Again since (0)(+)C r is a common ideal of R and Qo(R), dg and fb are in C no matter whether (/, g) is in (R : J) or not; i.e., the only thing that matters is that f d is in D. Hence J must be of the form I(+)C and I must be a divisorial ideal of D that contains a finitely generated ideal from the set S(C). D Theorem 4.4. Let D be a Mori domain and let C be a D-module. Then R = D(+)Qo(C) is a Qo-Mori ring. Proof. By the previous theorem, each divisorial ideal of R is of the form I(+)Qo(C] for some divisorial ideal I of D that contains a finitely generated ideal from the set S(C). Since D is a Mori domain, R satisfies both ACC and the restricted DCC on semiregular divisorial ideals. D As a special case we have the following. Corollary 4.5. Let D be a Mori domain and let P be a subset ofSpec(D). For each PQ 6 P, let Ka denote the quotient field of D/Pa. Then R = D(+) £ Ka is a Qo-Mori ring. Proof. Let B = $2Ka. Then by [9, Theorem 11], we have that B = Q0(B}. The result now follows from the Theorem 4.4. D

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Acknowledgements Several results in this paper were established while the author was in Italy during the spring of 2002. He acknowledges the support he received from Istituto Nazionale di Alta Matematica-GNSAGA which helped make his visit possible. The author also wishes to thank Stefania Gabelli, Marco Fontana and Francesca Tartarone of Universita degli Studi Roma Tre and Valentina Barucci of Universita "La Sapienza" for the hospitality and intellectual stimulation they provided during his visit. References [1] J. Arnold and J. Brewer, On flat overrings, ideal transforms, and generalized transforms of a commutative ring, J. Algebra 18 (1971), 254-263. [2] V. Barucci, Mori domains, in Non-Noetherian Commutative Ring Theory, S. Chapman and S. Glaz, Eds., 57-73, Kluwer, Dordrecht, 2000. [3] N. Dessagnes, Intersections d'anneaux de Mori—exemples, C. R. Acad. Sci. Canada 7 (1985), 355-360. [4] N. Dessagnes, Intersections d'anneaux de Mori—exemples, Port. Math. 44 (1987), 379-392. [5] R. Gilmer, Multiplicative Ideal Theory, Queen's Papers in Pure and Applied Mathematics, 90, Queen's University Press, Kingston, 1992. [6] J. Huckaba, Commutative Rings with Zero Divisors, Dekker, New York, 1988. [7] J. Lambek, Lectures on Rings and Modules, Chelsea, New York, 1986. [8] T. Lucas, The integral closure of R(X) and R(X), Comm. Algebra 25 (1997), 847-872. [9] T. Lucas, Strong Priifer rings and the ring of finite fractions, J. Pure Appl. Algebra 84 (1993), 59-71. [10] T. Lucas, The complete integral closure of R(X), in Factorization in Integral Domains, D.D. Anderson, ed., Lecture Notes in Pure and Applied Mathematics, 189, 401-415, Dekker, New York, 1997. [11] M. Nagata, Local Rings, Intersci. Tracts in Pure and Appl. Math. No. 13, Interscience, New York, 1962. [12] J. Querre, Sur une propriete des anneaux de Krull, C. R. Acad. Sc. Paris, 270 (1971), A 739A742. [13] J. Querre, Intersections d'anneaux integres, J. Algebra 43 (1976), 55-60. [14] N. Raillard (Dessagnes), Sur les anneaux de Mori, C. R. Acad. Sc. Paris 280 (1975), A1571A1573.

Minimal Prime Ideals and Generalizations of Factorial Domains Peter Malcolmson Department of Mathematics, Wayne State University, Detroit, Michigan (USA) petemfimath.wayne.edu

Prank Okoh Department of Mathematics, Wayne State University, Detroit, Michigan (USA) okohflmath.wayne.edu

Abstract. In a Noetherian domain or a factorial domain, every nonzero principal prime ideal has rank one and every non-zero element has only finitely many prime ideals minimal over it. In this paper we consider integral domains for which these and similar conditions hold. We refer to these conditions as FP (finiteness of primes) conditions. We determine which FP conditions are stable under localization and under adjoining a polynomial variable. The inclusion relations amongst the various FP domains are obtained. The largest class of these FP domains is defined by the property that every non-zero element is contained in only finitely many principal prime ideals of rank one. It contains every other class of generalized factorial domains in the literature.

Introduction All rings are commutative integral domains with identity. The ideal generated by a set S is denoted (S). We are interested in the domains in which various sets of prime ideals containing x are finite for every non-zero element x in the ring R. We refer to the conditions requiring this sort of finiteness as FP conditions. • A domain is called PF (primes finite) if every non-zero element is in only a finite number of prime ideals. • A domain is called MPF (minimal primes finite) if every non-zero element x is in only a finite number of prime ideals minimal over x. 2000 Mathematics Subject Classification. Primary 13F15; Secondary 13F05. Key words and phrases, factorial, generalizations, extension.

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• A domain is called ROPF (rank one primes finite) if every non-zero element is in only a finite number of rank one prime ideals. • A domain is called PPF (principal primes finite) if every non-zero element is in only a finite number of principal prime ideals. • A domain is called MPPF (minimal principal primes finite) if every non-zero element x is in only a finite number of principal prime ideals minimal over x. • A domain is called ROPPF (rank one principal primes finite) if every nonzero element is in only a finite number of rank one principal prime ideals. A PF domain is in particular a domain of finite character, see p. 97 of [FS2]. A PPF domain may also be called a PDF (prime divisors finite) domain, by analogy with IDF of [AAZ]. A domain is said to be a PPM-domain if every non-zero principal prime ideal has rank one. In such a domain the conditions PPF, MPPF, and ROPPF coincide. When R is Dedekind, every non-zero element x € R is contained in only finitely many prime ideals in R. This is the justification for the usage in [MO], in which we call PF domains and PPF domains by the names GD(2) domains and GD(1) domains, respectively. In [MO] we saw that PPF domains form a larger class than each of the generalizations of factorial domains in [AAZ]. We shall see in 1.3 that the class of ROPPF domains is even larger. Among the standard operations on domains (see the Table of Invariances on p. 622 of [B]), factorial domains and Noetherian rings are stable only under polynomial ring extensions and localization, see [B], [E], [M], and [N]. With the possible exception of MPF, all our FP domains and PPM-domains are stable under polynomial ring extensions. As for localization, all those classes defined via subsets of the set of principal prime ideals in R are not stable under localization. For PPF domains, this is due to [Cl]. However, PF, MPF, and ROPF domains are stable under localization. As remarked in [FS2], the class of valuation domains is closest to the class of Noetherian domains. This is reflected in valuation domains being MPF and PPF, and hence enjoying all the other FP-properties except PF (see [GH] or p. 516 of [FS2]). Neither Noetherian domains nor factorial domains are necessarily PF, see Theorem 36 of [K] or Proposition 2.4 of [MO]. A domain R is atomic if every non-zero non-unit element in R is a finite product of irreducible elements. We will observe that an atomic domain is a PPM-dornain. This is a special case of Krull's principal ideal theorem without any chain conditions. Another application of generalizations of factorial domains is in [MO] where an alternative proof of Zariski's version of the Nullstellensatz is given. 1. Relationships among the FP domains We begin with a criterion for a principal .prime ideal to be of rank one which will be used often in this paper. Lemma 1.1 (Exercise 5, p. 7 of [K]). Let P = (p) be a non-zero principal prime ideal and J(P) = r£°=1 Pn.

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(a) If Q is a prime ideal strictly contained in P, then Q C J(P)(b) If p is a not a zero-divisor, then J = pJ. Moreover J(P) is prime. (c) Suppose p is not a zero-divisor. Then the principal prime ideal P = (p) is of rank one if and only if J(P) = 0. Proposition 1.2. Let R be an atomic domain. Then R is a PPM-domain and the three classes of PPF, MPPF, and ROPPF domains coincide. Proof. Let P be a non-zero principal prime ideal with generator p. If J(P) ^ 0 then let x be a non-zero element. Then x — X i x % . . . X k where each x,, i = 1, 2 , . . . , k, is irreducible. But x e Pk+l implies that x^x2 ... xk = rph+1 for some non-zero element r S R. Since p divides some Xi for each copy of p, this leads to a contradiction. So J(P) = 0 and by Lemma 1.1, P has rank one. The rest of the proof follows from the definitions. D 1.1. Relationships

Theorem 1.3. We have the following diagram of implications: PF

;• MPF

> ROPF

PPF => MPPF => ROPPF. Moreover none of the implications are reversible. Proof. The implications follow from the definitions. The non-reversibility of the implications will follow from the First, Second and Third Examples, to follow. These examples are elaborate versions of the D + M-construction. For instance, Z + XQ[X] is not PPF, (see [MO], Example 1.8 and [MZ]). Let us choose some notation to simplify the examples. Let {X, Z, YI, Y^ • • • } be a set of indeterminates. Let Y denote the set of monomials in the elements of {y1;y2, . . . }. Put rady = {yl5 Y2, . . .,Y1Y2, . . . ,Y,Y, . . .yn , . Let f stand for {^ : y e Y}. Then ^Jy = {f : y £ 1.2. First Example Start with the ring RI = Z[X, Z, Yi, . . . , Yn, . . . , y, ^fy]. Let S denote the complement of USi(^i) m -^1- Let R be the localization of RI at 5, R = (Ri)s- We have the following inclusions of domains: Rv =

Z[X,Z,Y1,...,Yn,...]CR1CR.

Remark on the primeness of (X, y}. Note that if an element / in R is divisible by every positive power of Yn, then / must have ^ or X as a factor. This is the basis for showing directly that (X, y) is a prime ideal in this section and Section 3. The following lemmas describe the prime ideal structure of R. Lemma 1.4. For each positive integer n, the ideal (Yn) is a maximal ideal.

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Proof. This lemma requires proof because 5 is the complement of an infinite union of prime ideals. The prime ideals in R are contained in (Jili C^i) by ^ne construction of R. Suppose (Yn) C Q C U^ 1 (Yj), where Q is a prime ideal in R. Suppose there is an element / £ Q and / £ (Y n ). Therefore, / <E (Y m ), m ^ n. Thus Yn + / 6 Q, while Yn + / £ (Ym) U (Y n ). Hence Yn + / € (Yi),Z ^ {m,n} unless it is a unit. Thus we have an equation

(1)

Yn + f = Ym

for some gi £ R. By choice, / does not have Yn as a factor nor does Yn + f . Multiplying (1) by suitable elements s £ 5 and a monomial y 6 Y, the resulting equation in the polynomial ring R% is impossible because of discrepant powers of Yn. Hence (Yn) is a maximal ideal in R. D Lemma 1.5. For each positive integer n, J((Y n )) = (X, y}, and for each element yeradY, f £ J((Yn)). Proof. We begin by proving that y ^ J((Y n }) for all y 6 Y. Indeed y € Y implies that YJ 6 ,7({Yn}), for some i, by Lemma 1.4. By the same lemma, i — n. With Yn = y^<72 f°r some 52 S -R) we get the_contradiction that Yn is a unit. Now from y~ = Y£y^ we get that X e J((Y n )). Since Yn £ J((Y n }), y- £ J({Y n )) for ail positive integers i. Thus J((Y n )) D (X,Y). Let / £ J((Y n )). Then Y£ divides / for every positive integer k. This implies that / must have an ^--factor. Hence / 6 (X, y). Suppose — £ J((Y n }). Then — must have a factor of X or y^. This is a contradiction since the ideal (X) in RI cannot contain Z. O Lemma 1.6. The ideal (X, y) is a rank one prime ideal in R. Proof. Suppose we have a prime ideal P C (X, y). Assume that 0 ^ f £ P. Suppose we can write / = Y^fk with fk not divisible by Yn. By Lemma 1.4, Yn £ P; thus fk € P. But P C (X,y) C (Yn). This contradicts the assumption that Yn does not divide fk. Suppose instead Y^ divides / for every positive integer k. Then / must have a ^--factor. Hence X e P. From X = Y~ and Y ^ P we deduce that P = (X,y). D The next lemma follows from Lemma 1.1 and the preceding Lemmas. Lemma 1.7. The rank of each (Yn) is two. The only other prime ideal in R is

(*,&• Proposition 1.8. The ring R is ROPF and ROPPF and fails to satisfy any of the other FP conditions. Proof. Each principal prime ideal (Yn) contains the non-zero non-prime element X. Since Z <£ J((Y n )), it follows from Lemma 1.1 that each (Yn) is minimal over Z. Therefore R is neither MPF nor MPPF. Since R has only one prime ideal of rank one, R is obviously ROPF and ROPPF. D

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1.3. Second Example This is the First Example with Z = 0. Consequently we have the following analogues of the lemmas in that section with identical proofs. Lemma 1.9. For each positive integer n, the ideal (Yn) is a maximal ideal. Lemma 1.10. For each positive integer n, J ( ( Y n ) ) = (X, y). Lemma 1.11. The ideal (X, y) is a rank one prime ideal in R. Lemma 1.12. The rank of each (Yn) is two. The only other prime ideal in R is

(X,y)Proposition 1.13. The ring R is MPF, ROPF, MPPF, and ROPPF but satisfies neither PF nor PPF conditions. Proof. Since the element X is in (Yn) for each positive integer n, R is not PPF, hence not PF. Since (X, y) C (Yn} is a saturated chain of prime ideals, the prime ideal (X, y) is the only prime ideal minimal over every element in (X, y). Let f & R but / 0 (X, y). Then / is contained in (Yn) for only finitely many integers n. This can be seen by reducing mod (X, y) to get the localized factorial domain Z[YI,YZ, ...]j (where S is the reduction of S modulo (X, y)). We have proved that R satisfies the MPF condition and the MPPF condition. D 1.4. Third Example

This example, due to C. Weibel, is borrowed from Example 39 of [H] where it is used to illustrate phenomena other than FP properties. Consider T = C[-Xo, X\,...}. Let / be the ideal generated by all the elements X% — Xn-i, for n> 1. The ring of interest is R = T/I. Suppose (/} is a principal prime ideal in R = T/I. Then / e C[o; 1 ,X2,... ,xm] = C[xm], a polynomial ring, where Xi = Xi +1. But in C[xm], / = axm + b, for some complex numbers a and b. However, in R, we have that / = ax^+1 + b, which in turn factors nontrivially in C[a;m+i] C R. Therefore, R has no principal prime ideals. So R is vacuously PPF, MPPF, and ROPPF. The ring R is one-dimensional, since R is integral over the one-dimensional domain C[XQ], see [K], Theorems 44-46. Thus all the maximal ideals of R have rank one. The non-prime principal ideal (xm — 1) is contained in infinitely many maximal ideals (see [H], p. 71 for details). So R is not R.OPF, hence neither PF nor MPF. (As noted in [H] the ring R is Bezout. Thus a Bezout domain need not be MPF.) 1.5. Completion of the proof of Theorem 1.3.

The implications follow from the definitions. The three Examples complete the rest of the proof of Theorem 1.3. For instance; the First and Third Examples show that ROPF and PPF are independent, while ROPPF does not imply MPPF. D In [MO] it was pointed out that the class of PPF domains includes all the extensions of factorial domains in [AAZ] and [AM]. As a result of Theorem 1.3 we

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now have that the class ROPPF is larger yet. Also, an IDF-domain is ROPPF. By contrast a domain that is not a field which has no irreducible element is vacuously ROPPF but not IDF. Therefore the class of ROPPF domains is now the largest generalization of factorial domains. We shall see in 2.2 that not every domain is ROPPF. 2. Stability of the FP domains under the standard operations 2.1. Polynomial extensions Let / be an ideal in R. It generates an ideal IR[X] in .Rpf]. We note that / is prime (respectively principal) if and only if IR[X] is prime (respectively principal). We start with the following well-known result; see for instance [K]. Lemma 2.1. A prime ideal P in R is minimal over I if and only if PR[X] is minimal over IR[X}. Proposition 2.2. (a) Suppose R[X] satisfies an FP condition other than possibly ROPF. Then the same holds for R. (b) Suppose R[X] is a PPM-domain. Then R is a PPM-domain. Proof. Lemma 2.1 and the remark preceding it dispose of the cases when R[X] is PF, PPF, MPF or MPPF. Suppose R[X] is ROPPF. Let P be a principal prime ideal in R containing a non-zero element x € R with rank P = 1. Then by Lemma 1.1, J(P) = 0. This implies that J(PR[X}} = 0. Hence PR[X] is also of rank one. Therefore, if R[X] is ROPPF, so is R. This proof also shows that if R[X] is a PPM-domain, so is R. D Following [K], a ring R is an S-domain (respectively, strong S-domain) \iPR[X\ is of rank one (respectively rank n) for every rank one (respectively rank n, for any positive integer n) prime ideal P in R. Noetherian or Priifer rings are strong Sdomains, see p. 108 of [K], or see [Kb] or [MM]. Proposition 2.2 for ROPF rings is valid if the rings are S-domains. We do not yet know the status of Proposition 2.2 for the class ROPF for nonS-domains. However, if we define a rank one prime ideal P of R to be Seidenberg if the induced ideal PR[X] has rank one, then it is straightforward to verify that R[X] is ROPF if and only if the intersection of any infinite number of Seidenberg ideals of R is trivial. Let P be an ideal in R[X}. Define degP = min{deg/ : / e P}. Using this definition one can show the following property of degree (recall that R is always a domain in this paper.) Proposition 2.3. Let f be in distinct prime ideals PI, . . . , Pn with each Pi minimal over f . Then deg / > deg PI + deg P2 + • • • + deg Pn. Proposition 2.3 can be used to prove the next proposition. As D. Lantz pointed out to us Proposition 2.4 also follows readily by passing to A"[Jf], where K is the quotient field of R (see Theorem 36 in [K].) An ideal / in R[X] is said to be constant-free if / n R = 0.

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Proposition 2.4. An infinite number of distinct constant-free prime ideals in R[X] has trivial intersection. We can now prove the converse of Proposition 2.2. Theorem 2.5. (a) (Theorem 36 of [K] or Proposition 2.4 of [MO]) R[X] is PF if and only if R is a field. (b) Suppose R satisfies an FP condition other than PF and possibly MPF. Then the same holds for R[X]. (c) Suppose R is a PPM-domain. Then so is R[X]. Proof. Theorem 2.5(b) is proved for PPF in Proposition 1.4 of [MO]. In proving 2.5(b) we may, by Proposition 2.4, restrict to prime ideals P with P fl R ^ 0. Suppose R is ROPF. We have to show that if 0 ^ f(X] £ R[X], then / is contained in only finitely many prime ideals in R[X] of rank one. Consider the leading coefficient a of /. The element a is contained only in a finite number of rank one prime ideals Qi,Q2, • • • , Qn in R because R is assumed to be ROPF. We claim that the set of prime ideals P of R[X] that contain / with P f~l R ^ 0 is a subset of {QiR[X], ~ Q 2 R [ X ] , . . . , QnR[X]}.' Let / be in some prime ideal P of R[X] (with P H .R / 0). Now, 0 ^ (P h R)R[X] C P. Since both prime ideals are rank one prime ideals, we have (P n R)R[X] = P. Thus all coefficients of / are in P n R. In particular a e P n R. Therefore, P n R is Qi for some i e {1, 2 , . . . , n}. Therefore, P = QtR[X]. Suppose R is MPPF. Let 0 ^ f ( X ) € R[X], and let S = {P = (p), p e R[X] : P is a principal prime ideal minimal over f ( X ) } . Our goal is to show that S is a finite set. Let P = (p) S S. Recall that we are restricting to P with P n R ^ 0. Any non-zero element of P n R is a multiple of p, so p e R. Since pR[X] is minimaln over f ( X ] , we deducen from Lemma 1.1 nthat f(X) £

- But ir=i(p*w) 2 (rr=i(p£) )w Thus, Pick a

coefficient a of /(X) not in fC=i <>#)"• Since p S R and /(JQ e pfi[X], a e pfi, but a £ fl^iO'-R)"'- Thus P-R is minimal over o, by Lemma 1.1. Therefore every element in S gives rise to a principal prime minimal over a non-zero coefficient of f ( X ) . Since for prime elements pi and PJ in R, piR = pjR 4=> pi-R[-X] = pj-B[X], we deduce the finiteness of S from the hypothesis that R is MPPF. Since f(X) is an arbitrary non-zero element of R[X], R[X] is also MPPF. (c) Let (f(X)) be a non-zero principal prime ideal in R[X]. By Lemma 1.1, it is enough to show that J((f(X))) = 0. Suppose not. If deg/(X) > 0, then any non-zero element in J((f(X))) would be of unbounded degree. Hence f ( X ) 6 R. Since R is assumed to be a PPM-domain, J((f(X))) = 0. By Lemma 1.1, (f(X)) is of rank one. Thus R[X] is a PPM-domain. D

2.2. Stability under localizations This section provides examples of different types of FP domains, in particular their prime ideal structures. Recall that Rs denotes the localization of R at a multiplicative set S in R.

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Proposition 2.6. Let R be a PF, MPF, or ROPF domain and S a multiplicative set in R. Then RS is correspondingly a PF, MPF, or ROPF domain. Proof. We recall the following. Let P be a prime ideal of R disjoint from S. The ideal PE = {as"1 : a £ P, s 6 S} is a prime ideal of RS- If Q is an ideal of RS, Qc — Q n R is a prime ideal of R. The maps C and E are a one-to-one orderpreserving correspondence between all the prime ideals in RS and those prime ideals in R disjoint from S. We now consider PF, MPF, and ROPF in turn. PF: Suppose as"1 is in the prime ideals Qi, Q%, . . . of RS- Then a £ Qi n R = Pi. Since Pi = Pj <==£> Qi = Qj we conclude that R being PF implies that RS is PF. MPF: Let Q be a prime ideal in RS minimal over as"1. Then Q n R = P is minimal over a. For if a 6 PI C P. Then as"1 <E Pf C PE = Q. Thus PL being MPF implies that Rs is MPF. ROPF: Since a rank one prime ideal containing a non-zero element x is minimal among prime ideals containing x, the proof for ROPF is identical to that for MPF. D

Proposition 2.7. None of the classes, of PPF, MPPF, and ROPPF domains is stable under localization. Proof. We shall construct a domain that is PPF with a localization that is not ROPPF. This and Theorem 1.3 will prove the proposition. Example 2.8. In addition to the notation in the First Example, we imitate the Third Example by letting Y^] denote the set {Y^ , Y? , . . . , Y?* ,...}, where Yi is an indeterminate for 'each i = 1, 2, . . . . Let Z, Xi,X%, . . . be new indeterminates. Let Y denote the set of monomials in the elements of Ui^ii^i }• Start with the domain Ml

With the above notation and that of Section 1.1, the domain PL of Example 2.8 is the localization (R^s at the complement S in PI of LCi^')- where (^') denotes the ideal generated by all elements of Yi . The methods of Section 1.2 can be used to prove the following lemma. [-1 Lemma 2.9. (a) For each i = 1, 2, . . . , (Yi 2 ) is a prime ideal of R that contains the prime ideals (Xn, ^) for all n = 1, 2, . . . . (b) Each of the prime ideals (Xn, ^r), n = 1, 2, . . . contains the non-prime ideal V'SdX/'

The fractions guarantee that (Yt2 ) is not a principal ideal. Thus Lemma 2.9 and Theorem 1.3 imply that Lemma 2.10. The ring R is vacuously PPF, MPPF, and ROPPF.

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Now consider the multiplicative set Y in R which is the set of monomials in the elements of |J^1{Fi 2 } as before. In the localized domain Ry, the prime ideals (Xn, ^Y~}, n = 1,2,... are principal prime ideals containing Z. Hence Ry is not ROPPF. Hence by Theorem 1.3, RY is neither PPF nor MPPF. This completes the proof of Proposition 2.7. D J. Coykendall proved Proposition 2.7 for PPF domains, see [Cl] and [C2]. When we replace -^^ by j? (see notation in 1.1) in R in Example 2.8, the resulting domain has no principal prime ideal. Hence it is vacuously a PPM-domain. With Y as the multiplicative set above, the domain Ry now has (Xn, ^P-), n = 1 , 2 , . . . as principal prime ideals that contain the prime ideal (Z, j|). We have thus proved Proposition 2.11. The class of PPM- domains is not stable under localization. 2.3. Stability under subrings and quotient domains It is easy to see that the class of PF domains (GD(2)) is stable under quotients by prime ideals, see [MO]. For the behavior under quotients of the other FP domains we return to the Third Example. The non-MPF example R in the Third Example is a quotient of the polynomial ring over C in infinitely many variables. The latter may be regarded as a quotient of a UFD, hence a quotient of an MPF ring. Therefore a quotient of an MPF domain by a prime ideal need not be MPF. The maximal ideals in R are each of rank one. Since we saw that some non-zero element is contained in infinitely many maximal ideals, R is,not ROPF. Theorem 1.3 now tells us that neither MPF nor ROPF is stable under quotients by prime ideals. In order to deal with PPF, MPPF, and ROPPF we need a domain R that is PPF with a quotient domain that is not ROPPF. The domain is a variation of the examples in Section 1.1. We also use the notation from that section. Use a single indeterminate X and start with the ring R% = The ring R is the localization of #2 at the complement of (Ji^iC^i) m -^2- The following properties of R are proved as in Section 1.2. Lemma 2.12. For each n, the principal prime ideal \Yn} contains the element X and has rank one. Thus R is not ROPPF. When we rewrite R% as a quotient of a factorial domain R2 = Z[X0, Xlt . . . , YI, Y2, . . . }/(Xi - Xi+1Yi+l : i = 0, 1, 2, . . . ), we see that R can be regarded as the quotient of a localization of a factorial domain by a prime ideal. Since a factorial domain is PF, a quotient need not have any FP conditions. Since a field is PF, and since subrings of a field need not have any PF conditions, we have both parts of the following proposition. i Proposition 2.13. (a) No class of FP domains is stable under subrings. (b) Only the class of PF domains is stable under quotients by prime ideals.

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2.4. Stability under power series ring extensions. Our last example is a stripped-down version of the domains in Section 1.1. We shall construct a domain R that is PF with R[[T]} not ROPPF. By Theorem 1.3 we will have shown at one fell swoop that no class of FP domains is stable under power series rings. Let {X, Y} be a pair of indeterminates. Let RI = Z[X, Y, ^ : n = 1 , 2 , . . . ] and let R = (-Ri)s where 5 is the complement of (Y) in R^. The only principal prime ideal of R is (Y). The prime ideal (X,£. : n = 1,2,...) lies below (Y). Since R has only two prime ideals, it is trivially PF. In order to see that -R[[T]] is not ROPPF, we write X0 = X and X> = $-, i = 1 , 2 , . . . . So Xt = YXi+l. The element Y + nT is a prime element in R[[T]} with J((Y + nT}) = 0. Therefore, (Y + nT) is of rank one by Lemma 1.1. For every positive integer n, X0 = (Y + nT)(Xi - nX2T + n2X3T2 - n3X4T3 + • • • ) . lim^n then Y + nT and Y + mT are not associates in R[[T]], since Y + nT = (Y + mT)(l + kT H ) implies n = m + kY, which is not solvable in R. Thus A"0 is in infinitely many principal prime ideals of rank one. Using this and Theorem 1.3 we get Proposition 2.14. No class of FP domains is stable under power series extensions. We remark that it follows from [Cl] that no FP domain is stable under integral closure. References [AM] D. D. Anderson and B. Mullins, Finite factorization domains, Proc. Amer. Soc. 124 (1996), 389-396. [AAZ] D. D. Anderson, D. F. Anderson, and M. Zafrullah, Factorization in integral domains, J. Pure Appl. Algebra 69 (1990), 1-19. [B] N. Bourbaki, Commutative algebra, Elements of Mathematics, Springer-Verlag, 1989. [Cl] J. Coykendall, GD(1) and GD(2) are not preserved in integral closure, to appear in Int. J. Comm. Rings 2, no. 1. [C2] J. Coykendall, OD(1) is not preserved under localization, preprint. [E] D. Eisenbud, Commutative algebra, Graduate Texts in Algebra v. 150, Springer-Verlag, 1994. [FS1] L. Fuchs and L. Salce, Modules over valuation domains, Lecture notes in pure and applied mathematics v. 97, Marcel Dekker, New York and Basel, 1985. [FS2] L. Fuchs and L. Salce, Modules over non-Noetherian domains, Mathematical Surveys and Monographs v. 84, Amer. Math. Soc, 2001. [GH] R. W. Gilmer and W. J. Heinzer, Intersections of quotient rings of an integral domain, J. Math. Kyoto Univ. 7 (1968), 133-150. [H] H. C. Hutchins, Examples Of commutative rings, Polygonal Publishing House, 1981. [Kb] S. Kabbaj, Sur les S-domains forts de Kaplansky, J. Algebra 137 (1991), 400-415. [K] I. Kaplansky, Commutative rings, Revised Edition, University of Chicago Press, Chicago, 1974. [M] H. Matsumura, Commutative algebra, W. A. Benjamin, Inc. New York, 1970. [MO] P. Malcolmson and F. Okoh, Expansions of prime ideals, to appear in Rocky Mountain J.

Math. [MM] S. Malik and J. L. Mott, Strong S-domains, J. Pure and Appl. Algebra 29 (1983), 249-264. [MZ] J. L. Mott and M. Zafrullah, Unruly Hilbert domains, Canad. Math. Bull. 33 (1990), 106109. [N] M. Nagata, Local rings, Interscience Publishers, New York, 1962.

Factorizations of Monic Polynomials Stephen McAdarn The University of Texas, Department of Mathematics, 1 University Station C1200, Austin TX 78712-0257 mcadamSmath.utexas.edu

Richard G. Swan Department of Mathematics. The University of Chicago, Chicago IL 60637 swanfimath.uchi cage.edu

Abstract. We study two types of factorizations of monic polynomials over a commutative ring R; factorizations into irreducible polynomials, and factorizations into pairwise comaximal pseudo-irreducible polynomials. We show that both types of factorizations exist (and the latter are unique) for all nonconstant monic polynomials if and only if R has only finitely many idempotents. Connected rings are also characterized via such factorizations.

In this paper, R will be a commutative ring with 1. Definitions. Let & be a non-unit element of R. a) We will say that b £ R is pseudo-irreducible if it is impossible to write b = cd with c and d comaximal (i.e., (c, d) = R} non-unit elements. b) By a complete comaximal factorization of b we will mean a factorization b = ^1^2 • • • bn where the bi are pairwise comaximal pseudo-irreducible elements. c) We will say that 6 € R is irreducible if it is impossible to write b — cd with c and d non-units. (We mention that there is a more general study of factorizations over rings in which various concepts of irreducibility are considered. [AV] is a good entry to that area. As we will only be factoring monic polynomials, the above "standard" definition is adequate.) In this paper, we will prove the following facts concerning monic nonconstant polynomials in 1991 Mathematics Subject Classification. 13B2p. Key words and phrases, connected rings, idempotents, irreducible factorization, comaximal factorization, polynomials.

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a) (Theorem 13) If R is connected, then every nonconstant monic polynomial in R[X] has a factorization into monic irreducible polynomials (which may not be unique) and a unique complete comaximal factorization into monic polynomials. Furthermore, either of these statements characterizes R being connected. b) (Theorem 17) If R has only finitely many idempotents, then every nonconstant monic polynomial in R[X] has a factorization into irreducible polynomials (having idempotents for their leading coefficients), and a unique complete comaximal factorization (whose factors have idempotents for their leading coefficients). Furthermore, either of these statements characterizes R having only finitely many idempotents. c) (Corollary 22) Every nonconstant monic polynomial in R[X] has a unique irreducible factorization if and only if R is a direct sum of finitely many integrally closed domains. This paper was inspired by a clever observation in [HL]. A companion paper is [MS], which studies unique comaximal factorization of elements in a domain. (The closing remark of this paper points out a slight difference in terminology between this paper and that one.) Given a polynomial f ( X ) € ^[^], we must study when f ( X ) has a monic associate. That is, we must explore when there are polynomials u(X) and k(X] in R[X] with u(X) a unit, k(X) monic, and f ( X ) = u ( X ) k ( X ) . Lemma 1. // f ( X ) 6 -R[-X] has a monic associate, then it has a unique monic associate. Proof. Suppose g(X) and h(X) are both monic associates of f ( X ) , writing f ( X ) = u(X)g(X) = v(X)h(X) with u(X) and v(X) units in R[X]. As g(X) = ( u ~ l ( X ) v ( X ) ) h ( X ) , and as g(X) and h(X) are monic, we have that u~l(X)v(X) is monic. However, it is also a unit in R[X], and so must equal 1. Thus g(X) = h(X). D Lemma 2. Let R be a commutative ring, let f ( X ) 6 R[X], and let r > 0. The following are equivalent. i) f ( X ) has a monic associate of degree r. ii) The coefficient of Xr in f ( X ) is a unit of R while the coefficients of all terms of f ( X ) of degree higher than r are nilpotent elements of R. Proof, i) => ii): Suppose f ( X ) — u ( X ) k ( X ) with k(X) monic and u(X) a unit in R[X], and let deg k(X) = r. Let N be the nilradical of R. Since u(X) is a unit in R[X], clearly u(0) is a unit in R. Now for any prime ideal P of R, u(X) mod P is a unit in (R/P)[X], and so u(X) mod P must be a constant polynomial, showing all the coefficients of u(X) except the constant one are in P. As this holds for all prime ideals P, all coefficients of u(X) except the constant coefficient are in TV. We now see that in f ( X ) = u ( X ) k ( X ) , the coefficient of Xr equals u(0) plus an element of N, making it a unit of R. Also, all higher degree coefficients in that product are clearly in N.

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ii) =$• i): Assume (ii) holds, and let / be the ideal of R generated by the coefficients of terms of f ( X ) having degree higher than r (the problem being trivial if there are no such terms). As / is generated by finitely many nilpotent elements, / is nilpotent. We claim there is a unit v ( X ) € R[X] such that the coefficient of Xr in v ( X ) f ( X ) is a unit, while the coefficients of all terms in v ( X ) f ( X ) of degree higher than r are in I2. Supposing this claim is true, we can then iterate, finding an associate of f ( X ) having the coefficient of Xr a unit and having the coefficients of all terms of higher degree in Is, for s equaling any power of 2. However, for sufficiently large such s, Is = 0, and we will be done. It remains to prove the claim. As working modulo a nil ideal does not affect units, we may assume I2 — 0, and show some associate of f ( X ) has degree r and has a unit as its leading coefficient. We will induct in degf(X] — n, the case n = r being trivial. As we know the coefficient of Xr in f ( X ) is a unit, we may assume it is 1. We may write f ( X ) = (1 + g(X))Xr + h(X) with g(X) e IR[X], and with d e g g ( X ) = n - r and deg/i(JQ < r. We have (1 - g ( X ) ) f ( X ) = (I g2(X))Xr + (1 - g ( X ) ) h ( X ) = Xr~+ h(X) - g ( X ) h ( X ) . Since" deg 'h(X) '< r.'the coefficient of Xr in this polynomial has the form 1 plus a nilpotent element, and so is a unit of R. Furthermore, the terms of degree exceeding r all come from g(x]h(x) and so have coefficients in /. " Also, the degree of this polynomial is at most max(r, degg(X)h(X}) < max(r, (n — r) + (r — 1)) < n. By induction, some associate of (1 — g ( X ) ) f ( X ) has degree r and has a unit as its leading coefficient. However, since g(X) is contained in the nilradical of Upf], we see that 1 — g(X) is a unit of -R[-X], and so f ( X ) also has such an associate. D Corollary 3. The following are equivalent. i) R is reduced. ii) For f ( X ) £ R[X], f ( X ) has a monic associate if and only if the leading coefficient of f ( X ) is a unit of R. Proof. This follows easily from Lemma 2.

D

Corollary 4. Let I be a nil ideal of R. For f ( X ) £ -R[-X], f ( X ) has a monic associate in R[X] if and only if f ( X ) modulo I has a monic associate in (R/I)[X]. Proof. One direction is trivial. For the other, suppose f ( X ) modulo I has a monic associate. Then the trivial direction shows the same is true of f ( X ) mod N with N the nilradical of R. Since R/N is reduced, Lemma 2(i) => (ii) shows f ( X ) modulo N has a unit leading coefficient. If the degree of f ( X ) modulo TV is r, we now see that condition (ii) of Lemma 2 holds for that r, and so f ( X ) has a monic associate. D Recall that R is connected if it only has the trivial idempotents 0 and 1 . We will first study connected rings, and then extend the results to rings with only finitely many idempotents. Lemma 5. The follouiing are equivalent. i) R is connected.

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ii) If f(X)g(X) is monic in R[X], then f(X) and g(X) have monic associates. iii) Iff(X)g(X) is monic with (f(X),g(X)) = R[X], then f(X) andg(X) have monic associates. Proof, ii) =£• iii): This is immediate. iii) => i): Suppose (iii) holds and let e £ R with e2 = e. Let f(X) = eX2 + (1 _ e)X -(l-e) and g(X) = (1 - e)X2 + eX - e. Using ef(X) = eX2 and eg(X) = e(X -1), we easily see (/(X), g(X)) = R[X}. Also, f(X)g(X) = X3-X2 is monic. By (iii), f(X) has a monic associate. By Lemma 2(ii), we see that the leading coefficient e of f(X) must either be nilpotent or a unit. As e = e2, we see e equals 0 or 1. i) =>• ii): Let N be the nilradical of R and let overbars denote images modulo N. If f(X)g(X) is monic, then f(X)g(X) is monic. If we can show both f(X) and g(X) have monic associates, then Corollary 4 shows the same is true of f(X) and g(X). Therefore, it will suffice to show that (i) =4> (ii) holds in P[X]. However, since idempotents in R lift to idempotents in R [B, Chapter III, Prop. 2.10], R is both reduced and connected. Therefore, we will simply assume that R is reduced and connected. That (i) =>• (ii) now follows from (i) =>• (ii) of the next lemma. D Lemma 6. The following are equivalent. i) R is reduced and connected. ii) // f(X)g(X) is monic, then the leading coefficients of f(X) and g(X) are units. iii) If f(X)g(X) is monic with (f(X),g(X)) = R[X], then the leading coefficients of f(X) and g(X) are units. Proof, i) => ii): Suppose f(X)g(X) is monic and R is reduced and connected. We will show both f(X) and g(X) have units for their leading coefficients. Let b ^ 0 be the leading coefficient of f ( X ) . Consider a prime ideal P and let * denote images modulo P. Since f(X)g(X) is monic, neither f*(X) nor g*(X) can be zero, and we see degf(X)g(X) = deg(f(X)g(X))* = deg f* (X) + deg g* (X) (since R* is a domain). Note that since R is reduced and 6 ^ 0 , there exists some prime P with b <£ P. For any such P, degf*(X) = degf(X), and we see degf(X)g(X) - degf(X) = degg*(X). We call that number n, noting that n > 0 and is independent of the choice of P not containing b. Letting c be the coefficient of Xn in g(X), c* is the leading coefficient of g*(X) and so we have b*c* = 1*. Thus (be)*2 = (be)* whenever b £ P. On the other hand, if b € P, then clearly we also have (be)*2 = (be)*. That is, be is an idempotent modulo every prime ideal of PL. As PL is reduced, we see that be is an idempotent in PL. As R is connected, either be = 0 or be = I . As we know there is a prime P not containing b, and for that prime we already have (be)* = 1*, we see be ^= 0. Thus be = 1, and so b is a unit. Thus the leading coefficient of f(X) is a unit, and so the same must be true ofg(X). ii) =>• iii): This is trivial. iii) => i): Suppose (iii) holds. By Lemma 5(iii) =4- (i) (which has been proven), we see R is connected. It remains to show PL is reduced. Suppose to the contrary

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that 0 ^ a 6 R with a 2 = 0. Let f ( X ) = aX + 1 and g(X) = -aX2 + (1 + a)X - (1 + a). As 1 + a is a unit in R, we easily see ( f ( X ) , g ( X ) ) = R\X}. Now f(X)g(X] = X — (I + a) is monic but neither factor has a unit for its leading coefficient, contradicting (iii). D The equivalence of (i) and (ii) in Lemma 6 is also proved in [GH]. (A computational approach to the results in that paper is given in [Y].) The next corollary is also given in a remark in [HL], using different arguments from the present ones. Corollary 7. Let R be connected. Suppose p(X) and q(X) are in R[X] such that p(X)q(X] is monic. Then p ( X ) and q(X) have monic associates p!(X) andq'(X), and furthermore p ' ( X ) q ' ( X ) = p ( X ) q ( X ) . Proof. By Lemma 5(i) =>• (ii), p(X) and q(X) have monic associates, say p'(X) and q'(X). Clearly p'(x)q'(X) is a monic associate ofp(X)q(X). However, as p(X)q(X) is monic, it is a monic associate of itself. By Lemma I , p ' ( X ) q ' ( X ) = p ( X ) q ( X ) . D Corollary 8. Let R be connected and let f ( X ) £ R[X] "c nonconstant and monic. Then f ( X ) is irreducible if and only if there is no factorization of f ( X ) into two non-constant monic polynomials. Proof. This follows easily from the previous corollary.

D

Definition. Let 6 be a non-unit element of R. We say that b is pseudo-prime if whenever b divides cd with (c, d) = R, then b divides either c or d. Remark. Let g(X] be a nonconstant monic polynomial in R[X]. Using a degree argument, it is not hard to verify that if g(X] is pseudo-prime then it is pseudoirreducible. The next crucial lemma shows that if R is a connected ring, then the converse holds. (Also see Corollary 20.) Lemma 9. Let R be connected, and let g(X) be a nonconstant monic polynomial. The following are equivalent. i) g(X] cannot be factored into two comaximal nonconstant monic polynomials. ii) g(X) is pseudo-irreducible. iii) g(X) is pseudo-prime. Proof. The equivalence of (i) and (ii) follows easily from Corollary 7, while (iii) implies (i) is straightforward. Now suppose (ii) holds and suppose g(X) divides h ( X ) k ( X ) with ( h ( X ) , k ( X ) ) = R[X]. Let / = (g(X),h(X))R[X] and J = ( g ( X ) , k ( X ) ) R [ X } . As I and J are clearly comaximal, we see g(X)R[X] C / n J = IJ C g(X)R[X], so that JJ = g(X)R[X}. Therefore, I and J are invertible (i.e., rank 1 projectives). If S consists of all powers of g ( X ) , then Is = Js = R[X]s are free. Therefore (as pointed out in [HL]), [Ku, Chapter IV, Theorem 3.14] shows that / and J are free and hence principal. Say / = p(X)R[X] and J = q(X)R[X]. Since p(X)q(X)R[X] = g(X)R[X], there are polynomials u(X) and v(X) with g(X) = p ( X ) q ( X ) u ( X ) and p(X}q(X) = g ( X } v ( X ) , so that g(X)v(X}u(X] = g ( X ) . As g(X) is monic, we have v(X)u(X) = 1, making u(X) a unit of .RLY]. It

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does no harm to replace q(X) by q(X)u(X), so that we now have g(X) = p(X)q(X). Since p(X] and q(X] are comaximal (since / and J are), and since (ii) holds, one of p(X] or q(X) must be a unit of R[X\. Thus one of / or J equals R[X]. Therefore g(X) is comaximal to one of h(X) or k(X), and so divides the other. D Before concluding our study of connected rings, we must first explore direct sums. We note the trivial fact that R can be written as the direct sum of finitely many connected rings if and only if R has only finitely many idempotents. Lemma 10. Suppose A and B are rings with R ~ A ® B, and for g € R let the image of g be (5/1,55). Then g is pseudo-irreducible if and only if one of gA or gB is a unit and the other is pseudo-irreducible. Proof. (gAiQB*) = G?A, 1)(1,5.B) and these two factors are comaximal. If (gA,9s) is pseudo-irreducible then one of (gA, 1) or (1, gB) is a unit. By symmetry, suppose (gAi 1) is a unit. Then gA is a unit. We claim gB is pseudo-irreducible, first noting that since by definition, (gAi 93) is a non-unit, we must have that gB is a non-unit. Write gB = cd with c and d comaximai in B. That gives (gAi9B) = (dA, c)(I,o). As this is a comaximal factorization, one factor must be a unit, so one of c or d is a unit, proving the claim. Conversely, suppose gA is a unit and gg is pseudo-irreducible (the other case being similar). As (gA, 1) is a unit, it will suffice to show (1,55) is pseudo-irreducible. Suppose (1,5s) = (u,r)(v,s) with those factors comaximal. Clearly u and v are units in A, and therefore we must have r and s comaximal in B. As gB = rs and as gs is pseudo-irreducible, one of r or s is a unit, and so one of (u, r} or (v, s) is a unit. D Lemma 11. Suppose R = RI ® R% © • • • ® Rn with each Ri a ring. Let f ( X ) be a nonconstant monic polynomial in R\_X\. If f(X) has a factorization into pseudoirreducible polynomials (which we do not require to be pairwise comaximal), then the number of factors in that factorization is at least n. Proof. We induct on n, the case n = 1 being trivial. Suppose n > 1, and let A = Ri and B = R2 © • • • 0R n . Thus R ~ A® B, leading to R[X] ~ A[X] ®B[X}. For h e R[X], let the image of h in A[X] ® B[X] be (hA,hB). Suppose / is a nonconstant monic polynomial in R[X], and suppose / = Yidi '1S a factorization of / into pseudo-irreducible polynomials gi} having k factors. We must show k > n. We have /A — YlgiA and fB = YldiB- By Lemma 10, for each i we know that one of giA or giB is a unit and the other is pseudo-irreducible. Suppose that exactly r of the giA are pseudo-irreducible and exactly s of the 5^5 are pseudo-irreducible. Then r + s = k. Since under the isomorphism R ~ A 0 5, the image of 1 is (1,1), we see that fA € A[X] and fB e B[X] are both nonconstant monic polynomials. By induction, we see that r > 1 and s > h — 1. Thus k > n. D Corollary 12. i) Suppose R has infinitely many idempotents. Then no nonconstant monic polynomial in R[X] can be factored into pseudo-irreducible polynomials.

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ii) Suppose R has only a finitely many idempotents. Then R can be written as a direct sum of finitely many (say n) connected rings, and if f(X] is a nonconstant monic polynomial in R[X] such that f ( X ) has a factorization into pseudoirreducible polynomials, then that factorization involves at least n factors. Proof. If R has infinitely many idempotents, then for any n > 1 there is a decomposition of R as the direct sum of n rings. As n is arbitrary, Lemma 11 shows no nonconstant monic polynomial in R[X] has a factorization into pseudo-irreducible factors. This proves (i). As for (ii), we previously noted that R can be written as a direct sum of finitely many connected rings, and the rest follows from Lemma 11. D The next two results will complete our study of connected rings. Theorem 13. The following are equivalent. a) R is a connected ring. b) Any monic nonconstant f ( X ) £ R[X] has a unique (up to order) factorization as f ( X ) = f i ( X ) f 2 ( X ) - - - f n ( X ) with the f i ( X ) pairwise co-maximal nonconstant monic polynomials such that no f i ( X ) can be factored into two comaximal nonconstant monic pelynomials. c) Any monic nonconstant f ( X ] e R[X] has a factorization into monic nonconstant irreducible polynomials. d) There exists a monic nonconstant irreducible polynomial in R[X]. e) There exists a monic nonconstant pseudo-irreducible polynomial in R[X]. Proof, a) => b): Suppose (a) holds and let f ( X ) £ R[X] be nonconstant and monic. A degree argument makes it clear that a factorization of f ( X ) as in (b) always exists. Suppose there is a sdcond such factorization, say f ( X ) = g i ( X ) g z ( X ) • • • gm(X). By (a) and Lemma 9, the f i ( X ) and Qj(X] are pseudo-prime. Thus f i ( X ) divides some gj(X), and re-ordering, we may assume f i ( X ) divides g\(X). We similarly know g i ( X ) divides some f k ( X ) . Thus f i ( X ) divides f k ( X } . As the various f i ( X ) are pairwise comaximal, we must have k = 1. Thus f i ( X ) and g i ( X ) divide each other. Both being monic, they must be equal (using a degree argument). Canceling and using induction completes the proof. b) =4> a): Suppose (b) holds, and let e be an idempotent in R. We see that X(X — 1) = (X — e)(X — l + e) are both comaximal factorizations (using (1 — 2e) 2 = 1). As each of these four factors is linear, none of them can be further factored into two nonconstant monic polynomials. By (b) we must have e = 0 or e = 1. a) =4> c): Let f ( X ) be a monic nonconstant polynomial in .R[A"]. A degree argument makes it clear that f ( X ) can be factored into a product of nonconstant monic polynomials such that none of those factors can be further factored into the product of two nonconstant monic polynomials. By Corollary 8, each of those factors is therefore irreducible. c) => d) => e): These are immediate.' e) => a): Let g(X) be a monic nonconstant pseudo-irreducible polynomial in R[X}. Part (i) of Corollary 12 shows that R must have finitely many idempotents, and part (ii) of that corollary then shows R must be connected. D

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Remark. It is shown in [AZ, Theorem 3.2] that if a domain R is not integrally closed, then there exist irreducible but non-prime monic polynomials in .R[-X]. (Thus, the "non-pseudo" analogue of Lemma 9 ii) =>• iii) fails to hold.) It easily follows that for such R, we do not have uniqueness in part c of Theorem 13. (Lemma 21 (b) and Corollary 22 (i) below describe those rings R for which we do have uniqueness in part (c).) Corollary 14. Let R be connected and let f(X) € R[X] be nonconstant and monic. Then f ( X ) has a unique complete comaximal factorization, namely the factorization given in Theorem 13(6). Proof. Suppose f(X) = fi(X)fz(X) • • • fn(X) is the factorization given by Theorem 13(b). Obviously this is a comaximal factorization. In fact, by Lemma 9(i) => (ii) we see each fi(X) is pseudo-irreducible, and so we have a complete comaximal factorization of f ( X ) . Next suppose f ( X ) = gi(X)g2(X} • • • gm(X) is also a complete comaximal factorization of f ( X ) . By Lemma 5(i) =>• (ii), each gj(X) has a monic associate g'^X). Clearly Rg^X) is a monic associate of Ugj(X) = f ( X ) . As f(X) is a monic associate of itself, Lemma 1 shows Hs^^O = /PO- Clearly the g'j(X) are pairwise comaximal (since the gj(X) are) and no g'j(X) can be factored into two comaximal nonconstant monic polynomials (since gj(X) cannot be factored into two comaximal non-units). Thus the uniqueness of the factorization given by Theorem 13(b) shows that flSj'^) an<^ II/i(^0 are identical up to order. Since 9j(X) and g'j(X) differ by a unit, we see that Y[dj(X) and II/j(^0 are identical up to order and units, showing the complete comaximal factorization f(X) = Y[ fi(X) is unique. D We now use our knowledge of connected rings to deal with rings having only finitely many idempotents. Lemma 15. Let A and B be commutative rings. Then the following are equivalent. a) In A® B, the product Y[(ai> M *5 a complete comaximal factorization. b) For each i, one of ai or hi is a unit and the other is pseudo-irreducible. Also, the various a; are pairwise comaximal, as are the various &;. Proof, a) =>• b): Suppose (a) holds. As each (aj,6j) is pseudo-irreducible, the first statement in (b) is immediate from Lemma 10. Next suppose that a^ and a,- not comaximal. Then some maximal ideal M of A contains both of them, and so M® B contains (a^frj) and (a,j,bj). By (a), we must have i = j. This proves half of the second statement of (b), and the other half is symmetric. b) =4> a): Suppose (b) holds. By Lemma 10, each (a;, 6;) is pseudo-irreducible, and so we must only show these factors are pairwise comaximal. Suppose (ai,fc») and (a,j,bj) are not comaximal in A © B. We consider the case that a maximal ideal of the form M ® B contains both,of them (the other case being symmetric). In this case M contains both
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factorizations if and only if (a,b) 6 A (B B has a complete comaximal factorization. Furthermore, (a, 6) has a unique complete comaximal factorization if and only if a and b both have unique complete comaximal factorizations. Proof. First suppose a = f7 Oi (1 ii): Suppose (i) holds. Then we can write R — RI ® R% © • • • ® Rn with each Ri a connected ring. We see R[X] = Ri[X] © R2[X] 0 • • • ® Rn[X}. Let / be a nonconstant monic polynomial in .R[X], and let its image be (/i, /2, • . • , /n)- Let Fi be the n-tuple whose f-th entry is Jfi and having 1 in all other entries. Then we can factor / as Yl^i- AS the image of 1 € R is (1,1,..., 1), we see that each fi is a nonconstant monic polynomial in #i[.X"]. By Theorem 13(a) =^> (c), each fi can be factored into a product of nonconstant monic irreducible polynomials in Ri [X]. Thus each Fi is a product of factors of the form (!,...,!, g,!,...,!) with the g

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appearing in the i-th entry and with g monic and irreducible in Ri[X]. An easy exercise shows all such factors of Fi are irreducible, and so we have produced an irreducible factorization of / = FJ Fi. ii) => iii): This is immediate. iii) => i): Suppose g(X) G R[X] is monic, nonconstant, and has a factorization into irreducible polynomials. Then that same factorization factors g into pseudoirreducible polynomials. By Corollary 12(i), R has only finitely many idempotents. i) =>• v): Suppose (i) holds and let / be a nonconstant monic polynomial in J?[X]. With notation as in the proof of (i) => (ii), we have / = (/i, /2, • • • , fn) with each fi a nonconstant monic polynomial in Ri[X]. By Corollary 14, each /, has a unique complete comaximal factorization, and so by Corollary 16, / has a unique complete comaximal factorization. Thus (i) implies (v). v) => iv) =>• vi): These are immediate. vi) =>• i): This is by Corollary 12(i). D Theorem 18. Suppose R has only finitely many idempotents, so that R can be written as a direct sum of finitely many (say n) connected rings. Let f be a monic polynomial of degree d > 0 in R[X], and let f = /i/2 • • • fm be a factorization of f into pseudo-irreducible polynomials (such a factorization existing by Theorem 17(i) => ( i i ) ) . Then n < m < nd. Moreover, there are units Ui € R[X] for 1 < i < m, such that f = Il(wi/i) and such that the leading coefficient of each Uifi is an idempotent. Proof. Lemma 11 shows n < m. For the rest, we induct on n. For the case n = 1, the existence of the various Ui follows from Corollary 7 (inducting on m), and in this case we in fact have each Uifi monic. As / is the product of m monic polynomials Uifi, we must have'm < d = nd, completing the case n = 1. For n > 1, let A = RI and B = R^ © • • • © Rn, so that R[X] ~ A[X] ®B[X}. Letting the image of k(X) e R[X] be (kA,kB), we have ( f A , f s ) = T l ( f i A , f i B ) , so that fA =HfiA and /B = II f i B - As each ft is pseudo-irreducible, Lemma 10 shows that for each i, one of fiA or fiB is a unit and the other is pseudo-irreducible. Suppose the numbering is such that faA is pseudo-irreducible for 1 < i < r while fiB is pseudo-irreducible for r + 1 < i < m. Since FJ fiB is a factorization of fs (which is monic of degree d in -B[X]) into a product of r units and m — r pseudo-irreducible polynomials, induction shows m — r < (n — l)d. Similarly considering fA = Yl/iA, we see that r < Id. Thus m < nd as desired. We now produce the desired units Ui € R[X]. We have fA = F| fiA, and in this product, the first r factors are pseudo-irreducible polynomials while the last m — r are units in A[X]. By induction, there are elements gi e A[X] for 1 < i < r, such that gi = vtfiA for some unit Vi in A[X], the leading coefficient of each ^ is an idempotent of A, and fA = gig^- • • gr. Similarly, for r + 1 < i < m, there are elements hi € B[X] with hi — Wifig for some unit u>i in B[X], with the leading coefficient of each /i» being an idempotent of B, and with fg = frr+i/V+2 • • • hm. We have ( f A , fB) = (91,1)(ff2,1) • • • (ffr, l)(t /ir+i)(l, hr+2) • • • ( ! , hm). Forl
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(fi^,Wi) which is a unit satisfying u'^fi^, f i s ] = (l,^t)- Under the isomorphism R[X] ~ A[X] ® B[X], let u\ correspond to the unit Ui & R[X}. We know ( f i A , fiB) corresponds to /j. Thus Uifi corresponds to (gi, 1) for 1 < i < r, and to (1, hi] for r + 1 < i < m, and so a previous equation shows / = (111/1X^2/2) • • • (umfm)It remains to show the leading coefficient of each Uifi is an idempotent in R. We will do this for 1 < i < r, the other case being symmetric. Suppose Uifi has degree t and leading coefficient 6j. Since tij/i corresponds to (5,,!), we see that the coefficient of the degree t terms in gi 6 A[X] and 1 £ B[X] are fc^ and 6,3 respectively. These cannot both be zero, since 6; ^ 0. If i > 0, then clearly b^ = 0, so that biA ^ 0, showing that hi A is the leading coefficient of gi. However, we know that leading coefficient is an idempotent in A. As bi corresponds to (fr^O), we see bi is an idempotent in R, as desired. On the other hand, if t = 0, then Uifi = bi corresponds to (<^, 1) = (biA, !)• As again we have 6^ an idempotent in A, again we see that foj is an idempotent in R. D We illustrate some of the ideas in the previous proof with an easy example. Example. Let R = Z6 ~ Z2 © Z3 and let f ( X ) = X2 + 3X + 2. Under the isomorphism Z6[X] ~ Z2[X] @ Z3[X}, we have f ( X ) -» (X2 + X,X2 + 2) = (X,l)(X + l,l)(l,X + l)(l,X + 2). Each of these four factors is clearly irreducible. We see that under our isomorphism, 3X + 4 —> (X, 1), 2>X + 1 —> (X + 1,1), 4 ^ + 1 ->• (1,X + 1), and 4X + 5 -+ (l,X + 2). Thus in Z 6 [X], we have f ( X ) = X2 + 3X + 2 = (3A" + 4)(3X + 1)(4X + 1)(4X + 5) is an irreducible factorization. Note that 3 and 4 are idempotents in Zg. Since Z% and Za are fields, Corollary 22(b) below shows that this is the unique irreducible factorization of X2 + 3X + 2. Corollary 19. Let n > 1 be an integer. The following are equivalent. i) R is the direct sum of n connected rings. ii) n is the least integer such that there exist n irreducible polynomials in R[X] whose product is monic. iii) n is the least integer such that there exist n pseudo-irreducible polynomials in R[X] whose product is monic. Proof. Suppose R is the direct sum of n connected rings. By Lemma 11, the product of fewer than n pseudo-irreducible polynomials cannot be monic. It follows that the product of fewer than n irreducible polynomials cannot be monic. Therefore (i) implies half of both (ii) and (iii). On the other hand, Theorem 17(i)=»(ii) shows that X has a factorization into irreducible factors (which are also pseudo-irreducible), and Theorem 18 shows the number of factors in that factorization is exactly n. Therefore (i) implies both (ii) and (iii). Next, suppose either (ii) or (iii) holds. In either case, we are told there exist n pseudo-irreducible polynomials whose product is monic. By Corollary 12(i), R has only finitely many idempotents. Therefore, we can write R as the direct sum of n' connected rings. As we already know (I) implies both (ii) and (iii), we see that either (ii) or (iii) holds for both n and n'. Therefore, n = n' and so (i) holds. D

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Remark. The preceding three results are of particular interest when applied to the polynomial X. That case was done in [AV, Theorem 6.4], which says X is a product of n irreducible polynomials if and only if R is the direct sum of n connected rings. Corollary 20. Let g(X) e R[X] be monic. Then g(X) is pseudo-irreducible if and only if g(X) is pseudo-prime. Proof. If g(X) is pseudo-prime (and monic), it is easily seen g(X) is pseudoirreducible. For the converse, suppose g(X) is pseudo-irreducible. By Corollary 19(iii) =$> (i) (with n — I ) , we see that R is connected. By Lemma 9, g(X) is pseudo-prime. D Lemma 21. Let R be a commutative ring. a) The following are equivalent. i) R is a domain. ii) Every monic polynomial of degree 1 or 2 has a unique factorization into irreducible polynomials which are monic. b) The following are equivalent. i) R is an integrally closed 'domain. ii) Every nonconstant monic polynomial in R[X] has a unique factorization into irreducible polynomials which are monic. Proof, a) i) => ii) is clear, using that over a domain, a degree 2 polynomial has at most two roots. Now suppose (ii) holds. Then any X + c £ R[X] must be irreducible. Suppose (i) is false, and let ab = 0 with a and b nonzero elements of R. Then (X + a)(X,+ b} = X(X + a + b), which contradicts (ii). b) If (ii) holds, then part (a) shows R is a domain. Thus, we need only prove the result when R is a domain. That follows from [AZ, Theorem 3.2] (and is also shown in [M]). D Corollary 22. Let R be a commutative ring. a) The following are equivalent. i) R is a direct sum of finitely many domains. ii) Every monic polynomial of degree 1 or 2 in R[X] has a unique irreducible factorization. b) The following are equivalent. i) R is a direct sum of finitely many integrally closed domains. ii) Every nonconstant monic polynomial in R[X] has a unique irreducible factorization. Proof. The proof of (a) is essentially the same as that of (b), merely restricting the polynomial g in what follows to come from the set of monic polynomials having degree 1 or 2. We now prove (b). ii) => i): Suppose (ii) holds, but (i) does not. By Corollary 12(i), we see that R has finitely many idempotents, and so we can write R ~ RI ® R-% ® • • • ® Rn with each Ri connected. As (i) fails to hold, some Ri is not an integrally closed

Factorizations of Monic Polynomials

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domain. Without loss, say it is R±. Let g G -Ri[X] be a nonconstant monic polynomial. By Theorem 13(a)=*>(c), g has a factorization into monic irreducible factors. If such a factorization was unique for all such g, then Lemma 21 (b) would show RI is an integrally closed domain, giving a contradiction. Therefore, for some such g, there exist two distinct irreducible factorizations of g. Consider such a (monic) g, and let g = g\g-2.---gm be one of those irreducible factorizations. Suppose degg = d, and in Ri[X] 0 • • • © Rn\X], consider (g,Xd,Xd,.. .,Xd) = [life,!,. •-,!)][(!, X , l , . . . , l ) d ] - - • [ ( ! , ! , . . . , l , X ) d } . Since each Ri is connected, Corollary 8 shows X is irreducible in Ri[X], and we know each gj is irreducible in J?i[J5f]. It easily follows that the preceding is an irreducible factorization of (g,Xd,..., Xd). However, since we can replace g = Yldj with a different irreducible factorization of g in .Ri[X], we can also replace our irreducible factorization of (g, Xd,..., Xd) with a different one. Thus (g, Xd,..., Xd) has two irreducible factorizations. Now suppose under the isomorphism with /Z[Jt], (g,Xd,... ,Xd) corresponds to / 6 R[X]. Then / is a degree d polynomial which is monic (since its leading coefficient corresponds to (1,1,...,!)). A s / has two irreducible factorizations, we have contradicted (ii). i) => ii): Suppose (i) holds, and let / be a nonconstant monic polynomial in B[X]. By Theorem 17(i)=>(ii), we know / has a factorization into irreducible polynomials. We must show it is unique. We transfer attention to the isomorphic ring RI [X] © • • • © Rn [X], letting the image of / be (fi, / 2 , . . . , fn) • Now an arbitrary n-tuple is irreducible if and only if every entry except one is a unit, while the exceptional one is irreducible in the appropriate Ri[X}. Consider any irreducible factorization of (/i,... ,/«). Fixing some i, the product of the i-th entries of all the factors in that factorization constitutes a factorization of fi into a product of units and irreducible elements in Ri [X]. As Ri is an integrally closed domain, Lemma 21 (b) shows fi has a unique irreducible factorization (since over a domain, factors of a monic polynomial can always be taken to be monic). It easily follows that (/!,..., /„) has a unique factorization into irreducible factors. D Remark. We should point out a trivial discrepancy between the terminology of this paper and that of [MS]. In section 1 of [MS] we developed a concept analogous to UFD's. Now in a UFD, irreducible elements are traditionally defined to be nonzero non-units. Hence, in [MS], we took pseudo-irreducible elements to be nonzero non-units. However, in the present paper, we dropped the "nonzero" restriction, allowing 0 to possibly be pseudo-irreducible. The reason for this change is that it eases certain technical matters. For example, if we do not allow 0 to be pseudoirreducible, then Lemma 10 fails to hold for the pseudo-irreducible element (1,0) G Z © Z. That failure would complicate later arguments. As the main results of both [MS] and the present paper only concern nonzero elements, this discrepancy in terminology has no effect, other than easing those technical matters. References [AV]

D. D. Anderson and S. Valdes-Leon, Factorization in commutative rings, Rocky Mountain J. Math., 26 (1996) 439-480.

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[AZ]

S. McAdam and R. G. Swan

D. D. Anderson and M. Zafrullah, t-Invertibility HI, Comm. in Algebra, 21 (1993) 11891201. [B] H. Bass, Algebraic K-Theory, Benjamin, New York, 1968. [GH] R. Gilmer and W. Heinzer, On the divisions of monic polynomials over a commutative ring, Pac. J. Math., 78 (1978) 121-131. [HL] W. Heinzer and D. Lantz, Factorization of monic polynomials, Proc. Amer. Math. Soc. (to appear). [Ku] E. Kunz, Introduction to Commutative Algebra and Algebraic Geometry, Birkhauser, Boston (1985). [M] S. McAdam, Unique factorization of monic polynomials, Comm. in Algebra, 29 (2001) 4341-4343. [MS] S. McAdam and R. G. Swan, Unique comaximal factorization, (manuscript). [Y] I. Yengui, An algorithm for the divisions of monic polynomials over a commutative ring, (manuscript).

Monotone Complete Ordered Fields, Generalized Power Series, and Integer Parts Mojtaba Moniri Institute for Studies in Theoretical Physics and Mathematics (IPM), Tehran, Iran mojmonSipm.ir, monirijnQmodares.ac.ir

Jafar S. Eivazloo Department of Mathematics, Tarbiat Modarres University, Tehran, Iran [email protected], eivazl.jSmodares.ac.ir Abstract. We show existence of monotone complete ordered fields without integer parts. A criterion for monotone completeness of ordered fields of generalized power series, which could be of independent interest, is first established.

1. Introduction A discrete subring (including 1) of an ordered field is said to be an integer part (IP for short) in there whenever it approximates from below every element of the field within 1. There are ordered fields with no IP's, see [1]. On the other hand, it was proved in [4] that every real closed field has an IP. It is known and easy to see that ordered fields may have many IP's. Viewing the property of being real closed for an ordered field as being definably complete (i.e., all those-nonempty proper-cuts in the ordered field which are definable by polynomial inequalities there being realized), we are interested to determine whether ordered fields satisfying other (restricted and now topological) completeness notions of interest (obtained by weakening the full Dedekind completeness which itself characterizes the real field) must necessarily posses an IP or not. In this paper we construct monotone complete ordered fields with no IP's. Monotone complete ordered fields were introduced in [3]. They are ordered fields with no bounded strictly increasing divergent functions. In such ordered fields, 2000 Mathematics Subject Classification. 12J1£, 13F25, 13J05, 54H13. Key words and phrases. Ordered Field, Generalized Power Series, Monotone Complete, Integer Part.

425

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M. Moniri and J. S. Eivazloo

those nontrivial cuts which do not have a least upper bound in the field, are not traversed by strictly increasing nets of length equal to the cofinality of the field either. It is clear that there are no monotone complete ordered fields of countable cofinality except (those isomorphic to) R, in particular the IP-less ordered field constructed in [1] is not monotone complete. On the other hand, as implied by [6, Cor. 2.7], there are monotone complete ordered fields of any uncountable cofinality and so there exist plenty of monotone complete ordered fields not isomorphic to E. We will be dealing with ordered fields of generalized power series and will give a convenient monotone completeness criterion for them. Recall that for an ordered field F and ordered Abelian group G, the ordered field of generalized power series with exponents in G and coefficients in F is: [{FG}} = | ^ agt9 the support {g \ ag ^ 0} of ]P agtg is well ordered i. ^-n^n ncr: ' It is equipped with the natural addition, multiplication, and order defined by ]T\eG, os£5 > 0 if ago > 0, where go is the minimum of support of X^oeG a s^ s A well known necessary and sufficient condition for [[-FG]] to be real closed is that G be divisible and F be real closed, see [5, 6.10]. Observe that cf([[F G ]]) = cf(G) (where cf denotes cofinality). It is already known, see [2] (by the way we plan to present a general straightforward proof of this fact elsewhere), that all ordered fields of generalized power series satisfy the weaker notion of Cauchy-Scott completeness. Scott complete ordered fields were introduced in [7]. They are ordered fields with no divergent Cauchy nets of length equal to the cofinality of the field. We will consider the ordered field constructed in [1] as the field of coefficients and choose an appropriate ordered group of exponents so that the resulting power series field is monotone complete. This field, like its residue field, will fail to have an IP. The Laurent series over the rational ordered field, respectively over the ordered field constructed in [1], will appear earlier in the paper. They are both Scott complete monotone incomplete and posses, respectively fail to have, an IP. 2. A Monotone Completeness Criterion Theorem 2.1. For every ordered field F and nontrivial (totally) ordered Abelian group G, [[FG]] is monotone complete if and only ifG does not contain any bounded strictly monotone nets of length cf(G) and F does not contain any strictly monotone nets of length cf(G) (no matter whether bounded or not). Proof. Let cf(G) = A. Then, as mentioned before, cf([[F G ]]) = A also. (Only If) Assume for the purpose of a contradiction that (gi)i<\ is a bounded strictly decreasing net in G and so (t Si )i
Monotone Complete Ordered Fields

427

and by monotone completeness assumption, lim^A^it) and so lim^A^) would exist in [[T*1*3]]. But this can not happen since no net of elements in F converges in [[FG]} unless it is eventually stabilized (as the latter has F-infinitesimals). (If) Suppose (o"i)i i)(Vl < j}(gi i)(VZ < j ) ( y i , i = gij A a/^ = &i,i)- We show that there exists i < A such that for all 7 > i, we. have g/ ifc = gttK and o/ jft = aj iK . As A is a regular cardinal, there exists rj < A such that (V7 > 7?)(V7 < «)(5/,; = g^,i A a/ t ( = ar,ti). So the net (gi,K)r,0 . Now if (gl,K)ri C) ( (, we have (V7 > i) a/ jK = a i)K ) (in addition to gi,K — gi,*)- This finishes the Claim. Let a — 'Sj<\aijjtg'-3':i, where for each j < A, ij is a witness ordinal for the existential quantifier in the Claim above. Note that (Vji,j2 < A)(ji < jz —> fli^ji < 9iJ2,h < 9iJ2,h) and so (9ij,j)j<\ is strictly increasing. Hence a e [[FG]}. We show that lim.j0, there exists j < A such that g^^j > g. For all i > i j , by the Claim above, we have

3. Monotone Complete Ordered Fields with No IP's Proposition 3.1. For any ordered field F and totally ordered nonzero Abelian group G, the generalized power series field [[FG]] has an integer part if and only if F does so. Proof. The if part is well known (see [4]): if 7 is an integer part of F, then [[FG ]] (J) 7 is an integer part for [[T710]]. The only if part is also routine. D

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M. Moniri and J. S. Eivazloo

Example 3.2. (i) The Laurent series ordered field [[Qz]] with rational coefficients has an integer part. (ii) Let B be the ordered field constructed in [1] which has no integer parts. Then [[Bz]] has no integer parts either. The two ordered fields above are not monotone complete as the coefficient fields Q and B (like any other ordered field) have strictly increasing nets of length cf(Z) = u>. Replacing the ordered group of exponents there by a direct lexicographically ordered sum of continuum many copies of Z's, the resulting power series fields will still be monotone incomplete. In these cases, the exponents group has bounded strictly increasing nets like (
Monotone Complete Ordered Fields

429

[2] I. Kaplansky, Maximal Fields with Valuations, Duke Math. J. 9 (1942) 303-321. [3] H. J. Keisler, Monotone Complete Fields, in: Victoria Symposium on Nonstandard Analysis (A. Kurd and P. Loeb, eds.), Lecture Notes in Mathematics 369, pp. 113-115, Springer-Verlag, Berlin, 1974. [4] M. H. Mourgues and J. P. Ressayre, Every Real Closed Field Has an Integer Part, J. Symbolic Logic 58 (1993) 641-647. [5] P. Ribenboim, Fields: Algebraically Closed and Others, Manuscripta Math. 75 (1992) 115-150. [6] J. H. Schmerl, Models of Peano Arithmetic and a Question of Sikorski on Ordered Fields, Israel J. Math. 50 (1985) 145-159. [7] D. Scott, On Completing Ordered Fields, in: Applications of Model Theory to Algebra, Analysis, and Probability (W. A. J. Luxemburg, ed.), Holt, Rinehart and Winston, New York, 1969, pp. 274-278.

On Distance between Finite Rings Lkhangaa Oyuntsetseg Department of Mathematics (Doctor Course), Okayama University, Okayama 700-8530, Japan

Takao Sumiyama Center of General Education, Aichi Institute of Technology, Toyota 470-0392, Japan Abstract. By making use of structure constants, we can determine all finite rings which have a given finite Abelian group as their additive group. In this paper, we shall consider a sort of distance function between finite rings to determine wjiether they are isomorphic or not.

In what follows, by a finite ring we mean an associative ring (not necessarily with identity element) consisting of only finitely many elements. In [2], R. A. Beaumont presented a method to construct all rings which have a given Abelian group as their additive group. In [1], K. Baumgartner solved the problem to determine whether two rings with the same additive group are isomorphic. Following these ideas, in [5], J. Wiesenbauer presented a method to determine all finite rings which have a given finite Abelian p-group as their additive group. When R is a finite ring, the number of elements of R is called the order of R. A finite p-ring is a finite ring whose order is a power of a prime p. It is easy to see that a finite ring is a direct sum of finite p-rings. So, in what follows, we shall consider only finite p-rings. Let (x) denote the cyclic group generated by x. By C(m) we denote the cyclic group of order m. Let R be a finite p-ring whose additive group is where (xj) = C(p&i) (1 < i < n), and 1 < e\ < e2 < • • • < en is a nondecreasing sequence of positive integers. We can write n

(1)

XiXk = Y^oujkXj

(1 < i, k < n),

1991 Mathematics Subject Classification. Primary 16P10; Secondary 16Z05. Key words and phrases, finite rings, structure constants. 431

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L. Oyuntsetseg and T. Sumiyama

where a^k are integers such that 0 < aijk < pej - 1

(2)

(l
Then, by [5, Satz 2], it holds that (3)

aijk = 0 mod p^~e"

for 1 < k < j < n,

(4)

ej ei

for 1 < i < j < n,

aijk = 0 mod p '

and n

(5)

n

y~^ar-fci a fcjs = '^~\O'-iksarjk fc=l

mod p6j

(1 < i,j,r,s < n).

fc=l

Conversely, let p be a prime, and 5 ' ' ' © (£„)

be an additive group, where (xi) = C(pei) (1 < i < n), and 1 < e\ < 62 < • • • < &„ is a nondecreasing sequence of positive integers. If a^k (1
G= (Xi) ® (x2) ® • • • ® (xn)

where (xt) = C(pei)t (1 < i < n). Note that the exponent of G is p£n. A set {aijk}i Sn be the natural homomorphism. An element (ty) of 5n is said to be non-singular, if ((tij)) is an invertible element of Sn. When a = {aijk}i
On Distance between Finite Rings

433

isomorphic if and only if there exists a non-singular element (
(7)

n

n

]P aijktjs = ^2 ^2 tijtkra'jsr j-l

mod pe°

(1 < i,fc,s < n).

j = l r=l

Theorem 1 enables us to determine whether two ring structures are isomorphic or not. In this way, for a given positive integer N, we can construct all finite rings whose order is N (see [3]). However, it wastes time to compute the equation (7), as the number of nonsingular elements of Sn is very large. It seems that a sort of distance function is useful for economy of time. Let R denote the real number field. Let S be the set of all ensembles of structure constants for G. A function d: S x £ —> R is called a distance for G, if the following (i) - (iv) are satisfied. (i) (ii) (iii) (iv)

d(a, a') > 0 for any a, a' € S. d(a, a) = 0 for any a € S. d(a,a') = d(a',a) for any a, a' G S. If d(a, a') > 0, then Ga and G a < are not isomorphic.

Note that, in our terminology, d(a,a') = 0 does not imply a = a'. The triangular inequality is not necessarily satisfied, either. A distance d for G is said to be be faithful, if the following (*) is satisfied. (*)

If d(a, a') = 0, then GQ and G a < are isomorphic.

As a matter of fprm, at least one faithful distance always exists. For, we can define dQ: S x S -> R by . j0 do(a, a ) = < I 1

(if G0 and G a / are isomorphic) (if GO, and Ga> are not isomorphic).

However, it is obvious that such a distance cannot result in time saving. In what follows, we shall construct a distance, which can result in time saving, though it is not faithful. An element x of a ring R is said to be quasi-regular, if there exists an element y e R such that x + y — xy = 0. Proposition 2. Let V be the set of all vectors ( b i , b 2 , - - - ,bn) whose entries are non-negative integers such that 0 < 6j < pei — 1 (1< i < n). (1) Let gi,g2,-• • ,
9i = Y^ Tiixi i=i

(l - 1).

Let G' be the additive subgroup of G generated by gi,g2, • • • ,gd- Then there exists a one-to-one correspondence between G' and the set of all vectors

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L. Oyuntsetseg and T. Sumiyama

( & i j & 2 ) ' " > °n) of V such that there exist non-negative integers HI,u^^-•• ,Wd (0 < Ui < pen — 1, 1 < i < d) satisfying d

mod pe'

bj = y^Uirij

(1 < j' < n).

1=1

(2) I/ei a = {&ijk}i
mod p&i

2_^ bihkCtijk = bj + hj Proof. (1) Let

(1 < j < n).

n

(Q < 6 • < p6-*

•x •

1)

be an element of G'. We can write .d b=

We can assume 0 < Ui < p&ri — 1, as pBrl is the exponent of G. Then d

n

n

d

b=

Prom this equation, 'we see d

bj = ^2,uirij t=i

m

°d pej

(2) Let

be a quasi- regular element of Ga. Then there exists an element

of Ga such that bh — b + h, which is written as

j = l i,k=l

So we see bihkaijk = bj + hj

mod pej

(1 < j < n).

D

On Distance between Finite Rings

435

When a = {ceak}i
^ bihkdijk = bj + hj

mod pei

(l<j
i,k=\

By Proposition 2 (2), we see that v(a] is the number of all quasi-regular elements of G a . Let us define, for a, a' E S, d(a,a') = \v(a) - v(a'}\. We see that the function d defined above is a distance. However, this distance d has the weak point that d(a, a') = 0 whenever both Ga and Gai are nilpotent rings. We can improve this weak point as follows. When a = {aijk}i
bj = ^ uikaijk

mod p£j

(I < j < n).

i,k=l

As any element of G^ is a linear combination of XiXk (1 < i, k < n), by Proposition 2(1), we see that r(a) is the number of elements of G^. Let us define d* by d*(a, a') = |i/(a) - i/(a')| + T(Q) - r(a')|. We see that this function d* also is a distance. Though the distance d* defined above is not faithful, it can result in time saving. If d*(a,a') > 0, then we need not check the equation (7). Let g be the number of elements of G. To compute v(a), we must check quasiregularity for all b £ G Q , so the number of operations to compute v(a) is estimately g2. Similarly, the number of operations to compute r(a) is estimately n(j> e ") 2 . So, the number of operations to compute d*(a, a') is estimately 2g2 + 2n(pe"-)2, which is not greater than 2(1 + n)g2. On the other hand, to check (7) is to check all linear mappings from G to G. So, the number of operations to compute (7) is estimately 9n. Perhaps there can be other effective distances. Problem. Is there a faithful distance for G whose number of operations is not greater than 2(1 + n)
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L. Oyuntsetseg and T. Sumiyama

[3] Sumiyama, T. and Hirano, Y., Enumeration of finite rings, Proceedings of the Second JapanChina International Symposium on Ring Theory and the 24th Symposium on Ring Theory (Okayama University of Science, 1995), 123-126. [4] Sumiyama, T., On algorisms for finite rings, Math. Japonica 44, No. 2 (1996), 239-244. [5] Wiesenbauer, J., Uber die endlichen Ringe mit gegebener additiver Gruppe, Monatsh. Math. 78 (1974), 164-173.

Affine Pairs Ira J. Papick University of Missouri-Columbia, Department of Mathematics, Columbia, Missouri 65211 (U.S.A.) To Robert Gilmer on the occasion of his retirement

Abstract. The primary focus of this paper is on affine pairs, i.e., pairs of rings having the property that each intermediate ring is affine over the base ring. Several results concerning such pairs are discussed and a variety of other related "pairs" are considered.

1. Introduction In this paper we will briefly survey results concerning "pairs" of rings possessing certain algebraic prpperties, especially finiteness conditions. Our focus sets the stage for a concise discussion of some seminal work of Gilmer and Heinzer concerning special affine extensions of commutative rings. In particular, we will highlight a variety of their significant results pertaining to such extensions and indicate some important connections to the work of others. The treatment will be quite leisurely and certainly not totally exhaustive. A convention throughout this manuscript is that, unless otherwise stated, all rings are assumed to be integral domains that are not fields. 2. A pair of pairs In this section we will discuss some key results concerning normal pairs and Noetherian pairs. This fundamental work established a general framework to explore related pairs, and in particular, the development of Noetherian pairs provided a significant foundation for Gilmer and Heinzer's work on finitely generated intermediate rings [GiHe2]. The Krull-Akizuki Theorem asserts that, for a 1-dimensional Noetherian domain R with quotient field K and finite .algebraic extension L of K, the pair (R, L) is a Noetherian pair, i.e., each intermediate ring is Noetherian. More generally, 437

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the converse is also valid and follows directly from the existence of dominating valuation domains. (It is interesting to note that if R is Noetherian and each ring between R and K is coherent, then the dimension of R is also at most 1 [Pa3].) In the more general context of a commutative ring R with total quotient ring T(R), Davis characterized when (R,T(R)) is a Noetherian pair [Dal]. An interesting class of rings that emerged from his study were those rings having each overling normal. In the domain case he showed that rings with this property were precisely the class of Priifer domains. He further analyzed these rings in the context of zero divisors [Da2], and then investigated this class in a slightly different situation [Da3, Da4]. This latter direction was inspired by a question of Kaplansky, which in turn was motivated by Gilmer's survey address on Priifer-like conditions on the overrings of a domain [Gi3]. In particular, Davis provided a definitive structural analysis of normal pairs, i.e., a pair of domains (R,T), where R C T, is called a normal pair if each intermediate domain is normal in T [Da4]. It is worth noting and not difficult to see, that if (R,T) is a normal pair, then qf(R) = <$(T). Next we consider a few key results from Davis' study of normal pairs. Theorem 2.1—Local normal pairs [Da3, Proposition Ij. Let R be a local domain. Then (R, T) is a normal pair tf and only if for some prime ideal P of R, T = Rp, R/P is a valuation domain, and PT = P. Theorem 2.2—General normal pairs [Da3, Theorem 3]. Let R C T be an extension of domains. Then, (R, T) is a normal pair if and only if each intermediate ring is R-flat. Wadsworth introduced the notion of Noetherian pair of domains to generalize some work of Gilmer concerning integral domains with the property that each subdomain is Noetherian [Gi2]. We will indicate how Gilmer's results follow from Wadsworth's structural theorems on Noetherian Pairs. Wadsworth's initial characterization uses the standard finiteness condition utilized in the proof of the Krull-Akizuki Theorem [Ka]. Theorem 2.3—Structure of Noetherian pairs [Wai, Theorem 2]. (R,T) is a Noetherian pair if and only if for each intermediate ring S and each proper ideal I of S, S/I is module-finite over R. Note: The necessity is clear, since S/bS is a Noetherian ring for each nonzero b in 5, and this implies that each ideal of S is finitely generated. Corollary 2.4 [Wai, Remark following Theorem 2]. (/Z[x],T[xj) is a Noetherian pair if and only if R is Noetherian and T is module-finite over R. This follows from the previous theorem by taking S = T[x] and I = xT\x\. Theorem 2.5—Noetherian pairs, necessary conditions [Wai, Theorems 4, 9, and 10]. Assume (R,T) is a Noetherian pair with quotient fields K and L respectively, and let R' denote the integral closure of R in T. 1. IfR^K, then\L:K] < oo. 2. If R = K, then either L is algebraic over K (so T = L) or L is an algebraic function field in one variable over K.

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3. If R is local, dim(J?) > 2, and R is normal in T, then R = T. 4. // dim(-R) > 2, then T C (i~]R'f^), where the intersection is taken over all maximal ideals M of R' with ht(M) > 2. 5. Ifdim(R) > 1, then dim(R) = dim(T). Corollary 2.6—Gilmer's characterization of domains having each subring Noetherian [Gi2], [Wai, Corollary 6]. Let T be a domain with quotient field L. Also, Z denotes the integers and Q the rationals. 1. //char(T) = 0, then (Z,T) is a Noetherian pair <£> [L : Q] < oo. 2. //char(T) = p > 0, then (GF(p),T) is a Noetherian pair ^ L is algebraic over GF(p) or L is an algebraic function field in one variable over GF(p). Corollary 2.7. (R, R[x]} is a Noetherian pair (where x is an indeterminate over R) <=> R = K [Wai, Corollary 5] & Each intermediate ring T, R C T C R[x], is affine over R [Ea2]. Proof of first equivalence. Part 1 of the Theorem shows that R = K if (R, R[x]) is a Noetherian pair. Conversely, suppose. R = K, and let K C S C K\x\. Choose s e S \ K, and consider K[s] C S. Note that S is Noetherian by the Krull-Akizuki Theorem. D Remark 2.8. In Section 6 we will consider some work of Gilmer and Heinzer that broadens the previous corollary to the class of commutative rings with identity and initiates a comprehensive study of finitely generated intermediate rings [GiHe2]. The following result of Wadsworth extends the finiteness property present in the second equivalence of the previous corollary to the context of Noetherian pairs. It also provides natural motivation to further study pairs (R, T) having each intermediate ring affine over R. Corollary 2.9—Finitely generated intermediate rings [Wa2, Theorem A]. Let R C T, where R is a Noetherian domain with quotient field K and T is a domain affine over R. Then (R, T) is an affine pair if and only if either 1. R = K and tr. degR(T) < 1 or 2. R ^ K and (a) the integral closure R' of R in T is affine over R; (b) T C (p|^Z^f)nqf(_R'), where the intersection is taken over all maximal ideals M ofR' withht(M) > 2. Remark 2.10. The necessity follows from Theorem 2.5, since the assumptions imply that (R,T) is a Noetherian pair. In the case of dim(.R) = 1, T is algebraic over R' and so qf(fi') = qf(T). In the sections to follow we will explore more general contexts related to the above corollary.

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3. Pairs Defined by Finiteness Properties A pair of domains (R, T), where R C T, is called an affine pair if each intermediate domain is affine over R. A module-finite domain extension T of a Noetherian domain R provides a natural example of an affine pair (R, T). We have also seen that (K, K[x\) is an affine pair (K a field), and more generally Wadsworth characterized (previous corollary) when an affine extension R C T of a Noetherian domain R forms an affine pair (R, T). Additional examples of affine pairs can easily be constructed by observing that (.R, T) is an affine pair if and only if the ascending chain condition holds for intermediate rings between R and T. In particular, (R, T) is an affine pair if there are only a finite number of intermediate rings between R and T. For example, take R to be a finite dimensional valuation domain and T an overring ofR. Huckaba and Papick studied a special class of affine pairs called module-finite pairs, i.e., pairs (R,T) where each intermediate domain is module-finite over R. They proved: Theorem 3.1—Module-finite pairs and their conductors [HuPa, Theorem 2]. Let (R, T) be a module-finite pair, where R and T are commutative rings. Set I = (R:RT). 1. If R is a domain and I = (0), then R is a Noetherian domain, and so (R,T) is a Noetherian pair. 2. If I = (0), then R/P is a Noetherian domain for each prime ideal P of R. 3. If I ^ (0), T a domain and R is coherent, then each ideal of R containing I is finitely generated. In particular, R/I is a Noetherian ring. Example 3.2—Non-coherent module-finite pairs. Let k C K be an extension of fields such that [K : k] < oo. Let V be a valuation ring of the form K + M where M is not finitely generated. Then, R = k + M is not coherent [DoPa], whereas (R, V) is a module-finite pair. Module-finite pairs of domains first arose in the study of the structure of coherent pairs, i.e., a pair of domains (.R, T), where R C T, is called a coherent pair if each intermediate domain is coherent. Theorem 3.3—Connecting module-finite pairs and coherent pairs [Pa2, Corollary 14]. Let R be coherent and R C T a module-finite extension of domains. The following are equivalent: 1. (R,T) is a module-finite pair; 2. (R,T) is a coherent pair; 3. Each domain S, R C S C T, is a finitely presented R-module; 4. T is finitely presented as a module over each subdomain ofT containing R. Remark 3.4. It is interesting to note, in contrast to the Noetherian case, that if R is coherent and R C T is a module-finite extension, then (R,T) need not be a coherent pair or a module-finite pair. To see this, we can choose R to be a non-Noetherian coherent domain and T a module-finite extension of R, where R and T have different quotient fields. Since (R :R T) = (0) in this situation and

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R is non-Noetherian, then we know that (R,T) cannot be a module-finite pair (cf. Theorems.1). Moreover, using the previous theorem we also see that (-R,T) cannot be a coherent pair. Theorem 3.5—Connecting coherent pairs and Noetherian pairs [Pa5, Theorem 1]. Let R be a Noetherian domain. If (R, T) is a coherent pair, then (R, T) is a Noetherian pair. Remark 3.6. Recall that our general convention is that R is assumed to be a domain that is not a field. It is worth noting that the previous Theorem remains valid if R is a field, i.e., R = K. To see this, let S be an intermediate ring and without loss of generality assume S is not algebraic over R. Let s £ (S — R) such that s is transcendental over R. Then (R[s],T) is a Noetherian pair (previous theorem) and hence S is Noetherian. To a large extent, the structure of coherent pairs parallels the structure of Noetherian pairs. Corollary 3.7—Coherent pairs, necessary conditions [Pa5, Corollary 5]. Assume (R,T) is a coherent pair with quotient fields K and L. 1. IfR^K, then[L:K] < oo. 2. If R = K, then either L is algebraic over K (so T = L) or L is a finite algebraic extension of a purely transcendental extension of K of transcendence degree one.

Corollary 3.8—The coherent analogue of Gilmer's characterization of domains having each subring Noetherian [Pa5, Corollary 6]. Let L be the quotient field of T. The following are equivalent: 1. Each subdomain of T is Noetherian] 2. Each subdomain ofT is coherent; 3. i. L is finite algebraic over the rationals or ii. L is algebraic over a finite field or iii. L is an algebraic function field in one variable over a finite field. Corollary 3.9 [Pa5, Corollary 7). Let (R,T) be a coherent pair and S any ring between R and T. If P 6 Spec(.R), then &os(P) is a finite set. Moreover, if Q £ fibs(P), then [k(Q) : k(P)} < oo. (Here, k(Q) = SQ/QSQ and k(P) = RP/PRP.) Another important pair defined in terms of a finiteness property, and useful in the study of coherent pairs are INC pairs. A pair of domains (R,T), where R C T, is called an INC pair if for each intermediate domain S, the extension RCS satisfies INC [Pal]. Clearly each module-finite pair is an INC pair, whereas a non-finitely generated integral extension shows that the converse is not generally true. The following facts concerning INC pairs are easily deduced: (R, T) INC pair => T is algebraic over R; (R, T) INC pair <^> R C R[t] satisfies INC for each t in T <^> (R',T) INC pair, where R' denotes the integral closure of R in T «• (RP,TR-P] INC pair for each P 6 Spec(R). The next lemma connects coherent and INC pairs. Lemma 3.10 [Pa4, Lemma 2.3]. // (R,T) is a coherent pair, then (R,T) is an INC pair.

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Proof. If (R, T) is not an INC pair, then there is a t € T and prime ideals P C Q in JZ[t] such that P n f i = Q n ^ ! ^ ( 0 ) . We may assume R is local with maximal ideal M = P C\ R. Let 5 be the integral closure of R + P in R[t}. It is clear that P is a maximal ideal of S, t £ S, and P = (S 3'-1). Thus, as S is coherent, P is finitely generated. Hence, since tP C P, t is integral over 5, a contradiction. D Remark 3.11. The previous lemma is not generally valid if R is a field, e.g., (K,K[x\) is a Noetherian pair, but is not an INC pair. Our standard convention of assuming R is a domain and not a field is partially justified by the underlying importance of the previous lemma. 4. Coherent Normal Pairs In this brief section we will consider connections between normal and coherent pairs. Theorem 4.1—Relating normal and coherent pairs [Pa7, Theorem 3.1]. (R, T) is a coherent pair with R normal in T if and only if (R, T) is a normal pair with R coherent. Corollary 4.2—Factorization of coherent pairs [Pa7, Corollary 3.2]. Let R' denote the integral closure of R in T. Then, (R, T) is a coherent pair if and only if(R,R) and (R',T) are coherent pairs. Corollary 4.3—Characterizing Priifer domains in terms of pairs [Pa7, Corollary 3.3]. The following statements concerning the domain R with quotient field K are equivalent: 1. 2. 3. 4.

R is a Priifer domain; (R,K) is a normal pair; R is normal and (R, K) is an INC pair; R is normal and (R, K} is a coherent pair.

Remark 4.4. Gilmer and Mott [GiMo] studied domains having the property that each proper overring is Noetherian (respectively normal). In an analogous fashion, Papick showed that if each proper overring of R is coherent, then R' is a Priifer domain [Pa6]. 5. Coherent Afnne Pairs In this section we will primarily discuss affine pairs (R,K) where K = qf(-R) and R is assumed to be coherent [Pa2]. First a general theorem is needed. Theorem 5.1 [Pa2, Proposition 1}. If (R, K) is an affine pair, then 1. Each overring of R is semilocal; 2. R' is an open Priifer domain, i.e., R' is a semilocal Priifer domain and the prime ideals of R'M satisfy d. c. c. for each maximal ideal M of R'. Remark 5.2. In general, a domain R is 'open if the contraction mapping Spec(T) —> Spec(R) is an open mapping (in the Zariski topology) for each overring T of R, i.e., R C T C K.

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Theorem 5.3— Open Priifer domains [Pa2, Theorem 2 and comments following proof of Proposition 1] . Let R be a normal domain. (R, K) is an affine pair •& T = R[l/a] (a € R) for each averring T of R &• R C V is an affine extension for each valuation averring of R •& R is an open Priifer domain. Fontana and Viswanathan extended the previous theorem to the setting of arbitrary domains. Their work involved the introduction and analysis of a new concept called a Cohen overring. An overring 5 of a G-domain R is called a Cohen averring if 5 is not affine over R and is maximal with respect to this property (recall that R is a G-domain if K = R[l/a] for some a & R). It is interesting to note that Cohen overrings are local domains, and are valuation domains if and only if they are normal. To see the second part of the statement, assume T is a normal Cohen overring of R. Thus (T, K) is an affine pair with T local, and so by the previous theorem T is a valuation domain. Theorem 5.4 [FoVi, Theorem 14]. (R,K) is an affine G-domain and (R, R') is a module-finite pair.

pair <=> R' is a strong

Remark 5.5. Recall that a domain R is called a strong G-domain if each overring of R is of the form R[l/a] for some a e K — {0}. If R is integrally closed, then R is a strong G-domain if and only if R is an open Priifer domain [Pa2, Theorem 2], and thus Theorem 5.4 extends Theorem 5.3. In the context of general domains and arbitrary commutative rings, the study of affine pairs involves many interesting complexities. In the next section we will consider some significant work of Gilmer and Heinzer concerning affine pairs in the context of Noetherian rings and in more general settings. Before moving to that section, we complete this section with a few more results concerning coherent affine pairs. Lemma 5.6 — Ascent and descent of coherence (The coherent analogue of Eakin's Theorem [Eal]) [Pa2, Lemma 6], Let R C T be domains, where T is finitely presented as an R-module. Then R is coherent if and only if T is coherent. Theorem 5.7 [Pa2, Proposition 9]. If (R, K) is an affine pair with R a coherent domain, then 1. (R,K) is a coherent pair; 2. R[XI, . . . , xn] is a coherent ring. Proof of 2. The hypothesis implies that R' is an open Priifer domain and thus R![XI, . . . ,xn] is a coherent domain [Vas, Proposition 8.2 (b)]. The conclusion now follows from the previous lemma, since R'[XI, . . . ,xn] is a finitely presented . . , x n -module. D In general, coherence does not pass to polynomial rings. See Glaz's book, Commutative Coherent Rings [Gl], for a thorough discussion of this interesting topic.

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6. Afflne Pairs: A Variety of General Results The previous sections provided a broad sampling of research concerning pairs of domains sharing specific algebraic properties. A natural and especially interesting collection of pairs that emerged were the pairs (R, T) of domains having the property that each intermediate ring is affine over R, i.e, affine pairs. Recall that in Section 2, Corollary 2.7, we considered the equivalent statements (R,R[x}) is a Noetherian pair & R is a field <=> (R,R[x\) is an affine pair (R a domain), and also Wadsworth's generalization of this result. Motivated by these and other related results concerning "pairs" (some of which have been considered in the previous sections), Gilmer and Heinzer initiated a comprehensive study of affine pairs [GiHe2]. Their primary method for studying such "affine pairs" in the commutative ring case involved reducing to the domain case and then utilizing Wadsworth's results on Noetherian pairs. In this section we will detail a variety of their results within the category of commutative rings with identity. An especially useful result, having several interesting consequences, is the extension of the equivalence indicated in the previous paragraph. Theorem 6.1—Polynomial extension in one variable over a commutative ring R [GiHe2, Theorem 2.3]. (R, R[x}) is a Noetherian pair 4* R is a finite direct sum of fields O (R, R[x}) is an affine pair. Corollaries 6.2 [GiHe2, Corollaries 2.4, 2.6, and 2.7]. I. If (R,T) is an affine pair and T is not algebraic over R, then R is a finite direct sum of fields and tr.deg B (r) = l. 2. // (R,T) is an affine pair (R a domain) and there exists a t 6 T so that (R :R t) — (0), then R is Noetherian (cf. Section 3, Theorem 3.1). 3. If (R,T) is an affine pair of domains and R is not Noetherian, then T C qf(fl). As mentioned earlier, Wadsworth characterized affine pairs (R, T) for an affine extension R C T of domains over a Noetherian domain R. His proof exploited the elementary fact that any affine pair (R, T) over a Noetherian ring is a Noetherian pair. In a similar vein, Alamelu established the following result: Theorem 6.3 [Al, Theorem 2.4]. Let R be afield, T a commutative ring and R C T an affine extension. Then, (R, T) is an affine pair <& (R, T) is a Noetherian pair •<=> dim(T) < 1 and for each non-maximal prime ideal PofT,P is the only P-primary ideal. With these results in mind, it is natural to investigate conditions under which a Noetherian pair (R,T), with T affine over R, is an affine pair. To this end, we next mention an interesting result giveri by Gilmer and Heinzer. First recall that a domain R is called Japanese if the integral closure of R in any finite algebraic extension of its quotient field is module-finite over R.

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Theorem 6.4 [GiHe2, Theorem 3.6]. Let R be a Noetherian Japanese domain. If (R, T) is a Noetherian pair and T is a domain affine over R, then (R, T) is an affine pair. Dimension constraints play an important role in the analysis of Noetherian pairs and affine pairs. A summary of such constraints follows below. Theorem 6.5—Dimension constraints for Noetherian Pairs and afflne pairs of commutative rings [Wai, Theorem 10]; [GiHel, Corollary 1.8]; [GiHe2, Corollary 4.10]. 1. // (R,T) is a Noetherian pair, then dim(R) - 1 < dim(T) < dim(.R) + 1; if dim(R) > 1 or dim(T) > 1, then dim(R) = dim(T). 2. I f ( R , T ) is an affine pair, then dim(T) < dim(R) + 1. Note that by choosing R to be a suitable valuation domain, one can construct an affine pair (R,T), where dim(T) can take on any integer value between 0 and dim(R) < oo. The next example demonstrates that dim(J?) = n and dim(T) = n+1 is also achievable. Example 6.6. Let V — K(x] + M be an n-dimensional valuation domain of this form, where K is a field and x is an indeterminate over K. Set R = K + M and T = K[x] + M. It follows that dim(£) = n, dim(T) = n + I and (R, T) is an affine pair. In addition to focusing on conditions which imply that a Noetherian pair (R, T) with T affine over R is an affine pair, Gilmer and Heinzer also considered when (R, T) is an affine pair for each affine overring of R. Their efforts extended the work of Papick [Pa2] and Fontana-Viswanathan [FoVi] discussed in Section 5. We will underscore a few of their results within the categories of Noetherian rings and Priifer domains. First the Noetherian case. Theorem 6.7 [GiHe2, Proposition 5.12]. Let R be a Noetherian domain. Then, (R, T) is an affine pair for each affine overring •*=> dim(R) < 1 and the integral closure of RM is a finite R^-module for each maximal ideal M of R. If R is a Priifer domain and T is an affine overring of R, then T can be realized as the Nagata transform of a fractional ideal of R. The rings between R and the Nagata transform of an ideal of R behave nicely, and thus provide a valuable tool in the analysis of the Priifer case. Theorem 6.8 [GiHe2, Theorem 5.1]. Let R be a Priifer domain, T = R[ti, ...,tn] (ti 6 K = qf(.R)) and I = R n Rt±l n • • • n fit'1. 1. (R,T) is an affine pair <4> The finite type radical ideals of R/I satisfy the descending chain condition (d.c.c.). (A radical ideal I of a ring R is of finite type if I = yJ, for some finitely generated ideal J of R.) 2. (R,T) is an affine pair for each affine overring T of R •&• For each finitely generated ideal I •£ (0) of R, the finite type radical ideals of R/I satisfy d.c.c. This result has several appealing consequences. Corollaries 6.9 [GiHe2, 5.3, Corollaries 5.9, 5.10, Theorem 5.17]. 1. Let R be a valuation domain. Then, (R, T) is an affine pair for each affine overring T of R 4=>

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For each nonzero prime ideal P of R, the prime ideals of R that contain P satisfy d. c. c. <=> Each averring of R distinct from K = qf (R) is affine over R. 2. Let R be a Priifer domain with a.c.c. on prime ideals. Then (R,T) is an affine pair for each affine averring T of R 4=> There are only finitely many rings between R and T •& Each nonzero ideal of R is contained in only finitely many prime ideals of R. 3. If (R, T) is an affine pair for each affine averring T of R and R is normal, then R is a Priifer domain. 4. If R is a Priifer domain and (R, T) is an affine pair, then T — R[t] for some t&T. The following additional corollary addresses the non-normal case. Corollary 6.10 [GiHe2, Corollary 5.18]. If (R, T) is an affine pair for each affine overring T of R and the integral closure of RM is module-finite over RM for each maximal ideal M of R, then R' is a Priifer domain. Question 6.11. Does the previous corollary remain valid without the additional finiteness assumption? The answer to this question is yes in the case dim(.R) = 1, and (to this author's knowledge) has not yet been determined in the higher dimensional cases. For the dimension 1 part, assume that (R,T] is an affine pair for each affine overring T of R and dim(.R) = 1. Hence, for each maximal ideal M of R, (RM,S) is an affine pair for each affine overring S of RM- In particular, (RM,K) is an affine pair since dim(/?M) = 1- Therefore, (RM)' is a Priifer domain (cf. Section 5, theorem on open Priifer domains), and using elementary properties of localization, it follows that R' is locally (at maximal ideals) a valuation domain. We conclude this section by considering some results concerning proper affine pairs. A proper affine pair (R, T) is a pair of rings R C T where T is not affine over R, but each intermediate ring is affine over R. Examples 6.12 [GiHeS, Introduction]. 1. Let R be a valuation domain with prime spectrum M D PI D • • • D Pi = (0) (i.e., R is not a G-domain). It is easy to see that (R,K) is a proper affine pair. 2. Let R be a 1-dimensional Noetherian domain such that R is not modulefinite over R. Since R'/R is an Artinian ^-module, there exists an intermediate ring T that is minimal with respect to not being affine over R, and thus (R, T) is a proper affine pair. Remark 6.13. Gilmer and Heinzer use a different terminology to capture the notion of proper affine pair. Instead of calling (R, T) a proper affine pair, they say T is a J-algebra over R (Jonsson wo-generated algebra) [GiHeS]. This terminology is an extension of the notion of a Jonsson algebra from the discipline of universal algebra. The next theorem is comprised of a small sampling of some interesting necessary conditions concerning proper affine pairs.

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Theorem 6.14—Proper affine pairs, necessary conditions [GiHeS, Corollary 1.6, Theorem 2.3, Corollary 2.5, Theorem 5.2]. Assume (R,T) is a proper affine pair. 1. T is algebraic over R. 2. dim(T) < dim(.R). 3. If R is Noetherian and T is a domain, then T is Noetherian. 4. If R is a Noetherian ring, then T is integral over R. 5. If T is a domain and R is not Noetherian, then T is an averring of R. Assume additionally that R is a domain and T = K = qf(R). 6. Each averring S ^ K of R is not a G-domain. 7. R does not satisfy d.c.c. on prime ideals. 8. Each averring of R is semilocal. A bit of terminology is needed before stating the final theorem. A domain R is called propen (properly open) if the contraction mapping Spec(T) —> Spec(fi) is an open mapping for each overring T with R C T C K [Pal]. The next result follows from some work of Gilmer and Heinzer in combination with some work of Papick. Theorem 6.15 [GiHeS, Theorem 5.3}; [Pal, Theorem 4.16]; [Pa2, Theorem 2]. Let R be a normal domain. Then, (R, K) is a proper affine pair if and only if R is a propen not open Priifer domain (cf. Section 5, Theorem 5.3). Question 6.16. Is the analogue of the Fontana-Viswanathan result (Section 5, Theorem 5.4 and Remark 5.5) valid in this setting? More particularly, is it true that (R, K) is a proper affine pair •& R1 is a propen not open domain and (R, R') is a module-finite pair? References [Al] [Dal]

S. Alamelu, Subrings of affine rings, J. Indian Math. Soc. 42 (1978), 203-214. E. Davis, Over-rings of commutative rings — I, Trans. Amer. Math. Soc. 104 (1962), 52-61. [Da2] E. Davis, Overrings of commutative rings — //, Trans. Amer. Math. Soc. 110 (1964), 196-212. [Da3] E. Davis, Integrally closed pairs, Lecture Notes in Math. 311, Springer, New York (1972), 103-106. [Da4j E. Davis, Over-rings of commutative rings — ///: Normal pairs, Trans. Amer. Math. Soc. 182 (1973), 175-185. [DoPa] D. Dobbs and I. Papick, When isD + M coherent?, Proc. Amer. Math. Soc. 56 (1976), 51-54. [Eal] P. Eakin, The converse to a well-known theorem on Noetherian rings, Math. Ann. 177 (1968), 278-282. [Ea2] P. Eakin, A note on finite dimensional subrings of polynomial rings, Proc. Amer. Math. Soc. 31 (1972), 75-80. [FoHuPa] M. Fontana, J. Huckaba, and I. Papick, "Priifer Domains", Monographs and Textbooks in Pure and Applied Mathematics, 203, Marcel Dekker, Inc., New York, 1997, 328 pp. [FoVi] M. Fontana and T. Viswanathan, Analogues of a theorem of Cohen for overrings, J. Pure Appl. Algebra 29 (1983), 285-296. [Gil] R. Gilmer, "Multiplicative Ideal Theory", Pure and Applied Mathematics, 12, Marcel Dekker, Inc., New York, 1972, 609 pp.

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[Gi2]

R. Gilmer, Integral domains with Noetherian subrings, Comment. Math. Helv. 45 (1970), 129-134. R. Gilmer, Priifer-like conditions on the overrings of a domain, Lecture Notes in Math. 311, Springer-Verlag, New York (1973), 90-102. R. Gilmer and W. Heinzer, Noetherian pairs and hereditarily Noetherian rings, Arch. Math. 41 (1983), 131-138. R. Gilmer and W. Heinzer, Finitely generated intermediate rings, J. Pure Appl. Algebra 37 (1985), 237-264. R. Gilmer and W. Heinzer, On Jonsson Algebras over a commutative ring, J. Pure Appl. Algebra 49 (1987), 133-159. R. Gilmer and J. Mott, On proper overrings of integral domains, Monatsh. Math. 72 (1968), 61-71. S. Glaz, "Commutative Coherent Rings", Lecture Notes in Mathematics, 1371, Springer-Verlag, New York, 1978. J. Huckaba and I. Papick, Remarks on module-finite pairs, Proc. Edinburgh Math. Soc. 24 (1981), no. 2, 137-139. I. Kaplansky, "Commutative Rings", Allyn and Bacon, Boston, 1970. I. Papick, Topologically defined classes of going-down domains, Trans. Amer. Math. Soc. 219 (1976), 1-37. I. Papick, Finite type extensions and coherence, Pacific J. Math. 78 (1978), 161-172. I. Papick, A remark on coherent overrings, Canad. Math. Bull. 21(3) (1978), 373-375. I. Papick, Coherent overrings, Canad. Math. Bull. 22(3) (1979), 331-337. I. Papick, When coherent pairs are Noetherian Pairs, Houston J. Math. 5 (1979), 559564. I. Papick, A note on proper overrings, Universitatis Sancti Pauli XXVIII-2 (1979), 137-140. I. Papick, Coherent pairs, Boll. Un. Mat. Ital. 18-B (1981), 275-284. W. Vasconcelos, "The Rings of Dimension Two", Marcel Dekker, New York, 1976. A. Wadsworth, Pairs of domains where all intermediate domains are Noetherian, Trans. Amer. Math. Soc. 195 (1974), 201-211. A. Wadsworth' Hilbert subalgebras of finitely generated algebras, J. Algebra 43 (1976), 298-304.

[Gi3] [GiHel] [GiHe2] [GiHe3] [GiMo] [Gl] [HuPa] [Ka] [Pal] [Pa2] [Pa3] [Pa4] [Pa5] [Pa6] [Pa7] [Vas] [Wai] [Wa2]

When a Super-decomposable Pure-injective Module over a Serial Ring Exists Gena Puninski Department of Mathematics, The Ohio State University at Lima, 435 Galvin Hall, 4240 Campus Drive, Lima Ohio 45804, USA puninskiy.ISosu.edu

Abstract. We give a criterion for the existence of a super-decomposable pure-injective module over an arbitrary serial ring.

1. Introduction A module M is said to be super-decomposable if M does not contain indecomposable direct summands. The existence of a super-decomposable pure-injective module over a given ring R is a natural question in the theory of modules. The answer depends on a variety of ranks defined on the lattice of all pp-formulae over R and may be considered as a measure of complexity of the entire category of .R-modules. For instance, according to Trlifaj [10], over a von Neumann regular ring R, a super-decomposable (pure) injective module exists iff R is not semiartinian. Rather few cases are known where such a satisfactory answer can be obtained. For example Puninski (see [7, Ch. 12]) proved that over a commutative valuation ring V, a super-decomposable pureinjective module exists iff the Krull dimension of V is undefined. A further example we can mention is the first Weyl algebra A\ (F) over a field F of characteristic zero, where the existence of such a module was proved by Prest and Puninski [6] in case F is countable. Also, as follows from [8], every non-domestic string algebra over a countable field possesses a super-decomposable pure-injective module. 2000 Mathematics Subject Classification. 03C60, 16D50, 16L30. Key words and phrases. Serial ring, super-decqmposable module, pure-injective module. This paper was written during the visit of the author to the University of Manchester supported by EPSRC grant GR/R44942/01. He would like to thank the university for the kind hospitality.

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Let us describe the situation over a serial ring R. As we have just mentioned, a complete answer is known if R is commutative. Reynders [9] proved that a serial ring R with Krull dimension does not have a super-decomposable pure-injective module. Nevertheless, if R is a serial ring whose Krull dimension is undefined, it is not always the case that R possesses such a module. For instance, if R is an exceptional uniserial ring, then (see [7, Ch. 15]) R has neither Krull dimension nor a super-decomposable pure-injective module. In this paper we give a simple criterion for when a super-decomposable pureinjective module over a serial ring exists. In particular it will follow that if there is a super-decomposable pure-injective right module over a serial ring R, then there is a super-decomposable pure-injective left .R-module. It is known that, over an arbitrary ring R, if a super-decomposable pure-injective .R-module exists, then the lattice of all pp-formulae over R does not have width. We prove that for serial rings the converse is also true. Using this criterion we also show that a serial ring whose lattice of two-sided ideals has Krull dimension fails to have a super-decomposable pure-injective module — a result that seems to be nearly the best possible. 2. Preliminaries The notions from ring and module theory used in this paper are quite standard. Recall that a module M is said to be uniserial if the lattice of submodules of M is a chain, and M is serial if M is a direct sum of uniserial modules. As usual, a ring R is serial if both RR and xR are serial modules. This just means that the unit of R can be decomposed into orthogonal pieces \ = e\ + •••-{- en such that every (right) module &iR is uniserial and every (left) module R&i is uniserial. For instance, the ring Tn(F) of upper triangular matrices over a field F is serial (and artinian). Note that every serial ring is semiperfect. Let us recall some basic notions of the model theory of modules. Let R be a ring with a unit, and let LR be a language of right .R-modules. A positive-primitive formula (pp-formula) (x) is an existential L#-formula 3y = (yi,...,yk)(yA = xb), where A is a fc x I matrix over R, and b is a row of length I with entries in R. We use A | xb (A divides xb) as a shorthand for this formula. For instance, a x is an abbreviation for the pp-formula 3 y (ya = x) which expresses that x is divisible by a. Let M be a right .R-module, and let f ( x ) be a pp-formula. We write M \= 1/1 (ip implies VO if a x for all a, b € R. Also,

and T/J implies (p. After factorization by this equivalence relation the set of all pp-formulae becomes a partial order. In fact, this poset is a modular lattice, where the meet is conjunction (of pp-formulae),

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and the join is sum (of pp-formulae). Here ip + tf} is denned as the pp-formula 3 y ip(x — y) A tj)(y). For instance, it is easily seen that the pp-formula a x + xb = 0 is equivalent to the pp-formula ab xb. There is a duality (i.e. an anti-isomorphism) D between the lattices of right and left pp-formulae over R. In particular if

are pp-formulae, then ip —> ip iff Dtp —> Dip. What will be essential for us is that this duality swaps divisibility and annihilator formulae. More precisely, D(a \ x) is the left annihilator formula ax — 0, and D(xb = 0) is the left divisibility formula b x. A free realization of a pp-formula ip is a pair (M, rn) where M is a finitely presented module, m € M is such that 1) M (= (m) for some pp-formula ?/>, then ip —> ijj. Every pp-formula has a free realization. For instance, (R/aR, b) is a free realization of the formula 3 y (ya — 0 A yb = x). A pp-type is a set of pp-formulae closed with respect to finite conjunctions and implications. If m is an element of a module M, then ppM(m] = {f f a PPformula and M \= if(m)} is a pp-type. Given a pp-type p, we will often distinguish its positive part p+ from its negative part p~, consisting of all pp-formulae which are not in p. Additional information about the theory of serial rings and the model theory of modules over them can be found in [7]. To make the paper uniform we refer to this book where it is possible. Another reference for model theory of modules is [5] and for model theory over a serial ring the paper of Eklof and Herzog [1]. We recall some facts that will be used further without special reference. Let e be an indecomposable idempotent of a serial ring R (hence both modules eR and Re are uniserial) and let M be either a pure-injective or a pure-projective module over R. Then, by [7, L. 11.4], Me is distributive as a left module over S, where S = End(M). Moreover if M is pure-injective and indecomposable, then Me is uniserial over 5. In particular, for arbitrary pp-formulae
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3. pp-formulae The following lemma describes important implications between pp-formulae over a serial ring (to be more precise we describe a basis of open sets of the Ziegler spectrum over R — see [9]). Lemma 3.1. Let e be an indecomposable idempotent of a serial ring R, a, c G Re, b, d G eR. Then the following are equivalent: 1) a | x /\xb = 0 —> c x + xd = 0; 2) a G Re or d G bR or cb < ad. Proof. 2) =£• 1). If a £ Re, then a \ x —> c \ x, therefore the required implication holds true. Similarly, if d G bR then xb = 0 —> xd = 0, and we obtain the required conclusion. Finally, suppose that cb < ad, i.e. ad G RcbR. By [5, Cor. 4.35], it suffices to prove that in every indecomposable pure-injective module M over R we have an inclusion Ma 0 (xb = 0)(M) C Me + (xd = 0)(M). If Mcb = 0, then cb < ad yields Mad = 0, hence Ma C (xd = 0)(M), which suffices. Otherwise Mc6 ^ 0 therefore Me is not contained in (xb = 0)(M). Since Me is a uniserial S-module, (xb = 0)(M) n Me C Me, which is as desired. Note also that if a ^ Re, d £ bR, then c G ,Ra, b G dR therefore ad < cb. Together with cb < ad this yields ad ~ cb. 1) =>• 2). Since Re is uniserial, there is an indecomposable idempotent f £ R such that Ra = Rfa. Then the formulae a \ x and fa \ x are equivalent. Similarly bR = bgR for an indecomposable idempotent g G R. Then the formulae xb — 0 and z&g = 0 are equivalent. Also RabR = RfabgR. Thus we may assume that a G fRe, c G f'Re, b G e.R<7 and d G eRg' for indecomposable idempotents f,f',g,g' G .R. We can also assume that a (£ Re, d £ bR. Therefore c = c'a, c' G f'Rf D Jac(fl), and b = db', b' G g'Rg n Jac(fl). Clearly (fR/abR, a) is a (free) realization of a | x A 0:6 = 0. This formula implies c\ x + xd — 0, which can be written as cd \ xd. Then ad = hcd in fR/abR for some h G /-R, i.e. (1)

ad = abv + /icd

for some h G fRf, v G p.R ab. But ab = adb' > ad, therefore ab ~ ad. Multiplying (1) by b' on the right we obtain ab = abvb'+hcb, that is ab(l—vb'} — hcb. Since 1 — vb' is invertible, it follows that ab > cb. Since cb = c'ab > ab, we have ab ~ cb. Thus cb ~ ab ~ ad yields cb ~ ad. D The following claims are obtained from Lemma 3.1 by substitution. Corollary 3.2. Let e be an indecomposable idempotent of a serial ring R, a, c G Re, b G eR. Then the following are equivalent:

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1) a | x /\xb = 0 —> c | x; 2) o e EC or 6fi = efi or c6 < a. Proof. Since a | x A x& = 0 is an e-formula, the implication 1) in Lemma 3.1 can be written as a x A xb = 0 —> (c \ x + xd = 0) A e | x, and the latter is c | x + (e | i A xd — 0). Take d = e in Lemma 3.1. Then e \ x/\xd = 0 is equivalent to x = 0. Therefore c | x + (e | x A xd = 0) is equivalent to c \ x. Finally d = e € bR in 2) means that bR = eR. D Corollary 3.3. Let e be an indecomposable idempotent of a serial ring R, a G Re, b,d G e/?. T/ien the following are equivalent: 1) a x/\xb = Q-> xd = 0; 2) d e 6fi or ad — 0. Proof. Take c = 0 in Lemma 3.1. Then c \ x is equivalent to x = 0, hence c x + xd = 0 is equivalent to xd = 0. Now a G Re means that a = 0 (therefore ad = 0). Also cb < ad is the same as 0 < ad i.e. ad = 0. D Corollary 3.4. Lei e be an indecomposable idempotent of a serial ring R, a, c G Re, d G eR. Then the following are equivalent: 1) a x —> c | x + xd = 0; 2) a e fie or ad = 0. Proof. Take fe = 0 in Lemma 3.1. Then xb = 0 is equivalent to x = x, hence a | x A xb — 0 is equivalent to a x. Now d & bR means d = 0 (therefore ad — 0). Also 0 = cb < ad implies ad = 0. D Corollary 3.5. Let e be an indecomposable idempotent of a serial ring R, c e Re, b,d £ eR. Then the following are equivalent: 1) xb = Q -» c | x + xd = 0; 2) Re = fie or d e fofi or d> < d. Proof. Note that 1) is equivalent to e | x A xb — 0 —» c [ x + red = 0. Now take a = e in Lemma 3.1. Then a e fie means that e G fie, i.e. Re = Re. D Now we are in a position to analyze any implication (p —> ij) between e-formulae. Indeed we can write (p as a sum of conjuncts £^ Oj | a; A 2:6; = 0 and write ijj as a conjunction of sums AJ(CJ | x + zdj = 0). Clearly,

iff GJ x A x6j = 0 —» Cj | x + xdj = 0 holds for all i,j. Now we may apply Lemma 3.1 to every such implication. Let us give an example of such an analysis. Lemma 3.6. Let e be an indecomposable idempotent of a serial ring fi, a,c € fie, b, d € efi. Then the following are equivalent: 1) a x + xb = 0 —> c | x + xd = 0; 2) a G fie or ad = 0, and Re = Re or d G bR or cb < d.

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Proof. Clearly. 1) is equivalent to a x —> c \ x+xd — 0 and xb = 0 —> c x+xd = 0. By Corollary 3.4, the first implication is the same as 'a e Re or ad = 0'. To conclude, apply Corollary 3.5 to the latter implication. D 4. pp-types In this section we describe e-types over an arbitrary serial ring. The main point is to analyze implications of the form A"=1a.j&i | xbi —> ab \ xb. For a commutative valuation domain, by [7, Ch. 12], every such implication becomes trivial: it holds iff ttjfej | xbi —> ab | xb holds for some i. The following lemma is a weak form of this fact. Lemma 4.1. Let e be an indecomposable idempotent of a serial ring R and let a, ab \ xb. Then there are i,j such that Oj6.j | xbi A a.jbj xbj —> ab xb. Proof. Up to equivalence of pp-formulae we may assume that a € fRe, Oj G fiRe, b e eRg, and bi G e.R —> X^i-^1^*- Note that D Dpi+Dipj. By duality we obtain

9jj A <£,• —>
n

Now we are going to give a 'geometric' description of the implications of ppformulae in Lemma 4.1. Recall that Re is linearly ordered by o <; c if c € Ra. Similarly eR is linearly ordered by 6 < r d if d € bR. Thus, given a € -Re, c e Ra, b e e.R, d & bR (i.e. a ab rrfr and a& | xfe —* ad xd. We will refer to these implications as 'trivial.' If a formula a& xb is drawn as a point (b, a) on the plane eR x fie, then trivial implications act to the 'right and down:' 0 ab xb

a
e b Let us describe one nontrivial implication.

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Lemma 4.2. Let e be an indecomposable idem/potent of a serial ring R and let a, c £ Re, b, d £ eR be such that a £ Re, b £ dR and ab < cd. Then ab \ xb/\cd \ xd is equivalent to ad \ xd. ad xd a

o •*<. A

t\

c <,

o

ab xb Q

ab ~ cd

cd xd f1

Proof. Using distributivity, we have (a x + xb — 0) A (c | x + xd = 0) = (a | x A c x) + (a x A xd = 0) + (c | x A xb = 0) + (xb = 0 A xd = 0). Since a £ Re, the first sumrnand is just a \ x, and then the second summand can be removed. Also b £ dR yields that the last summand is xd = 0. Thus our original conjunction can be written as (a | x+xd = 0) + (c | x/\xb = 0), Since ab < cd, by Lemma 3.1 we obtain c \ x f\xb = 0 x + xd = 0, which gives the desired conclusion. D An equivalence of pp-formulae as given in Lemma 4.2 will be called a fusion of ab xb and cd xd. Note that some nontrivial implications of pp-formulae can be realized as fusions. For instance, let d cd \ xd is a fusion. ab \ xb

cb
a o

o

cd xd c <

}

o •< 'f

o cb \ xb

d

b

This remark is a part of a more general claim which completely describes the implications between e-formulae over a serial ring. Lemma 4.3. Let e be an indecomposable idempotent of a serial ring R, a, a» £ Re, b,bi £ eR. Then every implication Af =1 Oj6i | xbi —> ab \ xb can be obtained as a composition of trivial implications and fusions. Proof. If n = 1 then by Lemma 3.6 the above implication is equivalent to ai £ Ra or aib = 0 and

R& = Re or b £ b\R or abi < b.

If Ra = Re then ab xb (which is the same as a x + xb = 0) is true everywhere. Thus we may assume that Ra C Re.

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Suppose that ai € Ra. Then aibi xb\ —> a&i | xbi is a trivial implication. If 6 & biR, then we have a trivial implication abi \ xbi —> a& | xb. Otherwise b a6 xb by fusion:

o

a <

o •< A

b

I

o

bl

Thus we may assume that ai ^ Ra i.e. ai <; a. Then ai& = 0. If b G biR then ai&j | xbi —* »i& | xb is a trivial implication. The latter formula is xb = 0 which implies ab xb trivially. Otherwise b 1. By Lemma 4.1 we may assume that n = 2 i.e. ai&i xbi A a 2 & 2 | xb2 —> ab 26. Let ipi = a^ xbi, i = 1,2 and let y> = ab | xb. We may also assume that (fn does not imply a 2 6 2 :

61 By distributivity we obtain: 1) ai x —> 2) a2 x A: 3) xb = 0

i = 0) A (a2 x + xb2 = 0) j = 0) + (xbi = 0 A xb2 = 0)

=0 + xfc = 0.

From 1) by Corollary 3.4 we obtain (*)

ai € Ra i.e. a <,

or

= 0.

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Super-decomposable Pure-injective Modules

Since 951 does not imply (p and a\ \ x —-> (p by 1), it follows that xbi = 0 does not imply a | x + xb = 0. Then by Corollary 3.5 we obtain Ra C Re,

b ^ b±R i.e b
and

ab\ > b.

From 3), by Corollary 3.5, (keeping in mind that Ra C Re) we obtain °r

b G &2-R i-e 62
(***)

a

&2 < &•

Since y>2 does not imply (p and x&2 = 0 —> (p by 3), it follows that a2 \ x does not imply a x + xb = 0. Then, by Corollary 3.4, we obtain 02 ^ Ra i.e ai
and

a?b =£ 0.

Also from 2), by Lemma 3.1, (keeping in mind that a2 ^ Ra, b ^ &i-R) we obtain ob\ ^ a20. Let us return back to (*). If a\b = 0 and a\ ^ Ra, then a e .Rai, therefore a& = 0. Then hi efc^Ryields afoj = 0. Then ab^ < a2b yields a2b = 0. a contradiction. Thus by (*) a\ £ Ra i.e. a
ab xb o •<•• A

a 02

Otherwise b £ b^R therefore by (***) we obtain ah? < b. Then 6 < a6i < ab2 < 6, a contradiction.

D

We identify an e-type p(x) with the set of formulae ab \ xb it contains. Also, we identify each of these formulae with the corresponding point (6, a) on the plane eR x Re. Let / = I(p) = {b £ eR \ xb = 0 e p} and J = J(p) = {a 6 Re such that ab xb e p for 6 e eR implies xb = 0 € p}. It is easy to check that / C eR is a right ideal of R and J C Re is a left ideal of R: 0

J

e o e

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Set /• = eR \ I, J* = Re \ J.

Lemma 4.4. Let p be an e-type over a serial ring R, I = I(p) and J = J(p)- Then s*r* < sr for all s £ J, s* € J*, r £ I, r* £ /*. Proof. Otherwise, we obtain sr ~ s*r* (since sr £ Rs*r*R). We may assume that s*r* | xr* £ p. Indeed, otherwise s*r* \ xr* £ p". Since s* € J*, there is r' £ /* such that s*r' \ xr' £ p. Then clearly r' £ r*R, s*r' ~ sr, and we may replace r* byr'. Thus s*r* | xr* £ p. Since r £ /, it follows that s*r* | xr* A sr xr £ p. By Lemma 4.2 (i.e by fusion), this formula is equivalent to sr* xr*, which yields sr* | xr* £ p. Since r* £ /*, the definition of J yields s ^ J, a contradiction. D Recall that a pp-type p is called indecomposable if its pure-injective envelope N(pj is an indecomposable (pure-injective) module. Over a serial ring, by [7, L. 11.6], an e-type is indecomposable iff from ab \ xb £ p for a £ Re. b £ eR. it follows that either a x £ p or xfe = 0 £ p. Thus an indecomposable e-type p looks like a one-step ladder:

J

In particular, p is completely determined by the pair (J, I). Conversely, given a left ideal J C Re and a right ideal / C eR, we may define an e-type p = p ( J * / I ) in the following way. For a £ Re, b £ eR set ai> | xb £ p iff a £ J* or 6 £ /. If p is consistent, it is indecomposable, and all indecomposable e-types can be obtained in this way. Corollary 4.5. The pp-type p ( J * / I ) is consistent iff s*r* < sr for all r £ J, r* £ / * , s e J, s* £ J*. 5. Super-decomposable pure-injective modules Recall that a module M is super-decomposable if it fails to have an indecomposable direct summand, and a pp-type p is called super-decomposable if its pure-injective envelope N(p) is super-decomposable. Let p be a pp-type. A pp-formula ip £ p~ is large in p if for any if>i,f>2 G p~~ such that (f> —> fi, £ P such that () + (tf>2 A V>) £ p~. Note that if the relevant part of the lattice of pp-formulae is distributive, then the latter condition can be replaced by y>\ + y>2 £ p~. By [5, Thm. 9.16], a pp-type p is super-decomposable iff p contains no large formula. Lemma 5.1. Let p be a pp-type such that
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Proof. Let m G N(p) realize p, and assume that N(p) has an indecomposable direct summand. Then N(p) = N © K, where N is indecomposable. Decompose m = n + k, n G N, k G K, in particular n, k =£ 0. Since n G N(p), there is a formula ip(x,y) such that /V(p) |= t^(m,n) and JV(p) |= -> V"(w,0). Forming the conjunction with y(a:). Let $(rc) = tp(x,0). Projecting TJ)(m,n) onto K, we obtain 7f |= 0(k), hence N(p) \= 0(k). Since N(p) \= ->6>(m), it follows that TV f= -,#(n). Now it is easy to see that p and q = ppi^(n) are associated over 0, i.e. for every pp- formula £ with 6* —> £, £ G p iff £ G q. Since A?" is indecomposable, 9 is large in p and 6 —> 1/3, a contradiction. D Now we are in a position to describe serial rings with a super-decomposable pure-injective module. Theorem 5.2. Let R be a serial ring. Then the following are equivalent: 1) there exists a super-decomposable pure-injective module over R; 2) there exists an idempotent e G R (which may be chosen from the original list ei, . . . , en) and embeddings of linear orders / : Q —» Re, g : Q —•* eR such that f ( q ) 2). Let M be a super-decomposable pure-injective module over R and 0 ^ m e M. Then me ^ 0 for some indecomposable idempotent e e R (clearly e can be chosen from the list ei, . . . , e n ). Thus the e-type p = ppM(me) is super-decomposable. By [7, Fact 12.9] p contains no large formula. Set / = I ( p ) , Step 1. There is' ab \ xb e p~ such that a e J*, b e I*. Since the e-type p is nonzero, the zero formula

—> i and 1^2 are incomparable, therefore we may assume that a^ <; a^ and 61 r 6 such that 06' [ xb', a'b \ xb G p~ and a'6' x6' G p:

J a <

P

a' ' i

O

O

P

O

P

P+

I i 1

I

7

r

b b' Since if is not large in p there are pp-formulae ipi — aibi xb^, ip^ = 0-2^2 \ ^62 in p~ such that —>
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61 tpi, ip2 by Lemma 3.6 we obtain ai, a2 < a, therefore a 1,03 € J*. If 6 < r 61,62 then the required formulae are 062 262 and aib xb: j c

\

a < 0,2 <

\ >

0

O

•

I

62

€ p Then For instance, since 6 < r 61, we have ai6 xb ai6 xb € p~ . Otherwise 61
a <

o ......... >• o

6

62

Clearly 062 | xb% € p~ and 062 | x&2 + <2i6 | 16 = ai&2 x&2 G p. It remains to prove that £^6 xb £ p~ . Otherwise aib \ xb e p. But since ai6 < bi then (fusing with 61 x6i) we obtain 0,161 | a;6i 6 p, a contradiction. Continuing this construction we obtain a continually refined configuration:

• o

C2 o

o

o

•

ii

a!2

o

0^3

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Super-decomposable Pure-injective Modules

Let us show how to find the required maps from Q to Re and eR. We check for instance, that cidi < C2d2. Otherwise c\d\ ~ C2d2. Then by Lemma 4.2, by fusion, we obtain c2di \ xdi G p, a contradiction. Thus c^di < c2d2 < 03^3, and so on. For instance, f ( i ) = Ci, g(i) — di, i = 1,2,3 define maps from the three element chain 1,2,3 C Q to Re and eR. To extend this map to 3/2 6 Q (which is between 1 and 2) we use (di, c2) as a 'root' for the next step of the construction. 2) =r- 1). Let us define the set of pp-formulae and their negations p = p+ (J^p~~ in the following way. For a € Re, b 6 eR, we put ab xb e p if there are q > q' G Q such that f ( q ) <; a and b
:

n+

For instance, f ( q ) g ( q ) xg(q) G p for every q G Q, and I(p] = {b £ eR \ b > g(q) for all q G Q}. Let us check that p is consistent. By Proposition 4.1 it suffices to check that p is closed with respect to trivial implications and fusions. Look first at trivial implications. Let ab xb G p+ —» abd xbd G p~. Then there are q > q' G Q such that f ( q ) cb xb, a >i c are dealt with similarly. Now we consider fusions. Let 2 is equivalent to 0361 j xt>i by fusion. Let us assume that the latter formula is in p~, i.e. that there are q > q' e Q such that f ( q ) <; a2 and bi q2 and b\ 92 and /(?i) r g(q%), which implies 0.262 > f(l2)g(92}Thus we obtain a\b\ < f(q\)g(q\) < f(q2)g(q_i) < a2b2, a contradiction. Thus p is consistent. Now we prove that p is super-decomposable. By Lemma 5.1 it suffices to prove that p contains no large formula below e \ x. Otherwise ab \ xb e p~~, for some a e Re, b G eR, is large in p. Since ab \ xb G p~, there are q > q' G Q such that f ( q ) <; a and b qi > q'-

G. Puninski

462

a <

b

g(q'} g(q\)

g(q)

By the definition of p, since qi > q', we obtain f(qi)b xb £ p . From q > q\ it follows that ag(qi) \ xg(q{) S p ~. Clearly 06 xb —> ag(qi) xg(qi and f ( q i ) b xb. n Also the sum of these formulae is f(qi)g(qi) \ xg(qi) e p, a contradiction. 6. Conclusions Is the property of having a super-decomposable pure-injective module left-right symmetric? In general it is known that this property is symmetric for countable rings. Corollary 6.1. Let R be a serial ring. If there exists a super-decomposable pureinjective right module over R, then there exists a super-decomposable pure-injective left module over R. Proof. It is clear that condition 2) in Theorem 5.2 is left-right symmetric.

D

Reynders [9, Cor. 3.11] proved that any serial ring R with Krull dimension (in the sense of Gabriel-Rentschler) fails to have a super-decomposable pure-injective module. The following result is somewhat stronger. We say that a ring R has weak Krull dimension if the lattice of two-sided ideals of R has Krull dimension. Thus R fails to have weak Krull dimension iff the lattice of two-sided ideals of R contains a subset with the ordering of the rationals. Corollary 6.2. Let R be a serial ring with weak Krull dimension. Then there is no super-decomposable pure-injective module over R. Proof. Suppose that a super-decomposable pure-injective module over R exists. Then by Theorem 5.2 we obtain a densely ordered set Rf(q)g(q)R, q £ Q> of twosided ideals of R, a contradiction. D Nevertheless we do not know the answer to the following question. Question 6.3. Let R be a serial ring such that the weak Krull dimension of R is undefined. Does there exist a super-decomposable pure-injective module over R? By [5, Th. 10.9, 10.13], if R is a ring with a super-decomposable pure-injective module, then the lattice of all pp-formulae over R does not have width. The converse has been proved only for countable rings'. For serial rings the situation is better. Proposition 6.4. Let R be a serial ring. Then the following are equivalent:

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1) there exists a super-decomposable pure-injective module over R; 2) the width of the lattice L of all pp-formulae over R is undefined. Proof. We have just mentioned that 1) =£> 2) is true for any ring. 2) =3- 1). Let R be a serial ring such that L does not have width. By the definition of width, L contains a countable dense 'diamond' sublattice (see [5, Ch. 10]). It is quite easy to construct a countable elementary subring R' of R such that the lattice L' of all pp-formulae over R does not have width (we should include in R' all the elements of R which occur in this dense subset and also all elements witnessing implications between formulae in this subset). It is clear that R is serial if we include all the idempotents &i in R. Since L' does not have width and R' is countable, there is a super-decomposable pureinjective module over R. Then Theorem 5.2 yields that for some i there are functions / : Q —> Re^ and g : Q —> &iK which satisfy condition 2) of this theorem. Since R is an elementary subring of R, using the same functions we verify 2) over R. Applying this theorem again, we obtain a super-decomposable pure-injective module over R. D The author is indebted to Mike Prest and Philipp Rothmaler for helpful comments.

References [1] P. Eklof, I. Herzog, Model theory over a serial ring, Ann. Pure Appl. Math., 72 (1995), 145-176. [2] A. Facchini, Module Theory: Endomorphism rings and direct sum decompositions in some classes of modules. Birkhauser, Progress in Mathematics, 167 (1998) [3] G. Gratzer, General Lattice Theory. Academic Verlag, 1978. [4] C.U. Jensen, H. Lenzing, Model Theoretic Algebra. Gordon and Breach, Algebra, Logic and Applications, 2 (1989) [5] M. Prest, Model Theory and Modules. Cambridge University Press, London Math. Soc. Lecture Note Series, 130 (1987) [6] M. Prest, G. Puninski, Some model theory over hereditary noetherian domains, J. Algebra, 211 (1999), 268-297. [7] G. Puninski, Serial Rings. Kluwer Academic Publishers, 2001. [8] G. Puninski, Super-decomposable pure-injective modules exist over some string algebras, Preprint, 2001. [9] G. Reynders, Ziegler spectra of a serial ring with Krull dimension, Comm. Algebra, 27(6) (1999), 2583-2611. [10] J. Trlifaj, Two problems of Ziegler and uniform modules over regular rings, pp. 373—383 in: Abelian Groups and Modules, Lect. Notes Pure Appl. Math., 182 (1996)

Path Coalgebras of Quivers with Relations and a Tame-wild Dichotomy Problem for Coalgebras Daniel Simson Faculty of Mathematics and Computer Science, Nicholas Copernicus University, ul. Chopina 12/18, 87-100 Toruri, Poland simsonfimat.uni.torun.pi Dedicated to Otto Kerner on the occasion of his sixtieth birthday

Abstract. Path A'-coalgebras of quivers with relations over a field K, string coalgebras, coalgebras of tame comodule type and a tame-wild dichotomy for .R"-coalgebras over an algebraically closed field K are defined and studied in the paper.

1. Introduction Throughout this paper we fix a field K. The notion of a path .RT-coalgebra C(Q, fJ) of a quiver with relations (Q,fi) introduced in [37, p. 135] and [38] is investigated in the paper. It is shown in Theorem 3.14 that the coalgebra C(Q,£l) is basic, the left Gabriel quiver of C(Q, fi) is Q and there exists a K-lmeax isomorphism comod-C l (Q,fi) = nilrep/(<5,n) between the category of finite dimensional right C(Q, fi)-comodules and the category nilrep^(Q, fi) of nilpotent .ft'-linear representations of finite length of the quiver Q satisfying the relations in fi. Given any quiver Q, a correspondence between subcoalgebras H of the path coalgebra KQ and the relation ideals fi of the path .ftT-algebra KQ of Q is studied. As a consequence, we show in Theorem 4.9, that if K is algebraically closed and C is a basic /C-coalgebra such that its Gabriel's quiver Q is locally finite and has no oriented cycles, then there is a coalgebra isomorphism C S C(Q,f7), where fi is a relation ideal of the path /C-algebra KQ. The classes of JC-coalgebras of tame comodule type and of wild comodule type defined in [37] and [38] are studied in Section 5, under the assumption that K is an Partially supported by Polish KBN Grant 2 PO 3A 015 21.

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algebraically closed field. In particular, we prove in Theorem 5.4 that the wildness of a coalgebra is a "local-global property", which means that a coalgebra C is of wild comodule type if and only if C contains a finite dimensional subcoalgebra H of wild comodule type. Hence we conclude that any coalgebra C of wild comodule type is not of tame comodule type, which is a week form of the tame-wild dichotomy for coalgebras. Unfortunately, the tame wild-dichotomy for coalgebras formulated in 5.7 remains an open problem. At the end of Section 5 we study tame coalgebras of discrete comodule type. A special class of path /ST-coalgebras C(Q, fi) of quivers with relations are string coalgebras defined in Section 6. It is shown in Theorem 6.2 that any string coalgebra is of tame comodule type. Throughout this paper we use our coalgebra representation theory notation and terminology introduced in [37] and [38]. The reader is referred to [24] and [42] for the coalgebra and comodule terminology, to [13] and [14] for the category theory terminology, and to [1], [2], [3], [18] and [35] for the representation theory terminology and notation. In particular, given a ring R with an identity element we denote by Mod(.R) the category of all unitary right .R-modules and by mod(.R) the full subcategory of Mod(J?) formed by finitely generate'd .R-modules. If R is a K-algebia,, we denote by Mod (R) the full subcategory of Mod(.R) formed by the locally finite dimensional .R-modules, that is, the modules that are directed unions of finite dimensional right .R-submodules. Given a right R-module M we denote by soc M the socle of M, that is, the sum of all simple .R-submodules of M. Given a coalgebra C, we denote by C0 C C\ C . . . C Cm C ... C C the coradical filtration of C. We recall that C = U~=0 Cm.

2. Preliminaries on coalgebras, comodules and the associated pseudo compact algebras Let C be a ^T-coalgebra, with comultiplication A and counity e, where K is a field. We recall that a left C-comodule is a vector space M together with a ^-linear map SM '• M —> C ® M such that (A<8>idM)#M = (idc® <^M)^M and (£<8)idM)^M is the isomorphism M = K ® M. A AT-linear map / : M —> M' between left C-comodules is a C-comodule homomorphism i f S ^ ' f = (idc®/)^M • Lsft C-subcomodules of the C-comodule C are called left coideals of C. We denote by C-Comod the category of all left C-comodules, and by C-comod the full subcategory of C-Comod formed by C-comodules of finite .^-dimension. The corresponding categories of right Ccomodules are denoted by Comod-C and comod-C, respectively. A .R'-coalgebra C is said to be indecomposable, if C is not a direct sum of two subcoalgebras, or equivalently, if the category C-Comod is not a direct sum of two non-trivial subcategories. If C is a coalgebra, then the convolution pseudocompact JC-algebra associated with C is defined in [37, (3.3), (3.4)] to be the vector space C* = Hom^C, K) dual to C equipped with the convolution multiplication and the profinite topology given

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by C* = lira,, C* /H^, where H runs through all finite dimensional subcoalgebras of C (see also "[16], [44]). If M is a left C-comodule and C* is the pseudocompact /C-algebra of C then the jK"-linear map

defines a right C*-module structure on M (called the rational C*-module structure [24], [42]), where T : M®C* —> C*®M is the twist isomorphism and ev : C®C* —> .K" is the evaluation map c®(p H-> y(c). It is shown in [42] that this correspondence defines a categorical isomorphism C-Comod = Rat(C*) of the category of left Ccomodules and the category Rat(C*) of the rational right C*-modules in the sense of [42]. Throughout this paper we need the following useful observations (see [37] and [38]). Theorem 2.1. Let C be a K-coalgebra and C" = Homx(C, K) the associated pseudocompact K-algebra. (a) The map associating to any left C-comodule M the underlying vector space M endowed with rational right C* -module structure defines category isomorphisms (2.2)

C-Comod ^ Dis(C*)

and

C-comod <^ dis(C*)

where Dis(C*) is the category of discrete right C* -modules and dis(C') is the full subcategory o/Dis(C*) formed by finite dimensional modules. (b) There exist K -linear duality functors (2.3)

C'-Comod j=* C*-PC D2

and C-comod ^ comod-C. D

Let us list the main properties of comodule categories we need throughout this paper (see [24], [26] and [42] for details). Theorem 2.4. Let C be a K-coalgebra, where K is a field. (a) Every left C-comodule M is a directed union of finite dimensional subcomodules. (b) The Grothendieck category C-Comod is locally finite and C-comod is the full subcategory of C-Comod consisting of objects of finite length (see [14]). The category C-comod is a skeletally small abelian Krull-Schmidt K-category. (c) The coalgebra C, viewed as a left C-comodule, is an injective cogenerator in the category C-Comod. If M is a left C-comodule, then the left comultiplication map SM : M —> C ® M is a C-comodule embedding and the left C-comodule C ® M is injective. (d) The category C-Comod has enough injective objects and every injective object in C-Comod is a direct sum of injective envelopes E(S) of simple C-comodules S. Every simple left C-comodule is isomorphic to a subcomodule of the left comodule cC. (e) A direct sum ®^ Ep of any family of comodules Ep is injective in C-Comod if and only if each summand Ep is injective. (f) l.gl.dimC = r.gl.dimC.

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(g) The Yoneda K-linear map EM • Homc(M, C) ^> Homjf(M,K) = M*, H-> £
where Ic is a set, tj > I and S(j~) are pairwise non-isomorphic simple comodules for j & Ic- Since the Yoneda Jf-linear map ej^ : Homc(M, C) -^ Hom^-(M, K) = M*,

eip, is an isomorphism for any left C-comodule M then, for any j £ Ic, there are /iT-linear isomorphisms S(j)* = H.omc(S(j), C) = ~H.omc(S(j), soccC) = Homc(5'(j), S(j))*-'. Since dim^ S(j) is finite, we get the formula (26) V ' '

t3 =

dim g ^ (j) dimK Endc S ( j ) '

The formula (2.6) yields.

Corollary 2.7. Assume that the field K is algebraically closed and C is a Kcoalgebra. The following conditions are equivalent. (a) The coalgebra C is basic. (b) The coalgebra C is pointed, that is, every simple subcoalgebra of C is onedimensional. (c) Every simple C-comodule is one-dimensional. For every comodule M in C-comod we define the composition length vector (2.8)

length M = (m(j)) je/c e Z^,

where lf-Ic^ is the direct sum of Ic copies of the free abelian group Z and m(j) £ N is the number of simple composition factors of M isomorphic to the simple comodule S ( j ) . We recall from [37] that the rnap M ^ length M extends to a group isomorphism length :K0(C) ^1^Ic\ where K0(C) = K0(C-comod) is the Grothendieck group of the category C-comod.

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A ^"-coalgebra C is said to be hereditary if gl. dimC < 1, or equivalently, quotient C-comodules of injective C-comodules are injective. The coalgebra C is defined to be Ext-finite, if the dimension dij=dimExt1c(S(i),S(j))Fi

(2.9)

is finite for all i,j e Ic, where Fi = Endc S(i). Note that Ft is a division ring and the K- dimension /, = dim#- Fi of Fi is finite. An easy induction (on the length of comodules) shows that C is Ext-finite if and only if dim^Ext^X, Y) is finite for any pair of comodules X, Y in C-comod. To any hereditary Ext-finite /f-coalgebra C we associate the integral Z-bilinear form bc : TL^ x Z (/c) -> Z

(2.10)

called Euler bilinear form of C, defined by the formula

where x,y G Z^ /c ' and s° = dim^- Fi = dim/f Endc S'(i),

for all i & Ic,

s^ = sjdy = dim^ Ext^(S(i), S(j)),

for

all i,j e 7C.

The Euler quadratic form of C is the integral form (see [30] and [37, (8.11)]) (2.11)

qc : 1(Io} -> Z

defined by the formula qc(x) = bc(x, x) = Y^ielo ^^ ~ ^ijelc sijxixjIt is clear that, , if the field K is algebraically closed, then the Euler forms be (2.10) and qc (2.11) associated to any Ext-finite hereditary /C-coalgebra C coincide with the Tits Z-bilinear form qQ : 1^Qa"> x Z^°) -> Z [37, (8.11)] and the Tits quadratic form gg : Z^°) —> Z [37, (8.12)], respectively, associated to the left Gabriel's quiver Q = CQ of C in [37, p. 137]. Note that Q0 = Ic, s? = 1 and s^ is the number of arrows from i to j in Q = cQ, for all i,j 6 QQ = Ic- Throughout we identify be with fcg, and qc with qQ. An extension of [37, Proposition 8.13] is the following useful fact. Lemma 2.12. Assume that K is an arbitrary field, C is a hereditary K-coalgebra and soccC = ©jg/ c S(j)tj *s the decomposition (2.5). (a) For any simple left C-comodule S ( j ) there exists an exact sequence 0 -» S ( j ) -» E ( j ) -

E o <"•>•> -> 0

in C-Comod, where E ( j ) = E(S(j)) is the injective envelope of S ( j ) and daj is the number (2.9). (b) If C is Ext-finite and be is the Euler bilinear form (2.10) then (2.13)

bc (length X, length Y} = dimK Homc (X, Y ) -

for any pair of comodules X, Y in C-comod.

dimKExtlc(X,Y),

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Proof, (a) Fix an index j £ Ic- Since C is hereditary then the quotient C-comodule E'(j) = E(j)/S(j) is injective. By Theorem 2.4 (d), there is a direct sum decomposition E'(j) = © a£ / c E(a)^i\ where taj > 0. For any i € Ic, from the short exact sequence 0 - S(j) -+ E(j) -> £'(j) -> 0 we derive /C-linear isomorphisms

which are also isomorphisms of right Fi-modules, where Fj = Endc 5(0 • ^ follows that == ty • dimHomc(S'(i), 5(i))F. = ty • 1 = t y and we are done. (b) From the definition of be easily follows that the required equality (2.13) holds for the simple C-comodules X = S(i) and Y — S(j) with arbitrary i, j g Ic. The general case reduces to the above one by an obvious induction on the length of comodules X and Y in C-comod, because both sides of the equality (2.13) are additive functions on C-comod with respect to each of the variables X and Y. D Remark 2.14. For an arbitrary basic /C-coalgebra C such that the number s^s^ is zero and the number s?,- =dim A :Ext£(S(0,S'(j)) is finite for all i,j 6 Ic, we define the reduced Euler quadratic form qc : l(ic) -4 z of C by the formula (2.15)

qc(v) =

°

~

compare with [5] . If gl. dim C is finite, a complete Euler quadratic form qc '. l(ic) _^ Z of C is defined in a natural way, as in [1], [31]. It would be interesting to find the .ff-coalgebras C for which the category C-comod has wild comodule type (see Section 5) if and only if the reduced Euler quadratic form qc of C is not weakly non-negative, that is, there exists a vector w € Z^0-1 with Wj > 0, for all j e (Ic), such that qc(w) < 0, compare with [1], [2], [5], [29], [31], [41]. 3. Path coalgebras of quivers with relations Following Gabriel [15] and [16], a quiver Q = (Qo,Qi) is an oriented graph (in general infinite) with the set QQ of vertices and the set Qi of arrows. We associate to Q the quiver Q* = (Qo,Q*) opposite to Q, where QQ = Qo and Q$ consists of arrows of the form /?* : j —» i corresponding to the arrows (3 : i —> j in Q\. A K-lmear representation of the quiver Q = (Q0, Q\) is a system X = X

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where X- is a K-vector space and 93/3 : Xi —> X-j is a .ftT-linear map for any /3 : i —-> j. A representation X of Q is of finite length if .Xj is a finite dimensional K-vector space for any j, and Xi = 0 for almost all indices i. A morphism / : X —> X1 of representations of Q is a system / = (/i)ieg0 of K-linear maps /, : Xi —> X,', i 6 Qo, such that ifpfi = fj j in Qi (see [3], [35, Chapter 14]). The vector dimJt = (dimK-Xj)jeQo e Z^°) is called the dimension vector of X. We denote by Homg(X, Y") the J^-linear space of all morphisms from X to Y. We denote by Rep^-(Q) the Grothendieck /^-category of .^-linear representations of Q and by Rep^(Q) the full Grothendieck ./f-subcategory of Rep^(Q) formed by locally finite dimensional representations, that is, directed unions of representations of finite length. Finally, we denote by rep^(Q) 2 rep^(Q) the full subcategories of Rep^-(Q) formed by finitely generated objects and by locally finite dimensional representations, or equivalently, by representations of finite length. It follows that the category Rep^- (Q) is locally finite and rep^- (Q] consists of all objects of Rep^(Q) of finite length. By an oriented path in Q = (Qo,Qi) of length m > 1 starting from the vertex i — IQ and ending at the vertex -j = im we mean a formal composition

(3.1) of arrows / ? i , . . . , f3m. To any vertex i e QQ we attach a stationary path 77^ starting and ending at i. The stationary path at j in the quiver Q* opposite to Q is also denoted by rjj. If LJ = fii/32 • • • f3m is the path (3.1) in Q we set w* = /3^l/3^n_l • • • J3^ and we view it as a path in Q*. We denote by Qm the set of all oriented paths in Q of length m > 0. , A path K-algebra of a quiver Q is the graded K-vector space

(3.2)

KQ = KQ0 ® KQi © • • • 0 KQm ® • • •

equipped with the obvious addition and multiplication, where KQm = (see [17, 4.2], [3], [35, Chapter 14]). Given two vertices a, b € Qo of Q we denote by KQ(a, b) the subspace of . generated by all oriented paths from a to b, and by KQm(a, fo) the subspace of KQ(a, 6) generated by paths of length m. It is clear that KQ (3.2) is a graded /f-algebra, the stationary paths 77.;, i 6 Q0, form a complete set of primitive orthogonal idempotents of KQ, and there is a right ideal decomposition KQ = 0 i€ g 0 f]iKQ. If Qo is finite then the element SieQo ^ ^s ^e identity of KQ. If Qo is infinite the algebra KQ has no identity element. It is clear that the dimension of KQ is finite if and only if Q is finite and has no oriented cycles. It is well-known that there are equivalences of categories RepK(Q) = Mod(ATg) and rep^(<5) ^ mod(KQ) (see [17, 4.2], [35, Section 14.1]). Note that, for each m > 1, the K-vector space (3.3)

KQ>m J= 0

KQj

is the two-sided ideal of KQ generated by all paths of length m.

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The path K-algebra KQ endowed with the direct sum decomposition (3.2) can be viewed as a graded /C-coalgebra with the comultiplication A : KQ —> KQ ® KQ and the counity e : KQ —> K defined as follows (see [9], [45]). Given a stationary path r^ at i we set A(?7i) = 77$ <8> f]i and e f a ) = 1. Given any path uj — J3\02 • • • /3m (3.1) of length m > 1 starting from the vertex i = io and ending at the vertex j = im we set m-l

AH

= 77; ® w + w ® Tfe + ^ (/3i/32 " • & ) « > (&+i • • • An) s=l

and

e(w) = 0, where ® = ®jf. We call .fCQ together with the above coalgebra structure the path /C-coalgebra of the quiver Q with coefficients in K. It is clear that, for each m > 0, the K-vectoi space

(3.4)

KQ<m = KQ0 ® KQ! 9 ... 0 KQm

is a subcoalgebra of the path coalgebra KQ. It is easy to see that if Q is a finite quiver without oriented cycles and Q* is the quiver opposite to Q, then the finite dimensional K-algebra (KQ*)* = ]iomK(KQ*,K), K-dual to the /C-coalgebra KQ*, is the path algebra KQ and KQ*-comod ^ mod(^Q) ^ rep x (Q) (see [24], [42] and [35, Section 14.1]). In [37] and [38] we have defined a .JC-linear representation X of Q to be nilpotent (or a small representations of Q, in the sense of Gabriel [16, Section 7.4]) if there exists an TO > 2 such that the composed _K"-linear map v

A

JO

VPi

v

* -*•»!

f/32

v

^

A

im

is zero for any path /31(52 • • • /3m (3.1) in Q of length m. We denote by nilrep^(<3) the full subcategory of rep^-(Q) formed by all nilpotent representations of finite length, and by Rep^1 (Q) the full subcategory of Rep^(Q) formed by all locally nilpotent representations of finite length, that is, directed unions of representations from nilrep^(Q). It is easy to see that the path ff-coalgebras KQ and C = KQ* are basic, the one-dimensional vector space S(j) = r]jK is a simple left subcomodule (and a simple subcoalgebra) of C, KQ$ = soccC = ®J€QO S(j) and the left C-comodules S(i) and S(j) are not isomorphic for i ^ j. Note also that, for each j £ Qo, the indecomposable left ideal (KQ*)rjj of the path /^-algebra KQ* is an indecomposable injective left coideal of the coalgebra KQ*, soc(KQ*)r/j = S(j) and KQ* = (£)j£Q* (KQ*)r]j is a left coideal decomposition of the path coalgebra KQ*. The following result is a consequence of [45, Proposition 4.11] and [37, Proposition 8.1]. Proposition 3.5. Let Q be a quiver, Q* the quiver opposite to Q, and K a field. The coalgebras KQ*, KQ are basic and there exist K-linear category isomorphisms (3.6)

KQ*-comod ^ nilrep^(Q) = comod-KQ

and Comod-KQ ^ Rep^\Q).

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The following definition is due to Gabriel [17, 4.2] (see also [35, Section 14.1]). Definition 3.7. A quiver with relations (or a bound quiver) is a pair (Q,£l], where Q is a quiver (in general infinite) and fi is a two-sided ideal of the path K-algebra KQ such that fl C KQ>%. Every such an ideal Jl is called an ideal of relations, or a relation ideal of KQ. It is easy to see that any relation ideal J7 of KQ has a decomposition Q(a,&),

where fi(a, 6) = fl n KQ(a,b)

and KQ(a, b) is the subspace of KQ generated by all oriented paths starting at a and ending at 6. If ( Q , f l ) is a quiver with relations, we define repK(Q,fl) 2 rep^(<3,£i) 2 nilrep^(Q, fi) to be the corresponding full subcategories of rep^(Q) 2 reP^((3) 2 nilrep^t (Q) formed by the A"- linear representations of Q satisfying all relations in ft (see [17, 4.2] and [35, Section 14.1]). Definition 3.8 (see [37, p. 135], [38]). Let K be a field and (Q,n) a quiver with relations. A path K-coalgebra of a (Q,"fJ) is the subcoalgebra

(3.9)

C(Q, Q) = {a G KQ; (a, fi) = 0}

of the path K-coalgebra KQ, where (— , — ) : KQ x KQ —> K is the standard nondegenerate symmetric K-bilmeai form defined by the formula (u, w) — 5UrW for all paths u, w in Q. Here 6UtW is the Kronecker delta. In the proof of Theorem 3.14 below, we show that the subspace C(Q,£l) of KQ is a subcoalgebra. Example 3.10. Let Q be a quiver and let J7 be a two-sided ideal of the path algebra KQ generated by a set X of zero-relations (oriented paths) of length m > 2. It is easy to see that the path coalgebra C(Q,fi) of (Q, fi) is the subcoalgebra KQ generated as a vector space by all oriented paths of Q containing no subpaths from X. In particular, if f2 is the ideal generated by the set X = Qm of all zero-relations of length m > 2, then f2 = KQ>m and the path coalgebra C(Q, fl) of (Q, fi) is the coalgebra KQ<m^ = KQ0 ® KQl ® . . . Example 3.11. Let £ be the quiver consisting of one vertex 1 and two loops 0i and /?2- Let fi = (Pifo — /?2/?i) be the two-sided ideal of the path algebra KC generated by the one element set X = {PiPi — flzfii}- Given a pair of integers ---- n --- m n > 0 and m > 0 we define the element /?i /?2 G K£ by the formula

— n --- m

where i6 and jt are non-negative integers. It is easy to see that pi p2

— m-—-n

= P2 Pi

and A(p\"p7m) = Er=o A* ® #T~tSm + E7=o A^ ® /32m"J'. It follows that

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the path coalgebra C(Q,Q,} of the quiver with relations (Q,£l) has the form

n,m>0

KftT^r

Hence we easily conclude that the path coalgebra C(Q,£l) is isomorphic with the polynomial ./C-coalgebra (K[Ti,T2], A,e) of two commuting indeterminates 7\ and T2, where e(T^T^) = 0 for n + m > 1, e(l) = 1 and A(T^T2m) = E"=oTi ® Example 3.12. (a) Let I = (I, X) be a locally finite poset. The incidence Kalgebra KI of / is the matrix subalgebra of the algebra of the square I by I matrices A = [Xpq] with coefficients \pq in K such that Apq = 0 for almost all indices p, q e / and Apg = 0 for all p ^ 9- The matrices ep?, with p ;< q, having the identity on the (p, q) entry and zeros elsewhere, form a /C-basis of KI. By the incidence .ff-coalgebra of / we mean the algebra KI equipped with the comultiplication A : KI —> KI <8> KI and the counity £ : KI —> .K" defined by the formulae: A(e p9 ) = ^ ept ® etq,

£(epq}.= 0 for p ^ q, and £(epp) = 1 for p € /.

Let Q7 — (Q7,, Q{) be the Hasse quiver of I, that is, QQ = I and there is a unique arrow from p to q in Q7 if and only if p -< g and there is no t 6 / such that p -< t -< g. We denote by fJ/ the two-sided ideal of the path .K'-algebra KQ1 of Q7 generated by all commutativity relations, that is, all differences w' — w" of paths w', w" of length m > 2 with a common source and a common terminal point. Note that the incidence coalgebra KI is isomorphic to the path coalgebra C(QI ,fi/) of (<5 7 ,Q/). For this we note that the /C-linear map 6 : KI —> C(<5, fi) attaching to epg the sum £pg ~ S w °f au oriented paths w in Q7 of the form (3.1) starting in p — i0 and ending with q — im is a ^-coalgebra isomorphism, because fif (p, q) = ^Cepq for all p, q e /, where ^(p, g) = {a e ^Q7(P, ?); (a, ^/) = 0}. (b) Let Q be a locally finite quiver without oriented cycles and let fi be a twosided ideal of the path algebra KQ generated by all commutativity relations. Then the path coalgebra C(Q, fl) of (Q, fi) is isomorphic with the incidence coalgebra of the poset IQ = (Qo, ^), where a ^ b holds if and only if there is an oriented path from a to b in Q. It is clear that the Hasse quiver of the poset IQ is the quiver Q. By a coalgebra C°p opposite to C we mean (C, A op ,£), where the comultiplication Aop : C —> C <8> C is the composition of A with the twist linear map C (& C -^ C ® C, £ ® y i —> y ® x. It is clear that there is a category isomorphism Cop-comod = comod-C. Definition 3.13. A relation subcoalgebra of a path Jf-coalgebra KQ is any subcoalgebra H of KQ satisfying the following two conditions. (a) The subcoalgebra KQ
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Given a quiver Q and a field K, a connection between relation ideals of the path algebra KQ and relation subcoalgebras of the path coalgebra C = KQ is described as follows. Theorem 3.14. Let K be a field, (<2,ft) a quiver with relations and (Q*,ft*) its opposite. (a) The subspace C(Q, ft) = ft1- (3.9) of the path K-coalgebra C = KQ is a basic relation subcoalgebra of KQ such that the Gabriel quiver fjQ °f H = C(Q,$l) is isomorphic with Q. The K-linear category isomorphism comod-KQ = nilrep^(Q) of (3.6) induces category isomorphisms

comod-C(Q,Cl) ^nilrep^(<2,ft) Comod-C(Q, ft) =* Rep^Q, ft). If H is a relation subcoalgebra of the path algebra KQ, then H^ — {a € KQ; (H, a) = 0} is a two-sided relation ideal of the path algebra KQ. (b) The map associating to each path u> of Q the path uj* of Q* defines an anti-isomorphism of the path coalgebras KQ and KQ*, and induces a coalgebra isomorphism of the coalgebra C(<3,ft) op opposite to C(Q,ft) with the path coalgebra C(Q*,£l*) of (Q*,ft*), and a K-linear category isomorphism C(Q*,n*)-comod = comod-C(Q,n). (c) Assume that Q is a locally finite quiver without oriented cycles. The map ft H-> C(Q, ft) defines a bisection between the set of relation ideals ft of the path K-algebra KQ and the set of relation subcoalgebras H of the path coalgebra KQ. The inverse map is given by H i—> H-1 . (d) Assume that Q is a locally finite quiver. The map ft H-* C(Q,Q.) defines a bijection between the set of homogeneous relation ideals ft of the Z-graded path K-algebra KQ and the set of graded relation subcoalgebras H of the %-graded path coalgebra C = KQ. The inverse map associates to any It-graded subcoalgebra H of C = KQ the homogeneous ideal H1- of the path algebra KQ. Proof. First we prove the statements (a) and (b). Assume that ft is an ideal of relations of the path algebra KQ. View the quiver Q as a directed union Q — U/3 Q^ •> where Q@ runs through all finite subquivers of Q. Given Q13 and any m > 2 we consider the finite dimensional factor algebras and 13

3

Rf}<m =

3

of KQ , where ft' = KQ/ n ft. Since KQ? is obviously a factor algebra of KQ, then there is a canonical jFC-algebra surjections

KQ = A

R>-^-> Rftim Consider on the .JiT-algebras A = KQ and R — KQ/£l the linear topologies with the bases {Ker fptm}p,m and {Ker/i(gim}/3im. It is clear that there is a continuous

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K-algebia, surjection A =l

of the pseudocompact .^-algebras. By [37, Theorem 3.6 (d)], there is a pair of DI duality functors CoalgK <—^ PCpc between the category CoalgK of .ftT-coalgebras D2 and the category PCx of pseudocompact /C- algebras, where D2(R) = homx(-R, K) is the space of all continuous /C-functionals for any pseudocompact K-algebra R. It follows that the continuous K-algebia, surjection e induces a coalgebra injection D2(R) = homK(R, K) C horn*: (A, K) = D 2 (A). It is easy to see that the embeddings R <—> R and A c-> A induce coalgebra isomorphisms homK(R,K) ^ homK(R,K) and hom K (A.^) ^ homK(A,K) ^ KQ* making the following diagram of coalgebra homomorphisms commutative D 2 (A) ^ homK(KQ, K) - >

KQ'

because the image of e° = hom/f(£, K) is ft1 = {y> € hom^-(/C<5, ^);v?(^) = 0} and obviously the isomorphism $ carries the subcoalgebra fi-1 onto C(Q*,n*). Since the obvious coalgebra anti-isomorphism KQ* = -fCQ, a;* >—» a;, yields the coalgebra anti-isomorphism C"(Q*,fi*) = (^(Q,^), then C^^fi) is a subcoalgebra of KQ and there exists a .RT-linear category isomorphism Cr(<5*,fi*)-comod = comod-C'(Q,n). Note also that, under the above identifications, the space C'0>m = Rl,m C (KQ*)* - K(Q?r C KQ" JT-dual to the finite dimensional .ff-algebra Rp^m is a finite dimensional X-subcoalgebra of the path coalgebras K(Q13)* C KQ* and (3.16)

C7(Q*,n*) = D 2 (fl) = hom^( lim m=2

that is, the path coalgebra C(Q*,£l*) is a directed union of the finite dimensional subcoalgebras C'p m. Now we prove (3.15). Since C(<2*,fi*)-comod ^ comod-C1(Q,fi), it is sufficient to show that there is a .RMiriear category isomorphism C(Q",fi*)-comod^ n i l r e p ( Q , Q). For this purpose we note that every comodule M in C(Q*,n*)-comod lies in the category C'p m-comod C C(Q*,fi*)"Comod for some finite subquiver Q@ of Q and

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m > 2. Since the coalgebra C'0m is finite dimensional and there is a K-algebra, isomorphism C'pm = R/3,m, then in view of Theorem 3.14, there are canonical /("-linear category isomorphisms ) ^ mod(R/3,m) ft'3 + #<2> m ) C nilrep^(g, ft) The composed functor induces the required category isomorphism (3.15). To finish the proof of (a) we note that if ft is a two-sided ideal of the path K-algebra KQ contained in KQ>2, then obviously ftx = C(Q, ft) contains KQ0 © KQi. Conversely, if H is a, subcoalgebra of the path coalgebra KQ containing KQo © KQi, then obviously H±={a€KQ;(H,a)=0} is a two-sided ideal of the path algebra KQ contained in KQ>2, that is, H1- is a relation ideal of KQ. Assume that ft is a two-sided ideal of the path /T-algebra KQ contained in KQ>2- Then the vector space decomposition (3.17)

KQ= (ft

KQ(a,V)

yields ft = ®^b€Qo ft(a, b), where ft(a, 6) = ft r\KQ(a, b). Let H = ft-1 = C(Q, ft). We claim that there is a vector space direct sum decomposition (3.18)

# = 0 ff(M), a.bGQo

where H(a,b) = Hr\'KQ(a,b). The inclusion "D" is obvious. To prove the inverse assume that h 6 H. Then h has a unique form h = ]T]a b€ g o foajb, where ha>b 6 KQ(a,b). It is clear that ( Q ( a , b ) , Q ( c , d ) ) = 0, if (a, 6) / (c,d). Since he H, then 0 = (h, ft(a, 6)) = ^ (hc,d, ft(a, &)) = (/i a>b , ft(a, b)) c,d€Q0

for all a, 6 e Q 0 - Hence (/i a ,b,ft) = IZc.deQo^.b'^^'^))'= (ha,b,Q(a,b)) = 0 and therefore /i aib € F n ^TQ(a, 6) = H(a, b) for all a, b e Q 0 - Then the equality (3.18) holds. This shows that H is a relation subcoalgebra of KQ. To prove the final statement of (a) assume that H is a relation coalgebra of the path algebra C = KQ. In order to prove that H^ is a two-sided ideal of the path algebra KQ we note that for all a,b,c e KQ the equality (a, be) — (A(a),6 ® c) holds, where the right hand side is the standard non-degenerate symmetric Kbilinear form on KQ (g> KQ. Then given a € H^ and 6 G /fQ we have (c, ab) = ( A ( c ) , a ® f o ) C (H,a)(H,b) =0 and (c,6a)(A(c),6 ® a) C (H,b)(H,a) =0 for all c £ H. It follows that ab,ba £ H1, that is, H^ is a two-sided ideal of the path algebra KQ (compare with [24, p. 57]). • To finish the proof of (a) we show that xQ = Q. We recall from [37, Proposition 8.1] that KQo C KQo ® KQi are the first two terms of the coradical filtration of KQ. It follows that HQ = KQo and HI = KQ0 © KQi are the first

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to terms of the coradical filtration of the coalgebra H = C(Q,£l). In particulax, H0 = KQ0 = ®.€QoS(j), where 5(j) = K^. Let J = J(ff'). By [21] (see also [24, Proposition 5.2.9]), J = H^H" and Hi = (JZ}^H* . It follows that J2 = (J2J-L-L is a closed ideal of H" and there_are (tf*/V)-(#V^)-bimodule isomorphisms (KQ-i)* ^ (Hi/H0}f ^ J/J2 ^ J/J2. Hence, by applying (8.8)-(8.10) to C = H , we conclude that dim#Extfl-(5(i),5(ji)) equals the number of arrows from i to j in Q. Consequently, we get nQ — QThe above arguments also prove that given a relation ideal fl of KQ and a relation subcoalgebra H of C = KQ: an element haj, & KQ(a,b) belongs to fi-1 if and only if (ha>b,Sl(a,b)) = 0, that is, the following equalities hold (3.19)

Q- L n J Ft:Q(a,&) = Q(a,6)- L ' >

and

H1- DKQ(a,b) = H(a,b)-La

for all a, b e Qo, where fl(a, b)±ab = {c& KQ(a, b}; (c, fl(a, 6)> = 0} and

ff (a, o)1"" = {r e A"Q(a, 6); (C(a, 6), r) = 0}. (c) Assume that the quiver Q is locally finite and has no oriented cycles. It follows that, for all a, b e Q0, the vector space KQ(a,b) is finite dimensional. In view of the equalities (3.18) and (3.19), given a relation ideal Q = 0a fcego fi(a, b) of /CQ the subcoalgebra ff = ft-1 of C = /CQ has the decomposition fi1 = 0 0>beQo ft(a,6)-L°b and the two-sided ideal H^ = (fi-1)-1 of KQ has the form

because (fi(a, 6)-1 )-L = fl(a, 6) for all a, 6 € Qo- Similarly, one shows that given a relation subcoalgebra H of C = KQ there is a coalgebra isomorphism (H-1-)^ = H. This finishes the proof of (c). (d) Assume that the quiver Q is locally finite. It follows that for all a, b & QQ and any m > 0 the vector space KQm(a, b) = KQmr\KQ(a, b) is finite dimensional and there is a vector space decomposition (3.20)

KQ(a,b) = 0 KQm(a,b). m=0

If fi is a homogeneous relation ideal Q of KQ then £l(a, b) = ©^=0 ^ m (a, 6) and tt(a,b)^b = ®™=onm(a,b)^, where Q m (a, b}^ = {c e KQm(a, b); (c, £lm(a, b)) = 0}. This shows that H = £l^~ is a graded subcoalgebra of C = KQ. On the other hand, if H = ®^=0 Hm is a graded relation subcoalgebra of C = #Q, then F(a,fc) = ©^ =0 ^ m (a»6), H(a,b)±"" - 0^=0 H m (a, 6)^- and H-1- is a homogeneous relation ideal of KQ, where Hm(a, b}^ = {r& KQm(a, b); (Hm(a, b), r} = 0}.

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It follows that (an(a1&)-L"V"' = V>m(a,b) and (fl m (a,&)- L m)- L m S # m (a,6) for all a, 6 e Qo and m > 0. Hence we get (ft-1)-1 = ft and (^?'L)± = H. This finishes the proof of (d) and the theorem is proved. D 4. When is a coalgebra a path coalgebra of a bound quiver? It is well known that any basic finite dimensional algebra A over an algebraically closed field K is isomorphic to an algebra K(Q,Ci) — KQ/Q,, where Q is the Gabriel quiver of A and ft is an admissible ideal of KQ. In this section we study the problem, when a basic A'-coalgebra C is of the form C(Q,ft), where (<3,ft) is a bound quiver. Here we follow an idea of Gabriel [16, Sections 7 and 8] developed in [9], [25], [45], [37]. Assume that K is an algebraically closed field and C is a basic /C-coalgebra, or equivalently, C is pointed, by Corollary 2.7. We denote by CQ the coradical of C (that is, the sum of all simple subcoalgebras of C), and by C\ the degree one piece of the coradical filtration of C, that is, (4.1)

d = A-^C Co + C0 C} = Ker(C -^ C ® C -> (C/C0) <8> (C/C0))

(see [24] and [45, (15)]). We fix a left comodule decomposition (4.2)

where Ic is a set and S(j) are pairwise non- isomorphic simple comodules for j G IcSince K is algebraically closed, then Corollary 2.7 yields dim^ S(j) = 1 for all j e /c, each 5(j) is a subcoalgebra of C, (4.3)

Co = soccC'

and (4.2) is a coalgebra direct sum decomposition of C0. It follows that each subcoalgebra S(j) contains a group-like element gj such that

and the subspace G(C) of C generated by all group-like elements in C is just the coalgebra Co = soccC1. Since K is algebraically closed, then there exist a coideal J of C and a vector space decomposition (see [45, (4.5)]) (4.4)

C = C0 ® J.

We recall that the (left) Gabriel quiver cQ of C is defined as follows (see [9], [25], [37, Definition 8.6]). The vertices of CQ are the elements j of the set Ic (identified with the simple left C-comodules S ( j ) ) . The arrows from i to j are elements of a fixed basis of the K-vector space jEi = Ext^(5(z), 5(j)). The quiver cQ has an equivalent description as follows (see [16, 7.5] and [45, Section 4]). View the vector space (4.5)

W = Ci/C0

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D. Simson

as a Co-Co-bicomodule in a natural way. Since (4.2) is a coalgebra direct sum decomposition of CQ then, for each pair i,j 6 Ic, there exists an S(i)-S(j)-subbicomodule iWj of W such that

(4.6)

W=

is a direct sum decomposition of Co-Co-bicomodules, where the Co-Co-bicomodule structure on iWj is induced by its 5(i)-S(j)-bicornodule structure. This allows us to interpret the bicomodule W in terms of extensions between simple C-comodules, because by [16] and [45, Proposition 4.9] there is an isomorphism (4.7)

1 iWj^E^t c(S(i),S(j))

of Co-Co-bicomodules for all i,j e Ic- It follows that the arrows from the point i to the point j of the Gabriel quiver cQ of C can be defined as elements of a fixed basis of iWj. It follows that the space K(cQi) is isomorphic to W. Let TCoW = ®^L0 WUm is the (co)tensor coalgebra of the Cb -Co -bicomodule W (see [9, Remark 4.2], [27], [45, Section 4]). Here W°° = CQ and WDl = W. Then there is a coalgebra isomorphism K(cQ)^TCoW, such that K(cQo) @ K(cQi} = C0 ® W C TCoW. In view of (4.4), there is a natural projection j\:C-+W such that CQ C Ker/i. It follows from a universal property of Tc0W that /i extends to a unique coalgebra map (see [16], [9, Remark 4.2] and [45, Proposition 4.6]) (4.8)

f:C^TCoW*K(cQ).

Now we are able to prove the main result of this section. Theorem 4.9. Assume that K is an algebraically closed field. Let C be a basic K-coalgebra, Q = cQ the Gabriel quiver of C and KQ the path K-coalgebra of Q. (a) The coalgebra homomorphism f : C —> KQ given in (4.8) is injective, f ( C i ) — KQo ® KQi and f induces a full, faithful, K -linear and exact functor /„ : C-comod -> nilrep^(Q*). (b) The coalgebra C is hereditary if and only if f is an isomorphism. (c) // the Gabriel quiver Q = cQ of C is locally finite and has no oriented cycles, then f ( C ) is a relation subcoalgebra of the path coalgebra KQ, the space 17 = f(C)± is a relation ideal of the path K -algebra KQ, the homomorphism f defines a coalgebra isomorphism C = C(Q, fi) and a K -linear category isomorphism comod-C' = Proof. The statement (a) follows from [9, Theorem 4.2], [45, Section 4] and [37, 8.3]. An implicit form of it can be found in [16]. For the sake of completeness we give here an outline of the proof. It follows from the construction of/ that the restriction of / to GI is injective, and therefore / is injective (see [24, Theorem 5.3.1]). Since obviously KQ0®KQi C f(d) and dim Ar (^'<5o®-R'Qi) = dim*: /(Ci) = dimx C"i,

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then KQ0®KQi = f(C\}. The second part of (a) is a consequence of the category K-lme&r isomorphism KQ-comod = nilrep^(Q*) [37, 8.3]. (b) If / is an isomorphism then C = KQ is hereditary (see [37, Proposition 8.13 (b)]). Conversely, assume that C is hereditary. By applying the statement (a) and [7, Lemma 3] (or the arguments of Gabriel [16, Sections 7.5, 8.4, 8.5]) we conclude that / is surjective (see also [7] and [22] for alternative proofs). (c) It follows from (a) that the coalgebra C can be viewed as a subcoalgebra of KQ such that KQo © KQ\ C C. We show that C is a relation subcoalgebra of KQ by proving the equality (4.10)

C= (B C(a,6),

where C(a, b) = C f~l KQ(a, b). The inclusion "D" is obvious. To prove the inverse inclusion we take an element c £ C. In view of (3.17), c has a unique form c = ^a b£Q0 ca,b> where catb 6 KQ(a, b}. Since the quiver Q is locally finite and has no oriented cycles, then there exists a finite full subquiver Q' of Q such that c e KQ' and c ajfc £ KQ' for all a, b £ Q0. The path coalgebras KQ' and H = C fl KQ' are finite dimensional and the elements c, Qa^ belong to H for all a, b £ QQ. Note also that KQ'0 © KQ( C H C KQ'. It follows that fl = H-1 C KQ' is a relation ideal of the path algebra KQ' and, by Theorem 3.14 (c) applied to Q', the subspace fl1- = (-fr-1)1 = H of KQ' is a relation subcoalgebra of the path coalgebra KQ'. This means that H = ®a,b&Q' H(ai & )> where H(a, b} = H n KQ'(a, b). It follows that for any a, 6 € Q0 the 'element ca,b belongs to tf (a, 6) = HnKQ'(a,b) C C n KQ(a,b) = C(a,b), and therefore the element c = ^a h6go ca^ belongs to ®Q 6(EQo C(a, b). Thus the equality (4.10) holds. Consequently, C is a relation subcoalgebra of KQ and according to Theorem 3.14 (c) there is a coalgebra isomorphism C = C(Q,fl), where fl = C^ is a relation ideal of the path algebra KQ. The remaining statement of (c) is now a consequence of Theorem 3.14 (a). This finishes the proof. D Another consequence of Theorems 3.14 and 4.9 is the following useful fact. Corollary 4.11. Assume that K is an algebraically closed field. Let C be a basic K-coaigebra, Q = cQ the Gabriel quiver of C and KQ the path K -coalgebra of Q. I f C = Ci, the second term of the coradical filtration ofC, then there is a coalgebra isomorphism where fl = KQ>% = ® m >j KQm is the ideal of the path algebra KQ generated by all zero relations. Proof. It follows from Theorem 4.9 (a) .that C may be viewed as a subcoalgebra of the path coalgebra KQ and KQo © KQi = C\. Since C = C\ and obviously KQ0@KQi — KQ^2, then we get the equalities C = Ci = KQG@KQi = KQ^2 = fl-1 = C(Q, fl), where fl = KQ>2. ~ D

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5. On tame-wild dichotomy for coalgebras Throughout this section we assume that K is any field and C a JC-coalgebra. We fix a left comodule decomposition

(5-1) where Ic is a set, tj > 1 and S(j) are pairwise non-isomorphic simple comodules for j G IcFollowing Drozd [12] we have introduced in [37] tameness and wildness for coalgebras as follows (see also [35, p. 368] and [36]). Assume that K is an algebraically closed field. A .ftT-coalgebra C is of wild comodule type, if the category C-comod is of wild representation type, that is, there exists an exact representation embedding .K'-linear functor T : mod.Y?,(K) —> C-comod (see [36]), where

0

K

If, in addition, the functor T is fully faithful, we call C of fully wild comodule type, or strictly wild comodule type (see [12], [36]). A .ff-coalgebra C is of tame comodule type, if the category C-comod of finite dimensional left C-comodules is of tame representation type, that is, for every vector v G Ko(C) there exist C-.K"[t]-bimodules L^ , . . . , L^TV\ which are finitely generated free X[t]-modules, such that all but finitely many indecomposable left C"-comodules M with length M = v are of the form M = L^ ®K\, where s < rv, K^ = K [ t ] / ( t — A) and X & K. If there is a common bound for the numbers rv of such C-/C[i]-bimodules L^ , . . . , Z/r") in each vector v, the tame coalgebra C is called domestic (see [40, (2.1)], [35, Section 14.4]). Following [39] (see also [35, p. 368]) we have introduced in [37] the polynomial growth for tame coalgebras as follows. Assume that K is an algebraically closed field and let C be a /C-coalgebra of tame comodule type. We define two growth functions (5.3)

^,/x°,:^ 0 (C)-,N

as follows. Given a vector v € K0(C) we define ^lc(v) to be the minimal number rv of C-.K"[t]-bimodules L^l\ . . . , Lfr^ satisfying the conditions in the definition of tame comodule type. We define ^(v) to be the minimal number of isoclasses of indecomposable discrete C-comodules M with length M = v, that is, Ccomodules M which are not of the form M = L^ K\, where s < nlc(v), K\ = K[t\/(t -X),\£K and I/1), . . . , Z>c(")) is a minimal family of C-^j-bimodules satisfying the conditions in the definition of tame comodule type. We define a tame /f-coalgebra C to be of discrete comodule type if the growth function //J, : Ko(C) —> N of C 'shown in (5.3) is zero, that is, for every vector v £ Ko(C] the number of the isomorphism classes of indecomposable Ccomodules X with length X = v is finite (compare with Vossieck [43]).

Path Coalgebras of Quivers

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A tame .ff-coalgebra C is said to be of polynomial growth if there is a formal power series

with t — (tj)j£i0 and non-negative coefficients gj l t ...,j m & Z such that fj,^(v) < G(v) for all vectors v = (v(j))jeic 6 K0(C] ^ 1^Ic) such that ||u|| := Eje/ c U C?) ^ 2 By [37], the tame comodule type, the domestic comodule type, the discrete comodule type and the polynomial growth are invariant under Morita equivalence of coalgebras. Following [32, Section 7] and [33] a .Ff-coalgebra C is defined in [37] to be left pure semisimple if the Grothendieck category C-Comod of left C-comodules is pure semisimple, that is, every left C-comodule is a direct summand of a direct sum of finite comodules. It follows from [28] that the definition is not left-right symmetric. It was shown in [37] that every left pure semisimple coalgebra C is of tame comodule type and the growth function (j^, : N^ c ^ —> N is zero, that is, C is of discrete comodule type. Moreover, for every indecomposable non-projective comodule Z in C-comod there exists an almost split sequence 0 — > X —> Y —> Z —> 0 in C-comod. The following theorem extends [37, Theorem 6.10] and shows that the wild comodule type of a coalgebra C reduces to the study of finite dimensional subcoalgebras of C. Theorem 5.4. Let K be an algebraically closed field and let C be a basic Kcoalgebra. The following conditions are equivalent. (a) The coalgebra Ci is of wild comodule type. (b) There exists a finite dimensional subcoalgebra H of C of wild comodule type. (c) The coalgebra C is a directed union of finite dimensional subcoalgebras of wild comodule type. Proof. Since C is basic then the left C-comodule soc cC has a direct sum decomposition soc^C = ©jg/ c S(j)i where Ic is a set, S(j) are simple comodules and S(i) ¥ S(j) for all i ^ j. Let Q = CC be the (left) Gabriel quiver of C. Since K is algebraically closed then, according to Theorem 4.9 (a), there is a /sT-coalgebra embedding C <—> KQ such that the degree one piece C\ (4.1) of the coradical filtration of C is isomorphic to the subcoalgebra KQo © KQ\ of the path .ff-coalgebra KQ of Q. (a)=>(b) Assume first there exist i,j € /c such that dim^- ExtJ;(S'(i), S(j)) > 3. It follows that there are at least three arrows starting from the vertex i and ending at j in the quiver Q. Assume that i ^ j. Then the three arrows quiver

Q' : i o =| o j is a subquiver of Q. If fii, ft, ft : i —> j are the arrows of Q', then the subcoalgebra Ci of C contains the coalgebra

H' = KQ' = Kei © Kej © K/3: © Kfa © K/33

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of dimension 5, where &i and BJ are the stationary paths at i and j, respectively. Since the dual convolution K-algebia H* of H is the self-dual wild /f-algebra

of dimension 5, then the subcoalgebra H' = KQ' of C\ of dimension five is of wild comodule type. Assume that i = j. Then the three loop quiver

at vertex i is a subquiver of Q. If 71, 721 73 '• i ~^ i are the loops of Q" at i, then the subcoalgebra C\ of C contains the coalgebra

H" = KQ'^ = Ka ® Kn © ^72 ® #73 of dimension 4. It is easy to see that the dual convolution K-algebra, H"* of H" is the self-dual K- algebra

of dimension 4. Let modsp(T3(K)) be the full subcategory of mod(r3(.K")) consisting of all modules having no simple injective direct summands. It is clear that mod S p(r3(/Q) is equivalent to the full subcategory of iepK(Q') consisting of all K-linear representations V = (Vj, Vj,(ppl,

Vj are .RMinear maps such that the intersection Ker ip^ fl Ker (ppt Pi Ker t/?^ is zero. Define the ^T-linear functor F : modsp(r3(K)) -» mod(A) ^ comod-tf", by attaching to any K-linear representation V = (Vi,Vj, (p^ , (p^ ,
for s = 1,2,3. It is easy to check that F is exact, carries indecomposable modules to indecomposable ones, and F(V) = F(V) implies V = V for any pair of modules V and V of modsp(T3(K)). This means that F is a representation embedding functor and therefore the ^-algebra A of dimension 4 is of wild representation type. It follows that the subcoalgebra H" of Ci is of wild comodule type. Consequently, (b) holds if there 'exits a pair of indices i,j 6 Ic such that Assume now that dim^-Ext^(5(i), 5(j)) < 2 for all i,j e Ic- It follows that dim^Extc(S1, S') is finite for every pair S1, S' of simple left C-comodules and,

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according to [37, Theorem 6.10], the coalgebra C contains a finite dimensional subcoalgebra of wild comodule type. This finishes the proof that (a) implies (b). The implications (b)=^(c)=>(a) are obvious. D The first part of the proof of Theorem 5.4 yields. Corollary 5.5. Let K be an algebraically closed field and C a basic K-coalgebra. If there exists a pair S, S' of simple left C-comodules such that dim#- Ext^(5, S') > 3, then the subcoalgebra C\ of C contains a subcoalgebra H of wild comodule type such that 4 < dimKH < 5. Proof. Since C is basic then the left C-comodule soc cC has a direct sum decomposition soccC = ®- e / c S ( j ) , where Ic is a set, S ( j ) are simple comodules and 5(i) ^ S ( j ) for all i ^ j. Since every simple left C-comodule is isomorphic with a comodule of the form S ( j ) , then the proof of Theorem 5.4 applies and the corollary follows. D As a consequence we get a weak version of tame-wild dichotomy for coalgebras (compare with [12]). Corollary 5.6. Let K be an algebraically closed field. If C is a basic K-coalgebra of tame comodule type, then C is not of wild comodule type. Proof. Assume to the contrary that C is of wild comodule type. By Theorem 5.4, there exists a finite dimensional subcoalgebra H of C of wild comodule type. By [37, Theorem 6.11] and our assumption, H is of tame comodule type. Since the category isomorphism H-comod = dis(.H"*) = mod(H*) given in [37, Theorem 4.3] preserves tame representation type and wild representation type, then we get a contradiction with the tame-wild dichotomy theorem of Drozd [12] applied to the finite dimensional algebra H*. This finishes the proof. D We hope that the following tame-wild dichotomy holds. Tame-Wild Dichotomy 5.7. Any K-coalgebra over an algebraically closed field K is either of tame comodule type, or of wild comodule type, and these types are mutually exclusive. This is a coalgebra analogue of the well-known tame-wild dichotomy for finite dimensional J^-algebras proved by Drozd in [12]. We are able to prove that the tame-wild dichotomy holds only for some classes of coalgebras (see [37, Sections 6 and 9]). Note that in view of Corollary 5.6, it remains to prove that any non-tame comodule type coalgebra is of wild comodule type. Corollary 5.8. Assume that K is an .algebraically closed field and C is a basic K-coalgebra of tame comodule type. Then dimxExt^(5, S') < 2 for every pair S, S' of simple left C-comodules and the Gabriel quiver cQ of C does not contain a subquiver of the form o —\ o.

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Proof. Assume to the contrary that there exists a pair 5, S' of simple left Ccomodules such that dimjr Ext^(5, 5") > 3. It follows from Corollary 5.5 that C contains a finite dimensional subcoalgebra H of wild comodule type, and according to Theorem 5.4 the coalgebra C is of wild comodule type. Since of C is of tame comodule type, we get a contradiction, because of Corollary 5.6. In order to complete the proof, we note that if i o —} o j is a subquiver of cQ, then dim/s:Extp(5'(i), S(j}) > 3 and again we get a contradiction. D We recall that the Second Brauer-Thrall Conjecture for finite dimensional algebras asserts that a finite dimensional A'-algebra A over an infinite field K is of finite representation type if and only if there exists a positive integer do such that for each d > do the number of isomorphism classes of indecomposable A-modules of dimension d is finite (see [35, p. 4]). We finish this section by some observations on tame coalgebras of discrete comodule type related with a coalgebra analogue of the second Brauer-Thrall conjecture. Proposition 5.9. Let K be an algebraically closed field and let C be a basic Kcoalgebra of tame comodule type. If C is of discrete comodule type, then (a) every finite dimensional subcoalgebra of C is of finite comodule type, and (b) C is a directed union of finite dimensional subcoalgebras of finite comodule type. Proof. Assume that C is tame of discrete comodule type. Let H be a finite dimensional subcoalgebra of C. It follows that the category embedding H-comod •—> C-comod induces an embedding of Grothendieck groups Ko(H) <—-> Ko(C). Since C is of tame comodule type, then so is H, and obviously the growth function H\j : K0(H) —> N is the restriction of the growth function plc : K0(C) —> N, which is zero by our assumption. Hence p,lH = 0, and therefore for each positive integer d > 1 the number of isomorphism classes of the indecomposable #-comodules of dimension d is finite. In view of the category isomorphism H-comod = mod(H*), for each d > 1 the number of isomorphism classes of the indecomposable .ff "-modules of dimension d is finite. By applying the Second Brauer-Thrall Conjecture (proved in [4]) to the finite dimensional ^T-algebra H* we conclude that H* is of finite representation type, and therefore the coalgebra H is of finite comodule type. Hence the proposition follows. D We do not know if the converse implication of Proposition 5.9 holds and we state it as the following open question. Question 5.10. Assume that K is an algebraically closed field and C is a basic K-coalgebra, which is a directed union of finite dimensional subcoalgebras of finite comodule type. Is C tame of discrete comodule type? Remark 5.11. (a) In view of the Corollary 5.13 below, the answer to Question 5.10 is affirmative for hereditary coalgebras C. (b) By 37, Proposition 6.14], Question 5.10 has an affirmative answer for coalgebras C such that every vector v 6 Ko(C) admits a finite dimensional support

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subcoalgebra in C in the sense of [37, Section 6] (see below for the definition). By Theorem 6.2 proved below, this is the case if C is a string coalgebra. We recall from [28] and [37, 9.1] that a quiver Q is said to be a locally Dynkin quiver, if Q is locally finite and the underlying non-oriented graph of every finite subquiver Q' of Q is a Dynkin diagram A n , Dn, Ee, Ey or Eg. It is clear that a connected quiver Q is a locally Dynkin quiver if and only if Q is finite and the underlying non-oriented graph of Q is any of the Dynkin diagrams A n , O n , Ee, ET, Eg, or else Q is infinite and, up to orientation, has the form A^ , ooA^, D^ presented in [37, Tables 9.2]. This happens if and only if the Tits quadratic form qq : 1^®°* —> Z of Q (see Section 2) is positive definite. An immediate consequence of Theorem 4.9 and [37, Theorem 9.4] we get the following characterisation of hereditary coalgebras of tame comodule type. Theorem 5.12. Assume that C is an indecomposable hereditary basic K-coalgebra over an algebraically closed field K. Let Q = cQ be the left Gabriel quiver of C. Then the following conditions are equivalent, (a) (a') (b) (c)

C is domestic of tame comodule type. C is of tame comodule type. Either Q is a locally Dynkin quiver, or Q is an extended Dynkin quiver. Q is locally finite and the Euler quadratic form (2.11)

of C (see also [37, (8.12)]) is positive definite or positive semidefinite. (d) C is not of wild comodule type. (e) C is not of fully wild comodule type. In [23], the above theorem is generalised to the case K is an arbitrary field, that is, not necessarily algebraically closed. The following corollary gives an affirmative answer to Question 5.10 for hereditary coalgebras C and is a coalgebra analogue of the Gabriel's characterisation of hereditary algebras of finite representation type given in 15]. Corollary 5.13. Let C be indecomposable hereditary basic K-coalgebra over an algebraically closed field K and let Q = cQ be the left Gabriel quiver of C . The following conditions are equivalent. (a) C is tame of discrete comodule comodule type. (b) The Gabriel quiver Q = cQ of C is a locally Dynkin quiver. (c) Q is locally finite and the Euler quadratic form qc = IQ '• 1^lc^ —> Z of C (see (2.11)) is positive definite. (d) Q is locally finite and the Euler quadratic form qc = qq of C is weakly positive, that is, q(v) > 0 for all non-zero vectors v 6 N^ / c ^. (e) C is a directed union of finite dimensional subcoalgebras of finite comodule type. If C is tame of discrete comodule type and Q = cQ, then the map length : C-comod -> Z (/c) ^ Z (Qo)

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defines a bijection between the isomorphism classes of indecomposable C-comodules in comod-C and the positive roots of the Euler quadratic form qc '• l[Ic) —* Z of C, that is, the vectors v & N^IC^ such that qc(v) = 1. Proof. By Theorem 4.9, the hereditary coalgebra C is isomorphic with the path coalgebra KQ, where Q = cQ is the left Gabriel quiver of C. Then the corollary is a consequence of Theorem 5.12, the results in [37, pp. 142-143] and a well-known Gabriel's theorem [15] (see also [1, Chapter VII], [3, VIII. 5-6] and [18, Chapter VII]). D Following Crawley-Boevey [10], we introduce the following definition. Definition 5.14. A left C-comodule M is denned to be generic, if dim^M is infinite and M is of finite endo-length, that is, M is a module of finite length, when viewed as a left module over the endomorphism X-algebra Endc M. It would be interesting to count generic C-comodules of a given endolength, where C is a /C-coalgebra of tame comodule type and K is an algebraically closed field. Moreover, in relation with a result in [10], the folio-wing question seems to be interesting. Question 5.15. Assume that K is an algebraically closed field and C is a basic indecomposable K-coalgebra. Is C of tame comodule type if and only if the endomorphism K-algebra Endc M of any generic C-comodule M is a PI ring! The class of coalgebras of tame comodule type needs more investigation. An important role in the study of tame coalgebras should play a tilting technique developed recently by Gregorio in [19] and [20], and the reduced Euler quadratic form (2.15), compare with [1], [2], [5], [29], [31], [41]. 6. String coalgebras Following [6] we introduce the class of string coalgebras and we prove that string coalgebras are of tame comodule type. Definition 6.1. A string AT-coalgebra is a path A'-coalgebra C = C(Q,fi) of a quiver with relations (Q,£l) satisfying the following conditions. (a) Any vertex of Q is a starting point of at most two arrows and is an end point of at most two arrows. (b) The ideal fl of the path algebra KQ is generated by a set fio of zero relations, that is, the elements of fi0 are paths in Q of length > 2. (c) Given an arrow (3 : i —> j in Q there is at most one arrow a : s —» i and at most one arrow 7 : j —> r in Q such that a/3 ^ £10 and (3j ^ fi0We recall from [37, Section 6], that a vector v e K0(C) = Z (/c > has a finite dimensional support subcoalgebra Hv of C if every indecomposable Ccomodule M in comod-C with length Alf = v lies in comod-^ C comod-C, that is, ind,,(comod-.ffu) = indu(comod-C<). Our main result of this section is the following theorem.

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Theorem 6.2. Let K be an algebraically closed field and C = K(Q, £1) a string K-coalgebra. (a) C is a graded subcoalgebra of the path K-coalgebra KQ (3.2) and has the form

(6.3)

C = KQ0 ® KQi © KQ% © KQ$ ® • • • 0 KQ% 0 • • - ,

where Q^ is the subset of Qm consisting of the oriented paths w of length m that do not contain any subpath from fi. (b) There is a K-linear category isomorphism comod-C = nilrep^(Q, 0). (c) Every vector v e Ko(C) admits a finite dimensional support string subcoalgebra C(Qv,£lv) ofC, where Qv is a finite subquiver ofQ. (d) The coalgebra C is of tame comodule type. Proof, (a) It follows from the definition that C = K(Q, fl) = fl"1- is a vector subspace of KQ spanned by all paths w € Q such that (w, fi) = 0. Hence (a) easily follows. (b) Apply Theorem 3.14 (a) to the coalgebra C. (cj By Theorem 3.14 we get KQ0 © KQl C C C KQ and soccC = KQ0. It follows that K0(C) = K0(KQ) ^ Z^. Fix a vector v = (vj)jeQo 6 K0(C] ^ 2(<3o) and set | j v j j = ^2^Qo Vj. We show that there exists a finite dimensional string subcoalgebra Hv of C such that every indecomposable C-comodule L in comod-C with length L = v lies in comod-Hv C comod-C. Let supp(u) = {ji, • • • ,jtv} be the set of all indices jr 6 Qo such that Vjr ^ 0. Let Qv be a full subquiver of Q such that Qv = {jlt... ,jtv}. By the property (a) of Q stated in Definition 6.1, the quiver Qv is finite. By (b), there is a .ftT-linear category isomorphism F : comod-C —=—> nilrep^(Q, fi), and therefore, given L in comod-C, we have length L = dimF(L) and ||dirnF(L)|| = dim^ L, where d'imF(L) is the dimension vector of F(L). If L is an indecomposable comodule in comod-C such that v = length L, then F(L) is an indecomposable representation in nilrep^(Q", fi") C nilrep^(Q, fi) such that dimF(L) — v\ — (vj1,... ,Vjt^), the restriction of v to supp(v), where £lv = £lr\KQv. It follows from [6, p. 161, Theorem] that F(L) is isomorphic with a string representation M(u) of (Q, fl) defined in [6, p. 160] for a string u in (Q, fi), or with a band representation M(w, tp) of (Q, J7) defined in [6, p. 160] for a band w in (Q, fl). From the construction of M(u) and M(w. dim# M(u] and any path w" of Q of length > dimKM(v,tp). Since F(L) ^ M(u) or F(L) ^ M(v,p), then F(L)u} = 0 for all paths a; of Q of length > dim^ L = \\v\\. This shows that for every indecomposable comodule L in comod-C such that v = length L the representation F(L) in nilrep^Q",^) C nilrep^(Q,n) is annihilated by all paths LJ of in Q of length > ||v||. In other words, the representation F(L) lies in v ,n« + KQIM_1) = ribep%(Q?,W + KQIM_J C nikep£(Q«,rr). Consider the path subcoalgebra Hv = C(QV,&) = {y e KQ°; (y,V>+KQIH_J

= 0}

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of K(QV,W) C K(Q,ti) = C, where fi" is the ideal of the path algebra KQV generated by W + KQ^.i.i^ It follows that dirnfc- Hv is finite, because Qv is finite and Hv C KQ% ® KQ\ ® ... ® KQ?.,i_r Moreover, the category isomorphism F restricted to comod-.fP defines a category isomorphism Fv : comod-Hv —» iep}f(Qv,£lv). It follows that every indecomposable comodule L in comod-C such that v = length L lies in comod-Hv. Since obviously Hv is a string subcoalgebra of C of the form required in (c) then we are done. (d) Let v e Ko(C) = Z^ 0 ). By (c), there exists a finite dimensional string subcoalgebra Hv = C(Qv,flv) of C such that indu(comod-fft,) = ind^(comod-C'), where Qv is a finite subquiver of Q. It follows that the finite dimensional K-algebra H* dual to Hv is a string algebra and therefore it is of tame representation type, by [11]. In view of the category isomorphism comod-Hv = H*-mod the coalgebra Hv is of tame comodule type. It follows that there exist K[t]-Hv-bimod\i\es L^\...,L^rv\ which are finitely generated free K[t]-modules, such that all but finitely many indecomposable right ^-comoduies AT with length N = v are of the form ./V S K\®L(a\ where s < rv, K{ = K [ t ] / ( t - X ) and A e K. This proves that C is of tame comodule type, because A'ftj-Hu-bimoduIes L' 1 ',... , L' rv ' are K\t}C-bimodules in a natural way and satisfy the conditions required in the definition of tame comodule type. This finishes the proof. D We finish the paper by three interesting open problems. Problem 6.4. Give a characterisation of right pure semisimple string /C-coalgebras. Problem 6.5. Give a characterisation of string /C-coalgebras that are tame of discrete comodule type. Problem 6.6. Give a characterisation of string .fC-coalgebras that are tame comodule type of non-polynomial growth. References [1] I. Assem, D. Simson and A. Skowronski, "Elements of Representation Theory of Associative Algebras", Vol. /: Techniques of Representation Theory, Londpn Math. Soc. Student Texts, Cambridge Univ. Press, London, 2003, to appear. [2] I. Assem, D. Simson and A. Skowronski, "Elements of Representation Theory of Associative Algebras", Vol. II: Representation-Infinite Tilted Algebras, London Math. Soc. Student Texts, Cambridge Univ. Press, London, 2003, to appear. [3] M. Auslander, I. Reiten and S. Smal0, "Representation Theory of Artin Algebras", Cambridge Studies in Advanced Mathematics 36, Cambridge University Press, 1995. [4] R. Bautista, P. Gabriel, A. V. Roiter and L. Salmeron, Representation-finite algebras and multiplicative bases, Invent. Math. 81(1985), 217-285. [5] K. Bongartz, Algebras and quadratic forms, J. London Math. Soc., 28(1983), 461-469. [6] M. C. R. Butler and C. M. Ringel, Auslander-Reiten sequences with few middle terms and applications to string algebras, Coram. Algebra 15(1987), 145—179. [7] W. Chin, Hereditary and path coalgebras,, Preprint, 2002, 2 pages. [8] W. Chin, M. Kleiner and D. Quinn, Almost split sequences for comodules, J. Algebra 249(2002), 1-19. [9] W. Chin and S. Montgomery, Basic coalgebras, AMS/IP Studies in Advanced Mathematics, 4(1997), 41-47.

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[10] W. Crawley-Boevey, Tame algebras and generic modules, Proc. London Math. Soc., 63(1999), 241-265. [11] P. Dowbor and A. Skowroriski, Galois coverings of representation-infinite algebras, Comment. Math. Helv. 62(1987), 311-337. [12] Ju. A. Drozd, Tame and wild matrix problems, in "Representations and Quadratic Forms", Akad. Nauk USSR, Inst. Matem., Kiev 1979, 39-74 (in Russian). [13] P. Freyd, "Abelian Categories", Harper and Row, New York, 1964. [14] P. Gabriel, Des categorie abeliennes, Bull. Soc. Math. France 90(1962), 323-448. [15] P. Gabriel, Unzerlegbare Darstellungen, I, Manuscripta Math. 6(1972), 71-103. [16] P. Gabriel, Indecomposable representations II, Symposia Mat. 1st. Naz. Aha Mat. 11(1973), 81-104. [17] P. Gabriel, Auslander-Reiten sequences and representation-finite algebras, Proc. ICRA II (Ottawa, 1979), Lecture Notes in Math. 831(1980), pp. 1-71. [18] P. Gabriel and A. V. Roiter, "Representations of Finite Dimensional Algebras", Algebra VIII, Encyclopaedia of Math. Sc., Vol. 73, Springer-Verlag, 1992. [19] E. Gregorio, Topological tilting modules, Comm. Algebra, 29(2001), 3011-3037. [20] E. Gregorio, Tilting theory for comodules, Abstracts, Algebra Conference "Venezia 2002". [21] R. G. Heyneman and D. E. Radford, Reflexivity and coalgebras of finite type, J. Algebra 28(1974), 215-246. [22] P. Jara, L. Merino, D. Llena and D. Stefan, Hereditary and formally smooth coaigebras, Preprint, 2002. [23] J. Kosakowska and D. Simson, Left pure semisimple hereditary coalgebras over a field, Toruri, 2002. [24] S. Montgomery, "Hopf Algebras and Their Actions on Rings", CMBS No. 82, AMS, 1993. [25] S. Montgomery, Indecomposable coalgebras, simple comodules and pointed Hopf algebras, Proc. Amer. Math. Soc. 123(1995), 2343-2351. [26] C. Nastasescu, B. Torrecillas and Y. H. Zhang, Hereditary coalgebras. Comm. Algebra 24(1996), 1521-1528. [27] W. D. Nichols, Bialgebras of type one, Comm. Algebra 6(1978), 1521-1552. [28] S. Nowak and D. Simson, Locally Dynkin quivers and hereditary coalgebras whose left comodules are direct sums of finite dimensional comodules, Comm. Algebra 30(2002), 405—476. [29] J. A. de la Pefia, Algebras with hypercritical Tits form, in "Topics in Algebra, Part I: Rings and Representations of Algebras", Banach Center Publications, Vol. 26, PWN Warszawa, 1990, pp. 353-369. [30] C. M. Ringel, Representations of K-species and bimodules, J. Algebra 41(1976), 269-302. [31] C. M. Ringel, "Tame Algebras and Integral Quadratic Forms", Lecture Notes in Math., Vol. 1099, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 19'84. [32] D. Simson, Pure semisimple categories and rings of finite representation type, J. Algebra 48(1977), 290-296; Corrigendum 67(1980), 254-256. [33] D. Simson, On pure semi-simple Grothendieck categories, I, Fund. Math. 100(1978), 211-222. [34] D. Simson, On the structure of locally finite pure semisimple Grothendieck categories, Cahiers de Topologie et Geom. Diff. 33(1982), 397-406. [35] D. Simson, "Linear Representations of Partially Ordered Sets and Vector Space Categories", Algebra, Logic and Applications, Vol. 4, Gordon & Breach Science Publishers, 1992. [36] D. Simson, On representation types of module subcategories and orders, Bull. Pol. Acad. Sci., Ser. Math. 41(1993), 77-93. [37] D. Simson, Coalgebras, comodules, pseudocompact algebras and tame comodule type, Colloq. Math., 90(2001), 101-150. [38] D. Simson, On coalgebras of tame comodule type, in: "Representations of Algebras", Proceedings ICRA-9, (Eds: D. Happel and Y.B. Zhang), Beijing Normal University Press, 2002, Vol. II, pp. 450-486. [39] A. Skowroriski, Group algebras of polynomial growth, Manuscripta Math., 59(1987), 499-516.

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[40] A. Skowronski, Module categories over tame algebras, in: Workshop on Representations of Algebras, Mexico 1994, Canadian Mathematical Society Conference Proceedings, AMS, Vol. 19, 1996, 281-313. [41] A. Skowronski, Simply connected algebras of polynomial growth, Compositio Math., 109 (1997), 99-133. [42] M. E. Sweedler, "Hopf Algebras", Benjamin, New York, 1969. [43] D. Vossieck, The algebras with discrete derived category, Preprint, Universitat Bielefeld, 2000.

[44] S. Warner, Linearly compact rings and modules, Math. Annalen 197(1972), 29-43. [45] D. Woodcock, Some categorical remarks on the representation theory of coalgebras, Comm. Algebra 25(1997), 2775-2794.

On Central Galois Algebras of a Galois Algebra George Szeto Lianyong Xue Department of Mathematics, Bradley University, Peoria, Illinois 61625, USA szetoShilltop.bradley.edu IxueShilltop.bradley.edu Abstract. Let B be a Galois algebra over R with Galois group G, C the center of B, Jg — {b € B j ox = g(x)b for all x 6 B} for each g € G, eg an idempotent in C such that BJg = egB, Kg = {g G G \ g(egc) = egc for each egc £ egC}, and Bg = ^2keK esJk- Then characterizations are given for Bg being a central Galois algebra with Galois group Kg. Consequently, the results of DeMeyer and Kansaki on central Galois algebras are generalized respectively.

1. Introduction Let B be a Galois algebra over a commutative ring R with Galois group G, C the center of B, and K = {g € G g(c) = c for each c £ C}. A natural question is whether B is a central Galois algebra with Galois group K. The answer is affirmative if R contains no idempotents but 0 and 1 ([2], Theorem 1). For any R, a central Galois algebra with Galois group K was characterized in terms of {Jg \ g 6 G} where Jg — {b e B \ bx = g(x)b for all x e B} ([5], Proposition 3). We note that for an g £ G, B Jg = egB for some central idempotent eg in C ([5]). Let Kg = {g £ G g(egc) — egc for each egc £ egC} and Bg = ^keKg egJk. Then Kg is a subgroup of G and Bg is invariant under Kg. The purpose of the present paper is to characterize a central Galois algebra Bg with Galois group Kg in terms of the Azumaya Galois extension (egB)L with Galois group Kg/L where L — {k € Kg | fc(a) = a for all a G Bg}, and the Azumaya Galois extension is in the sense of [1]. Consequently, our results derive a characterization of the central Galois algebra B\ (= ^2heK JH) with Galois group K (= K\), and the central Galois algebra B in which eg = 0 or 1 for any g G G, respectively; and 2000 Mathematics Subject Classification. 16S35, 16W20.

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so the results of DeMeyer and Kansaki on central Galois algebras are generalized respectively. 2. Basic Definitions and Notations Throughout, let B be a Galois algebra over a commutative ring R with Galois group G: C the center of B, K = {g G G \ g(c) = c for each c 6 C}, Jg = {b G B \ bx = g(x)b for all x & B} for each g & G, eg a central idempotent in C such that BJg = egB ([5]), G(eg) = {g £ G g ( e g ) = eg}, Kg = {g e G \ g(egc) = egc for each egc & egC}, Bg — Y^hzK e9^h ^or eac^ 9 e ^> an<^ ^9 = {o-& A\ax = g(x)a for all x G A} for a subring A of B. We keep the definitions of a Galois extension, a Galois algebra, a central Galois algebra, a separable extension, and an Azumaya algebra as denned in ([6]). An Azumaya Galois extension A with Galois group G is a Galois extension A of AG which is a CG-Azumaya algebra where C the center of A ([!]). 3. Central Galois Algebras Bg Keeping the definitions and notations in section 2, let Kg = {g G G \ g(egc) = egc for each egc G egC} and Bg = E/ieK ea^h for each g £ G. Noting that Kg is an Azumaya automorphism group of egB, we have that (egJk1}(GgJk1) = GgJ^k? for any k ^ , k% G Kg, and so Bg is a subring of egB and invariant under Kg. We shall characterize the central Galois algebra Bg with Galois group Kg (— Kg/L) where L —- {k G Kg | k (a) = a for all a G Bg] in terms of the Azumaya Galois extension (egB')L with Galois group Kg, and derive some consequences when eg = 1 and eff = 0 or 1 for any g & G, respectively. We begin with some properties of Kg, Bg, Lemma 3.1. The order of Kg is a unit in egC. Proof, Since B is a Galois algebra over R with Galois group G, there exists a c £ C such that TTG(C) = 1 by Proposition 5 in [5] on page 314. Clearly, G(eg) (= {g G G g(eg) = eg}) is a subgroup of G, and Kg is a subgroup of G(eg). Let {G(eg)hi hi £ G, i = 1, 2, • • • , n for some integer n} be the set of the right cosets of G(eg) in G and d = Y^=\hi(c}- Then Tr G(e 9 )(rf) = E se G(e s )5(^) = E ffe G(e 9 ) E"=i 5^»(c) = TrG(c) = 1. Noting that g(eg} = eg for each g € G(eg), we have that TrG(eg)(egd) — eg. Next, let {giKg \ gt e G(eg), i = 1,2, • • • , m for some integer m} be the set of the left cosets of Kg in G(eg). Noting that egC C (egB)Ks by the definition of Kg, we have that eg = TrG^gj(egd) — Y^iLi Y^keK 9ik(egd) = \Kg\ Y^iLi 9i(egd) where \Kg\ is the order of Kg. But eg is the identity of egC, so \Kg\ is a unit in egC. D Lemma 3.2. (egB)Ks is a separable algebra over egR. Proof. Since B is a Galois algebra over *R with Galois group G, B is a separable algebra over R. Hence egB is a separable algebra over egR ([3], Proposition 1.11, page 46). Since B is a Galois algebra over R with Galois group G again, egB is a

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Galois extension of (egB)Ks with Galois group Kg because eg £ CK" ([6], Lemma 3.7). Hence egB is a finitely generated and projective left (or right) module over (egB)K<>. Moreover, by Lemma 3.1, the order of Kg is a unit in egC, so (egB)Ks is a direct summand of egB as a bimodule over (egB)Ks. Thus (egB)Ks is a separable algebra over egR by the proof of Theorem 3.8 on page 55 in [3] for Beg is a separable algebra over egR. D Lemma 3.3. Let Z be the center of Bg. Then Bg and (egB)Ks are Azumaya algebras over Z such that Bg = VegB((egB)Ks) and VegB(Bg) = (egB)Ks. Proof. Since B is a Galois algebra over R with Galois group G, egB is a Galois extension of (egB)K<>. Hence VegB((egB)K^) = ®Y^k&K ^k9^ • But' by Lemma 3.3 in [6], J(kBes) = egjk for each k € Kg, so VegB((egB)K^ = ®Ek€K, egjk = Bg. By Lemma 3.2, (egB)Ks is a separable algebra over egR, so (egB)Ks is a separable subalgebra of the Azumaya algebra egB over egC. Thus Bg (= VegB((egB)K<>)) is a separable subalgebra of egB over egC, and VegB(Bg) = (egB)Ks by the commutator theorem for Azumaya algebras ([3], Theorem 4,3, page 57), This implies that Bg and (egB)Ks are Azumaya algebras over the same center Z. D Lemma 3.4. Let L = [k e Kg \ k(a) = a for all a e Bg}. Then (1) the order of L is a unit in egB and (2) (egB)L is a Galois extension of (egB)Ks with Galois group Kg (— Kg/L). Proof. (1) By Lemma 3.1, the order of Kg is a unit in egB. But L is a subgroup of Kg, so the order of L is a unit in egB. (2) Since B is a Galois algebra with Galois group G, egB is a Galois extension of (egB)G(e^ with Galois group G(eg) ([6], Lemma 3.7). Noting that L c Kg C G(eg) as normal subgroups, we have that (egB)L is a Galois extension of (egB)Ks with Galois group Kg. D Lemma 3.5. Let A be an Azumaya Galois extension of Aa with Galois group G. Then A = AG • VA(AG) = A° <8>Cc VA(AG) where C is the center of A. Proof. Since A is an Azumaya Galois extension of AP with Galois group G, AG is an Azumaya CG-algebra by definition and the skew group ring A * G is an Azumaya GG-algebra by Theorem 1 in [1]. Hence A*G is an Azumaya CG-aigebra containing an Azumaya subalgebra AG. Thus A * G = A° • VAtG(AG) = AG ®ca VA*G(A°) as Azumaya CG-algebras by the commutator theorem for Azumaya algebras ([3], Theorem 4.3, page 57). Since VA,G(AG) = VA(AG)*G, A*G S (AG ®caVA(AG)}* G. This implies that A 3* AG ®cc VA(AG] S AG • VA(AG), and so A - AG • VA(AG). D Next we show the main theorem. Theorem 3.6. Let eg ^ 0 for an g 6 Gi Then the following are equivalent: (1) Bg is a central Galois algebra over Z with Galois group Kg. (2) (egB)L = (Bg)(egB)K*.

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(3) (egB)L is an Azumaya Galois extension of (egB)Ks with Galois group Kg. Proof. (1) =£> (2) Since Bg is a central Galois algebra over Z with Galois group Kg, Bg ®z (egB)K° is a Galois extension of Z ®z (^gB)Ks with Galois group Kg <8> 1. But Bg ®z (egB)Ks is an Azumaya Z-algebra by Lemma 3.3, so Bg ®z (egB)K° ^ (Bg)(egB)Ko ([3], Theorem 4.4, page 58)_Hence (Bg)(egB)K^ is a Galois extension of (egB)K° with Galois group Kg (= Kg ® 1). Noting that Bg C (egB)L and (egB)K<> C (egB)L, we have that (Bg)(egB)K<> C (egB)L which are Galois extensions of (egB}Ks with Galois group ~K~g. Thus (egB)L = (Bg)(egB)Ks. (2) =» (3) By Lemma 3.4, (egB)L is a Galois extension of (egB)Ks with Galois group .Kg. By Lemma 3.3, Bg and (egB)Ks are Azumaya algebras over Z, so (egB)L = (Bg)(egB)K° = Bg ®z (egB)Ks which is an Azumaya algebra over Z. Noting that ZK* = Z, we conclude that (egB)L is an Azumaya Galois extension of (egB)Ks with Galois group Kg. (3) =>• (2) By hypothesis. (egB)L is an Azumaya Galois extension of (egB)Ks with Galois group Kj, so (es£)L = ((e55)L)^" • V (e!;B)i (((e g B) i )^) by Lemma 3.5. Since ((egB)L)K« = (egB)K» for L is a subgroup of Kg, it suffices to show that V(egB)L((egB)K<>) = Bg. In fact, since Bg = VesB((egB)Ks) by Lemma 3.3, and Bg C (egB)L C egB, we have that S9 = ^((e^B)*') C V r (ej|B) i.((e a B) Jf ») C K 9 B((e s B)^) = Bg. Thus V (esB)J o((e flJ B)^) = Bg. (2) -> (1) By Lemma 3.3, (Bg)(egB)K<> = Bg ®z (egB)K^ as Azumaya Zalgebras such that Bg = V(egB)L((egB)K°) = ® Z)fc6^-^(esB) } - By hypothesis, (egB)L = (Bg)(egB)Ks which is an Azumaya Galois extension of (egB)Ks with //_

Galois group Kg, so it can be check that Jl Bg ~ ® YskzTT J-

• Noting that Z

Kg

9

R^^l

^R"l

= J^

9

for each k € ^ ff . Hence

= Z and that Bg is an Azumaya Z-algebra,

we conclude that ji "' J-_{' = Z for all k e Kg. Thus Bg is a central Galois algebra fc fc with Galois group Kg ([4], Theorem 1). D When efl = 1, Theorem 3.6 derives a characterization of the central Galois algebra BI (= YlihtK ^) with Galois group K (= KI)\ and the central Galois algebra B in which eg = 0 or 1 for any g & G, respectively. We recall that K = {g £ G | g(c) = c for each c e C}. Corollary 3.7. Let BK = ^heK Jh and L = {k € K k(a) = a for all a e Then BK is a central Galois algebra with Galois group K if and only if BL = B which is an Azumaya Galois extension of BK with Galois group K/L. Proof. It is easy to check that K\=K and B\ = BK, so the corollary is the case eff = 1 of Theorem 3.6. D Corollary 3.8. B is a central Galois algebra with Galois group K if and only if eg = 1 or 0 for each g € G. Proof. (=>•) Since B is a central Galois algebra with Galois group K, we have B = is a Galois algebra with Galois group G, so B = ® ^ €G Jg.

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Thus ®Ys92K Ja = {°}- Therefore Jg = {0}, that is, eg = 0 for each g £ K. Noting that K = KI, we have that eg = 1 or 0 for each g 6 G. (•<=) Since 5 is a Galois algebra with Galois group G, B = ©X^eG-'s- But eff = 1 or 0 for each g 6 G by hypothesis, so B = £ h£jRri A = E^eft- Jh = #1 = BX where jB# is defined in Corollary 3.7. Hence B (= BK) is a central Galois algebra with Galois group K by Corollary 3.7 because BL = B = BK&K• O Corollary 3.9 ([2], Theorem 1). If C contains no idempotents but 0 and 1, then B is a central Galois algebra with Galois group K. Proof. Since C contains no idempotents but 0 and 1, eg = I or 0 for each g £ G. Hence by Corollary 3.8, B is a central Galois algebra with Galois group K. D Corollary 3.10 ([5], Proposition 3). B is a central Galois algebra with Galois group K if and only if Jg = {0} for each g $ K. Proof. By noting that KI = K and that Jg = {0} if and only if eg = 0, the corollary is an immediate consequence of Corollary 3.8. D Acknowledgement This paper was written under the support of a Caterpillar Fellowship at Bradley University. The authors would like to thank the Caterpillar Inc. for the support. References [1] R. Alfaro and G. Szeto, On Galois extensions of an Azumaya algebra. Comm. Algebra, 25(6): 1873-1882, 1997. [2] F. R. DeMeyer, Galois theory in separable algebras over commutative rings, Illinois J. Math., 10: 287-295, 1966. [3] F. R. DeMeyer and E. Ingraham, Separable algebras over commutative rings, Volume 181. Springer Verlag, Berlin, Heidelberg, New York, 1971. [4] M. Harada, Supplementary results on Galois extension, Osaka J. Math., 2: 343-350, 1965 [5] T. Kanzaki, On Galois algebra over a commutative ring. Osaka J. Math., 2: 309-317, 1965. [6] G. Szeto and L. Xue, The structure of Galois algebras. J. Algebra, Vol. 237, No. 1: 238-246, 2001.