A category of matrices representing two categories of Abelian groups

Journal of Mathematical Sciences, Vol. 154, No. 3, 2008

A CATEGORY OF MATRICES REPRESENTING TWO CATEGORIES OF ABELIAN GROUPS A. A. Fomin

UDC 512.541

Abstract. Every τ -adic matrix represents both a quotient divisible group and a torsion-free, finite-rank group. These representations are an equivalence and a duality of categories, respectively.

1. Introduction Great interest in the class of torsion-free, finite-rank Abelian groups first appeared when L. S. Pontryagin discovered its duality [20] in 1934. Then A. G. Kurosh [18], A. I. Malcev [19], and D. Derry [4] obtained a matrix description of these groups in 1937–38. The concepts of quasi-isomorphism and quasi-homomorphism were introduced by B. Jonsson [17] in 1957. The category QT F of torsion-free, finite-rank Abelian groups with quasi-homomorphisms as morphisms is a classical subject of investigations. The class G of mixed self-small groups G such that the quotient G/T (G) is divisible of finite rank was actively investigated by many authors [1,2,5,9,12,14,16,21] in the 1990s. The notion of quotient divisible group was introduced in [13] in 1998. The class of quotient divisible groups contains the class G as well as the classical quotient divisible torsion-free groups introduced by R. Beaumont and R. Pierce [3] in 1961. The category QD of quotient divisible groups with quasi-homomorphisms as morphisms was considered in [13]. Two mutually inverse functors of duality of d

d


RM



-

categories QD −→ QT F and QT F −→ QD were introduced there. In the present paper, a new category RM of special matrices with τ -adic entries is introduced and it is proved that this category is equivalent to the category QD and is dual to the category QT F. Thus, we obtain the following commutative diagram of dualities and equivalences of categories:

QD 

b
c

d d


b

-



c


-

QT F.

The dualities b and b
can be considered as a new version of the Kurosh–Malcev–Derry description. Herewith note that our approach is closer to the approach of Malcev [19], which is different from the more well known approach of Kurosh and Derry. In the last section, a new interpretation of the duality functors d and d
is given without using matrices (Theorem 6). The connection between this duality and the Pontryagin duality is found as well. Note that all basic constructions of this paper can be generalized to modules over Dedekind rings in the manner of [11]. The author expresses his acknowledgment to Prof. Otto Mutzbauer for fruitful discussions and for his essential contribution to this paper. ˆ p denote the ring of integers, the field of ˆ p , and Q All groups will be additive Abelian groups. Z, Q, Z ˆ p is ˆ =
Z rationals, the ring of p-adic integers, and the field of p-adic numbers, respectively. The ring Z p ˆ is called the ring of universal numbers. the Z-adic completion of Z, and K = Q ⊗ Z Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 3, pp. 223–244, 2007.

430

c 2008 Springer Science+Business Media, Inc. 1072–3374/08/1543–0430 

We use characteristics and types in the same manner as in [15], denoting the zero characteristic and the zero type by 0. Let τ and σ be types containing the characteristics (mp ) and (kp ), respectively. We define the operations τ ∧ σ = [(mp ) ∧ (kp )] = [(min{mp , kp })], τ ∨ σ = [(mp ) ∨ (kp )] = [(max{mp , kp })], τ + σ = [(mp ) + (kp )] = [(mp + kp )]. If (mp ) ≥ (kp ), then τ − σ = [(mp ) − (kp )] = [(mp − kp )], assuming herewith ∞ − ∞ = 0. We assume that the operations + and − are weaker than the operations ∧ and ∨, whence the expression τ − σ ∧ δ denotes τ − (σ ∧ δ). T (A) and Tp (A) denote the torsion part and the p-primary component of the torsion part of the group A, respectively. If {x1 , . . . , xn } ⊂ A, then x1 , . . . , xn is the subgroup generated by these elements and x1 , . . . , xn ∗ is the pure hull of these elements, i.e., the subgroup consisting of all elements that are in the subgroup x1 , . . . , xn after multiplication by some nonzero integers. In particular, T (A) ⊆ x1 , . . . , xn ∗ . If the group A is also a module over a ring R, then x1 , . . . , xn R denotes the submodule generated by these elements. A set of elements x1 , . . . , xn of a group or of a module is called linearly independent over Z if the equality a1 x1 + · · · + an xn = 0 with integer coefficients implies a1 = · · · = an = 0. All other notations are standard and are adopted from [15]. 2. τ -Adic Numbers ˆ p if mp = ∞. We define For a characteristic χ = (mp ), let Kp = Z/pmp Z if mp < ∞ and Kp = Z
h p the ring Zχ = Kp . For an element (α) ∈ Zχ , we denote by p the greatest power of p such that php p

divides αp in the ring Kp and pgp is the least power of p such that pgp αp = 0. If mp = ∞ and αp = 0, then 0 ≤ hp < ∞ and gp = ∞. In any case, hp + gp = mp for all prime numbers p. Since Q ⊗ Zχ = Q ⊗ Zµ for equivalent characteristics χ and µ, the following definition makes sense. Definition 1. Let τ be a type. The Q-algebra Q(τ ) = Q ⊗ Zχ for a characteristic χ ∈ τ is called ˆ the ring of τ -adic numbers. The ring or Z-module Zχ is said to be associated with Q(τ ). An element (αp ) ∈ Zχ is called a representative of the τ -adic number α = 1 ⊗ (αp ). The types type(α) = [(hp )] and cotype(α) = [(gp )] are called the type of α and the cotype or dual type of α, respectively. The ring Q(τ ) can be considered as a generalization of the ring Zpm . The type and cotype of α are well defined; they are analogues of the height and the order of α, respectively. In particular, Q(τ ) = K if τ = type(Q). Every τ -adic number α can be represented in the form α = rs ⊗ (αp ) with 0 = rs ∈ Q relative to an associated ring Zχ . If β = sr11 ⊗ (βp ) is the representation of the τ -adic number β relative to another associated ring, then α = β if and only if rs1 αp = r1 sβp for all prime numbers p with mp = ∞ and for almost all, i.e., all but finitely many, primes with mp < ∞. In particular, the rings Kp also coincide for this set of primes. Hence Q(τ ) depends on the type and not on the characteristic. Let χ = (mp ) ∈ τ be a characteristic with corresponding type τ . Then      ˆp ⊂ L : = ˆ p. Z Q Kp = Kp ⊕ Kp ⊕ Zχ = p

mp <∞

mp =∞

mp <∞

mp =∞

ˆ p by Q ˆ p . Considered as groups, Zχ and L Thus, L is obtained from Zχ by replacing the components Z have the same torsion subgroup, namely  Kp . T (Zχ ) = T (L) = mp <∞

The set ˆ p for almost all p with mp = ∞} S = {(αp ) ∈ L | αp ∈ Z 431

is a subring of L with torsion subgroup T (S) = T (L), which is an ideal, and Q(τ ) = S/T (S). This is a representation of τ -adic numbers without tensoring by Q. Thus, every τ -adic number, and not only those of the form 1 ⊗ (αp ), can be represented as an element of S. We call an element of S contained in α ∈ S/T (S) a representative of α, and all elements of S are representatives of τ -adic numbers. The ˆ Z-module S is not associated with Q(τ ) in general, but every representative of an element α of type 0 ˆ generates a Z-submodule of S associated with Q(τ ). For later use, we collect some mostly elementary properties of τ -adic numbers. These are direct consequences of the definition, and most of them can be found in [6] and in other papers. Let σ and τ be types and let α, β, . . . be τ -adic numbers. (1) type(α) ≤ τ and cotype(α) ≤ τ . (2) type(α) + cotype(α) = τ . (3) cotype(α) = τ − type(α). (4) type(α) ≥ τ − cotype(α). (5) The conditions α = 0, type(α) = τ , and cotype(α) = 0 are equivalent. (6) α divides β if and only if type(α)  ≤ type(β). (7) type(αβ) = type(α) + type(β) ∧ τ . (8) α is invertible if and only if type(α) = 0. (9) Every finite set of τ -adic numbers α1 , . . . , αn has a greatest common divisor gcd(α1 , . . . , αn ). (10) The greatest common divisor of α1 , . . . , αn allows a representation gcd(α1 , . . . , αn ) = β1 α1 + · · · + βn αn , (11) (12) (13) (14)

where β1 , . . . , βn are also τ -adic numbers. Every finitely generated ideal of the ring Q(τ ) is principal. Iσ = {β ∈ Q(τ ) | type(β) ≥ σ} is an ideal in Q(τ ). For all α ∈ Q(τ ), we have αQ(τ ) = Itype(α) . Q(τ ) = K/Iτ , where Iτ = {γ ∈ K | type(γ) ≥ τ }. cotype(α) ≤ σ ∧ τ if and only if type(α) ≥ τ − σ ∧ τ . 3. Homomorphisms and Quasi-Homomorphisms

The elements of the K-module Q ⊗ HomZˆ (M, M
) are called quasi-homomorphisms from M to M
, ˆ where M and M
are Z-modules. 3.1. Cyclic Modules. First, we are interested in K-module homomorphisms from Q(τ ) to Q(σ) and in ˆ quasi-homomorphisms of their associated Z-modules for arbitrary types τ and σ. We consider the ideal Iσ−σ∧τ = {β ∈ Q(σ) | type(β) ≥ σ − σ ∧ τ } = {β ∈ Q(σ) | cotype(β) ≤ σ ∧ τ } of the ring Q(τ ). ˆ Lemma 1. Let M and M
be Z-modules associated with the K-modules Q(τ ) and Q(σ), respectively, i.e.,
ˆ ˆ M = Zχ and M = Zµ , where χ ∈ τ and µ ∈ σ. Then, under natural identifications,   Q ⊗ HomZˆ (M, M
) = HomK Q(τ ), Q(σ) = Iσ−σ∧τ . ˆ p if mp = ∞ and let Proof. Let χ = (mp ) ∈ τ and µ = (kp ) ∈ σ. Let Kp = Z/pmp Z if mp < ∞ or Kp = Z
k



p ˆ Kp = Z/p Z or Kp = Zp . Let f : Kp → Kp defined by f (1) = αp ∈ Kp be the local component of some homomorphism in HomZˆ (M, M
), where M = Zχ and M
= Zµ . The order of αp
is less than or equal to pmin{kp , mp } . Hence αp
is divisible by plp , where lp = kp − min{kp , mp }. Note that lp = ∞ if and only if kp = ∞ and mp < ∞. The divisibility of αp
by p∞ implies αp
= 0. Thus, αp
∈ plp Kp
. On the other hand, if αp
is divisible by plp , then the order of αp
is less than or equal to pmin{kp ,mp } ˆ p -module homomorphism f : Kp → K
such that f (1) = α
. Identifying f and α
, we and there exists a Z p p p 432

get the equality HomZˆ p (Kp , Kp
) = plp Kp
. These local equalities imply that   HomZˆ (M, M
) = HomZˆ (Kp , Kp
) = plp Kp
. p

Tensoring by Q, we obtain

p

Q ⊗ HomZˆ (M, M
) = Iσ−σ∧τ ⊂ Q(σ).

Every element of Q ⊗ HomZˆ (M, M
) can be represented as r ⊗ f , where r ∈ Q and f ∈ HomZˆ (M, M
). We define a K-module homomorphism   η : Q ⊗ HomZˆ (M, M
) → HomK Q(τ ), Q(σ) as follows: η(r ⊗ f ) : q ⊗ x → rq ⊗ f (x) for every q ⊗ x ∈ Q(τ ), where q ∈ Q and x ∈ M . The homomorphism η is a K-module isomorphism. Identifying along it, we obtain   Q ⊗ HomZˆ (M, M
) = HomK Q(τ ), Q(σ) . The ideal Iσ−(σ−σ∧τ ) of the ring K coincides with the kernel of any K-module homomorphism g : K →   Q(σ), where type g(1) = σ − σ ∧ τ . Thus,   Im g = Iσ−σ∧τ ∼ = K/Iσ−(σ−σ∧τ ) = Q σ − (σ − σ ∧ τ ) . Note that σ − (σ − σ ∧ τ ) ≤ σ ∧ τ and the equality does not hold in general. Definition 2. Let α ∈ Iσ−σ∧τ ⊂ Q(σ) and β ∈ Q(τ ). By Lemma 1, let a multiplication ◦ be defined as where f ∈ HomK



β ◦ α = f (β) ∈ Q(σ),  Q(τ ), Q(σ) and f (1) = α.

There is an equivalent definition of this multiplication. Since Q(σ) = K/Iσ and Q(τ ) = K/Iτ by property (13) of τ -adic numbers, the elements are of the form α = α1 + Iσ and β = β1 + Iτ , where α1 , β1 ∈ K are universal numbers. Then β ◦ α = β1 α1 + Iσ . By type(α) ≥ σ − σ ∧ τ , this multiplication is well defined. Note the following properties of the multiplication ◦. (1) Iσ−σ∧τ is a module over Q(τ ) with   scalar multiplication ◦. (2) type(α ◦ β) = type(α) + type(β) ∧ σ. (3) Let β1 ∈ Q(τ1 ), β2 ∈ Q(τ2 ), β3 ∈ Q(τ3 ), type(β1 ) ≥ τ1 − τ1 ∧ τ2 , and type(β2 ) ≥ τ2 − τ2 ∧ τ3 . Then type(β1 ◦ β2 ) ≥ τ1 − τ1 ∧ τ3 and β1 ◦ (β2 ◦ β3 ) = (β1 ◦ β2 ) ◦ β3 . 3.2. Finitely Presented Modules. Recall that a K-module M is called finitely presented if there exists an exact sequence Kr −→ Ks −→ M −→ 0 for some natural numbers r and s. Theorem 1 ([8]). Finitely presented K-modules are of the form

m  i=1

Q(τi ) for some types τ1 , . . . , τm .

By Lemma 1, it is easy to describe homomorphisms of finitely presented K-modules. Definition 3. An arbitrary finite sequence of types Υ = (τ1 , . . . , τm ) is called a generalized type and MΥ denotes the set of (m × n)-matrices A = (αij ), where n is any natural number, such that the ith row (αi1 , . . . , αin ) of A consists of τi -adic numbers for each i = 1, . . . , m. 433

Let Σ = (σ1 , . . . , σk ) be another generalized type. We define the set MΥ Σ as a subset of MΣ consisting of (k × m)-matrices ∆ = (βij ) such that type(βij ) ≥ σi − σi ∧ τj for all 1 ≤ i ≤ k and 1 ≤ j ≤ m. A multiplication MΥ Σ × MΥ → MΣ is defined by ∆A = (γij ) ∈ MΣ , a (k × n)-matrix, where γij = βi1 ◦ α1j + · · · + βim ◦ αmj . This is the regular matrix product, where the entries are multiplied using ◦. Due to the associativity of the multiplication ◦, the multiplication of matrices is associative as well. If A ∈ MΥ , ∆ ∈ MΥ Σ , and , where Υ, Σ, and Ω are generalized types, then (∆ ∆)A = ∆ (∆A). ∆1 ∈ MΣ 1 1 Ω ˆ ˆ Replacing each K-module Q(τi ) by its associated Z-module in Theorem 1, we obtain a Z-module m  associated with the K-module Q(τi ). Theorem 2. Let M =

m 

i=1

Q(τi ) and N =

i=1

k 

Q(σj ) be finitely presented K-modules. Let M
and N


j=1

ˆ be Z-modules associated with M and N , respectively. Let Υ = (τ1 , . . . , τm ) and Σ = (σ1 , . . . , σk ) be generalized types. Then Q ⊗ HomZˆ (M
, N
) = HomK (M, N ) = MΥ Σ.
and N
= N
⊕ · · · ⊕ N
, where M
and N
are Z-modules ˆ associated Proof. Let M
= M1
⊕ · · · ⊕ Mm 1 j i k with suitable direct summands of the K-modules M and N . Then, identifying as in Lemma 1, we obtain     HomZˆ (Mj
, Ni
) = HomK Q(τj ), Q(σi ) Q ⊗ HomZˆ (M
, N
) = Q ⊗  i,j i,j Iσi −σi ∧τj = MΥ = HomK (M, N ) = Σ. i,j

Let us denote the isomorphism of Theorem 2 in a more explicit form. We choose natural bases so that m k   M= yi Q(τi ) and N = vj Q(σj ). Every matrix ∆ ∈ MΥ Σ corresponds to a K-module homomorphism i=1

j=1

ϕ : M → N , which is defined by the matrix equality (ϕy1 , . . . , ϕym ) = (v1 , . . . , vk )∆. Let x = α1 y1 + · · · + αm ym ∈ M , i.e., in matrix form x = (y1 , . . . , ym )[x], where [x] = (αi ) is the column of coordinates with respect to the basis y1 , . . . , ym . Then ϕx = (ϕy1 , . . . , ϕym )[x] = (v1 , . . . , vk )∆[x]. Denoting the column of coordinates of the element ϕx with respect to the basis v1 , . . . , vk by [ϕx], we finally obtain [ϕx] = ∆[x]. 3.3. The Defect of a Matrix. Now we consider a special case, where the generalized type Υ = (τ1 , . . . , τm ) is arbitrary but the generalized type Σ = (τ ) consists of only a single type τ . Let ∆ = (β1 , . . . , βm ) ∈ MΥ Σ , i.e., type(βi ) ≥ τ − τ ∧ τi , and A ∈ MΥ . The row of τ -adic numbers ∆A = β1 ◦ (α11 , . . . , α1n ) + · · · + βm ◦ (αm1 , . . . , αmn ) is called a τ -combination of the rows of A with coefficients β1 , . . . , βm . A τ -combination is called cancelled if gcd(β1 , . . . , βm ) = 1. The defect of a matrix A ∈ MΥ is the maximal type τ such that there exists a cancelled τ -combination of rows of A that is equal to zero, i.e., equal to (0, . . . , 0). Let I ∈ MΥ denote the unit matrix of size m × m, i.e., I = (δij ), where δij = 0 if i = j and δii = 1 ∈ Q(τi ) for all i. Theorem 3. For a generalized type Υ = (τ1 , . . . , τm ) and a matrix A ∈ MΥ , the following are equivalent. (1) A has defect zero. 434

(2) The columns of A generate the K-module M =

m 

Q(τi ).

i=1

(3) A is invertible on the right, i.e., there is a matrix B with universal entries such that AB = I ∈ MΥ . (4) For all types τ , only trivial τ -combinations are zero, i.e., all coefficients are zero. Proof. We consider the columns x
1 , . . . , x
n of the matrix A as elements of the K-module M = The columns y1 , . . . , ym of the matrix I ∈ MΥ form the natural basis of the K-module M = m  yi Q(τi ) such that x
i = α1i y1 + · · · + αmi ym , i = 1, . . . , n.

m 

Q(τi ).

i=1 m 

Q(τi ) =

i=1

i=1

(1) =⇒ (2). As in [8, 1.1], two K-modules x
1 , . . . , x
n K and M/ x
1 , . . . , x
n K are finitely presented. Suppose that the second one is different from zero. By Theorem 1, we have the isomorphism M/ x
1 , . . . , x
n K ∼ =

k 

uj Q(σj ),

j=1

where u1 , . . . , uk is a basis of this module and σ1 = 0. Applying the projection, we obtain a K-module homomorphism M → u1 Q(σ1 ), which gives a σ1 -combination with coefficients ∆ = (γ1 , . . . , γm ) by Theorem 2. The combination is equal to zero, A∆ = 0, because ϕ(x
1 ) = · · · = ϕ(x
n ) = 0. On the other hand, since the homomorphism ϕ is surjective, it follows that ϕ(β1 y1 +· · ·+βm ym ) = u1 . Hence (γ1 ◦ β1 + · · · + γm ◦ βm )u1 = u1 and γ1 ◦ β1 + · · · + γm ◦ βm = 1. Thus, the σ1 -combination ∆ = (γ1 , . . . , γm ) is cancelled and the defect of A must be greater than or equal to σ1 . This contradiction shows that M/ x
1 , . . . , x
n K = 0 and M = x
1 , . . . , x
n K . (2) =⇒ (3). The equalities yj = x
1 β1j + · · · + x
n βnj ,

j = 1, . . . , m,

with universal coefficients mean that the matrix B = (βij ) satisfies condition (3). (3) =⇒ (4). Let a τ -combination of rows of A with coefficients ∆ = (γ1 , . . . , γm ) ∈ MΥ τ be equal to the zero row, i.e., ∆A = (0, . . . , 0). Then 0 = ∆(AB) = ∆I = ∆ = (γ1 , . . . , γm ). Thus, all coefficients are zero. (4) =⇒ (1). If τ is the defect of A, then there is a cancelled τ -combination that is zero, and by (4) all coefficients must be zero. Thus, τ is zero. Note that a modification of this proof shows that the defect of A ∈ MΥ exists and is equal to σ1 ∨ · · · ∨ σk in our notation. A matrix A ∈ MΥ is said to be reduced if its defect is zero. Corollary 1. Rows of a reduced matrix have greatest common divisor 1. Proof. If all coefficients of a row are divisible by a number of a nonzero type, then the same is true after multiplication by a matrix on the right, contradicting Theorem 3 (3). 4. Equivalences and Dualities Recall that two categories A and B are equivalent (dual ) if there exist two covariant (contravariant)
functors f : A → B and f
: B → A such that f f
∼ idB and f
f ∼ idA . The notation  f f ∼ idA means
that for every object A of the category A there exists an isomorphism gA : A → f f (A) such that for 435

every morphism h : A → B of the category A the following diagram is commutative h

−−−−→ B A    g gA  B     f
f (A) −−−−−→ f
f (B) .

(4.1)

f
(f (h))

4.1. The Categories M and RM. We denote by Mat Q the set of all rectangular matrices with rational entries of arbitrary size. Let Υ = (τ1 , . . . , τm ) be a generalized type. Since the ring Q(τ ) is a Q-algebra for every type τ , every matrix A ∈ MΥ of size m × n can be multiplied on the right by a matrix L ∈ Mat Q of size n × l and the product belongs to the set MΥ as well, by definition. Definition 4. Objects of the category M are elements of the union MΥ over all generalized types Υ. Let Υ

A ∈ MΥ and B ∈ MΣ be two objects of the category M, where Υ and Σ are generalized types. Morphisms from A to B are pairs of matrices (∆, L), where ∆ ∈ MΥ Σ and L ∈ Mat Q, such that ∆A = BL. The product of the morphism (∆, L) : A → B by a morphism (∆1 , L1 ) : B → C of the category M is defined by (∆1 , L1 )(∆, L) = (∆1 ∆, L1 L) : A → C. The identity of the object A is the pair of identity matrices of suitable size. RM is the full subcategory of M consisting of all reduced matrices. We also need the following three categories. The category QT F is classical. Its objects are torsion-free Abelian groups of finite rank and morphisms are quasi-homomorphisms. The category QD was introduced in [13]. The objects of QD are quotient divisible groups. A quotient divisible group G is an Abelian group without torsion divisible subgroups that contains a finite-rank free subgroup F such that the quotient G/F is torsion and divisible. The rank of F is called the rank of the quotient divisible group G. A free basis of F is called a basis of the quotient divisible group G. Morphisms of the category QD are also quasi-homomorphisms. The third category L was introduced in [8]. The objects of this category are linear mappings g : U → M from finite-dimensional vector spaces U over the field of rational numbers to finitely presented modules M over the ring of universal numbers, which are considered as vector spaces over Q, such that the image of U under g generates M as a K-module. The morphisms from an object g : U → M to an object g1 : U1 → M1 are pairs (f, ϕ), where f : U → U1 is a linear mapping and ϕ : M → M1 is a K-module homomorphism, such that the following diagram is commutative: f

U −−−−→ U1    g1 g  M −−−−→ M1 . ϕ

Theorem 4. The category RM is equivalent to the category QD and dual to the category QT F. Proof. Let A be an object of RM, in particular, A ∈ MΥ , where Υ = (τ1 , . . . , τm ). Let A = (x
1 , . . . , x
n ) m  be written by columns. These columns are elements of a K-module M = yi Q(τi ) and they generate M n i=1  xj Q → M given by over K, because the matrix A is reduced. Thus, the linear mapping g : U = j=1 g(x1 ) = x
1 ,. . . , g(xn ) = x
n is an object of the category L. Let (∆, L) : A → B be a morphism of the category RM, where B = (z1
, . . . , zl
) corresponds analogously to an object l k   z i Q → M1 = vi Q(σi ). g1 : U1 = i=1

436

i=1

Then we define a K-module homomorphism ϕ : M → M1 by the matrix ∆ ∈ MΥ Σ , i.e., (ϕx
1 , . . . , ϕx
n ) = (∆x
1 , . . . , ∆x
n ). And we define a linear mapping f : U → U1 by the matrix L, where   f (x1 ), . . . , f (xn ) = (z1 , . . . , zl )L. Now we obtain

    ϕ g(x1 ) , . . . , ϕ g(xn ) = (∆x
1 , . . . , ∆x
n ) = ∆A = BL = (z1
, . . . , zl
)L.

On the other hand, applying the linear mapping g1 to the equality   f (x1 ), . . . , f (xn ) = (z1 , . . . , zl )L in U1 , we obtain

    g1 f (x1 ) , . . . , g1 f (xn ) = (z1
, . . . , zl
)L.

Hence ϕg = g1 f and the pair (f, ϕ) is a morphism of the category L. It is easy to see that the functor RM → L constructed above is an equivalence of the categories. Since the category L is dual to the category QT F by [8] and the category QT F is dual to the category QD by [13], the result follows, because a composition of two dualities is an equivalence. 4.2. Representation of Two Groups by One Matrix. We define two functors c : M → QD and b : M → QT F such that their restrictions on the subcategory RM are an equivalence and a duality, respectively, as mentioned in Theorem 4. Let A be an object of the category M, i.e., A ∈ MΥ , where Υ = (τ1 , . . . , τm ). First, we replace every type τi by a characteristic χi ∈ τi . Since a multiplication of the matrix A by a nonzero rational number is an isomorphism of the category M, we may assume,
= 1 ⊗ α , where α ∈ Z without loss of generality, that every entry of the matrix A is of the form αij ij ij χi
are representatives of τi -adic numbers. We replace all the entries αij by αij . A different choice of representatives, in general, leads to different pairs of quotient divisible and torsion-free, finite-rank groups, but they will be quasi-equal. Now we consider the following diagram: y1 .. .

x1 α11 .. .

x2 α12 .. .

... ...

xn α1n .. .

= 0 .. .

ym αm1 αm2 . . . αmn = 0   ...  x
2 . . . x
n . x
1

(4.2)

We emphasize that this diagram uniquely determines a quotient divisible group c(A) and a torsion-free, finite-rank group b(A), together with bases x1 , . . . , xn ∈ b(A) and x∗1 , . . . , x∗n ∈ c(A) such that c(A) and b(A) are dual in the sense of the dualities d : QD → QT F and d
: QT F → QD constructed in [13], and the bases are also mutually dual. Diagram (4.2) determines two systems of equalities simultaneously: x
1 = α11 y1 + α21 y2 + · · · + αm1 ym ,

x
2 = α12 y1 + α22 y2 + · · · + αm2 ym , ...

(4.3)

x
n = α1n y1 + α2n y2 + · · · + αmn ym 437

and α11 x1 + α12 x2 + · · · + α1n xn = 0, α21 x1 + α22 x2 + · · · + α2n xn = 0, ... αm1 x1 + αm2 x2 + · · · + αmn xn = 0.

(4.4)

We call (4.3) the QD-situation and (4.4) the QT F-situation. m m   ˆ yi Zχi is associated with yi Q(τi ), which is a K-module. The QD-situaThe Z-module M = i=1

i=1

tion (4.3) shows that x
1 , . . . , x
n ∈ M . Then K = x
1 , . . . , x
n Zˆ is a submodule of M . The pure hull H = x
1 , . . . , x
n ∗ in the additive group of K is the desired quotient divisible group c(A) with the basis x∗1 = x
1 , . . . , x∗n = x
n if the elements x
1 , . . . , x
n are linearly independent over Z. If rk(x
1 , . . . , x
n ) = r, where r < n, and, say, x
1 , . . . , x
r are linearly independent, then c(A) = H ⊕ ur+1 Q ⊕ · · · ⊕ un Q and the basis of c(A) is x∗1 = x
1 , . . . , x∗r = x
r , x∗r+1 = x
r+1 + ur+1 , . . . , x∗n = x
n + un . Considering the QT F-situation (4.4), we denote χi = (mp ) and αij = (αpij ) ∈ Zχi . The group b(A) is n  xj Q generated by all elements of the following form: a subgroup of the group j=1

(1) x1 , . . . , xn ; p (mod pmp ) if mp < ∞; (2) p−mp (api1 x1 + · · · + apin xn ), where apij are integers satisfying apij ≡ αij pk pk pk p k (3) p−k (apk i1 x1 + · · · + ain xn ), where aij are integers satisfying aij ≡ αij (mod p ), 0 < k ∈ Z, if mp = ∞.

The basis of the group b(A) is x1 , . . . , xn . Let B ∈ MΣ be another matrix in M with a generalized type Σ = (σ1 , . . . , σk ). Let (∆, L) : A → B be a morphism of the category M. Analogously, B determines a quotient divisible group c(B) with a basis z1∗ , . . . , zl∗ and a torsion-free, finite-rank group b(B) with a basis z1 , . . . , zl according to the diagram z1 z2 v1 β11 β12 .. .. .. . . . vk βk1 βk2  
z1 z2


. . . zl . . . β1l = 0 .. .. . . . . . βkl = 0 ...  . . . zl


Then the size of the matrix ∆ ∈ MΥ Σ is k × m and the size of the matrix L ∈ MatQ is l × n. We define quasi-homomorphisms f = b(∆, L) : b(B) → b(A) and f ∗ = c(∆, L) : c(A) → c(B) on the bases by the following matrix equalities:     x1 f (z1 )   ∗ ∗  ..   ..  f (x1 ), . . . , f ∗ (x∗n ) = (z1∗ , . . . , zl∗ )L.  .  = L . , f (zl ) xn The inverse functors c
: QD → RM and b
: QT F → RM have been constructed in [13]. In fact, matrices have intermediate links in the construction of the duality d : QD → QT F and d
: QT F → QD such that d = bc
and d
= cb
(see also [7, 8] for the functors b and b
and [10] for the functors c and c
). Here we 438

RM



-

distinguish matrices as a special category RM or M and we extend the functors c and b to the whole category M. This will be useful for applications. Thus, we have the commutative diagram

QD 

b
c

d d


b

-



c


-

QT F.

The commutativity of the diagram means that every way by the arrows is isomorphic in the sense of diagram (4.1) to the shortest way. The shortest way from a category to itself is the identity. 4.3. Equivalent Matrices in the Category M. Matrices A ∈ MΥ and B ∈ MΣ of generalized types Υ = (τ1 , . . . , τm ) and Σ = (σ1 , . . . , σk ), respectively, are called equivalent if the matrix B can be obtained from the matrix A using a finite number of “elementary” operations of four kinds: (1) permutation of rows; (2) addition or elimination of a τ -adic combination of the remaining rows. The generalized type of the matrix B is (τ1 , . . . , τm , τ ). The first m rows of B coincide with the rows of A and the last (m + 1)th row of B is a τ -adic combination of the rows of the matrix A. This operation is the passage from A to B or from B to A; (3) cancellation by a common divisor of a row. Let the ith row of the matrix A be represented
, . . . , α ◦ α
), where α ∈ Q(τ ), type(α) = τ ≤ τ and in the form (αi1 , . . . , αin ) = (α ◦ αi1 i i in


, . . . , α
), the remaining rows of B αi1 , . . . , αin ∈ Q(τi −τ ). Then the ith row of the matrix B is (αi1 in coincide with the corresponding rows of A, and the generalized type of B is (τ1 , . . . , τi −τ, . . . , τm ); (4) multiplication of a row by a τ -adic number. Let τ be a type satisfying τi = τ − σ for a type σ. Let α ∈ Q(τ ) and type(α) = σ. Then type(α) ≥ τ − τi and the multiplication (α ◦ αi1 , . . . , α ◦ αin ) is well defined. The operation is the replacing of a row (αi1 , . . . , αin ) of the type τi by the row (α ◦ αi1 , . . . , α ◦ αin ) of the type τ . Elementary operations of the kind (1) and (2) are invertible. Elementary operations of the kind (3) and (4) are mutually invertible. Hence the equivalence of matrices is, in fact, an equivalence relation. Theorem 5. Let A ∈ MΥ and B ∈ MΣ be equivalent matrices. Then c(A) = c(B) and b(A) = b(B). Proof. It is sufficiently obvious that elementary operations do not change b(A). Therefore, they do not change c(A) either. The second statement of the theorem is based on the correspondence between τ -adic relations of the group b(A) and locally cyclic subgroups of the group b(A)/ x1 , . . . , xn , which will be established in the next section. Namely, for a direct decomposition (5.6), we obtain a reduced matrix (5.8). The matrix (5.8) can be added to the matrix A; then the matrix A can be eliminated, and all of this is realized by elementary operations. 5. Connection with the Pontryagin Duality Let C be a locally cyclic periodic group, i.e., C is isomorphic to a subgroup of the group Q/Z. The group C can be represented in the form C ∼ = R/Z, where Z ⊂ R ⊂ Q. The characteristic of 1 in R is called the characteristic of the locally cyclic group C. 1 ∈ R, such that m We choose a generator y(m) of each cyclic subgroup C[m], where 0 < m k y(m) = y(k) for every positive divisor k of the integer m. Such a set of elements    1  ∈R (5.1) y = y(m)  m 439

is said to be a generating set of the group C. A generating set (5.1) of elements of a locally cyclic group C gives a homomorphism   1 y 0 : C → Q/Z by y 0 y(m) = + Z ∈ Q/Z. (5.2) m Lemma 2. Let C be a locally cyclic group of the characteristic χ with a generating set of elements y. ˆ Moreover, Then Hom(C, Q/Z) = y 0 Zχ and it is a cyclic module over the ring Z. Hom ˆ (Hom(C, Q/Z), Q/Z) ∼ = C. Z

ˆ Proof. Z-module homomorphisms f : y 0 Zχ → Q/Z are in one-to-one correspondence with elements f (y 0 ) k 1 of the form m + Z ∈ Q/Z, where m ∈ R. Since our goal is a new interpretation of the duality d
: QT F → QD, we need to summarize constructions that have been useful in [13]. We do it below for the reader’s convenience. Let A be a torsion-free, finite-rank group with a maximal linearly independent set of elements x1 , . . . , xn and F = x1 , . . . , xn . Then we get a short exact sequence 0 −→ F −→ A −→ A/F −→ 0.

(5.3)

n xn Let z + F ∈ (A/F )[m], where z ∈ A. Then z = a1 x1 +···+a . The integer coefficients a1 , . . . , an m are uniquely determined modulo m by the element z + F . Thus, an element z + F uniquely determines a relation in the Zm -module A/mA: α1 x1 + · · · + αn xn = 0, (5.4) where αi = ai + mZ ∈ Zm for i = 1, . . . , n. We use here and below the same notation for the elements xi and for their images in the group A/mA to avoid complications. Tensoring the sequence (5.3) by Zm , we obtain an exact sequence:

0 −→ Tor(Zm , A/F ) −→ Zm ⊗ F −→ Zm ⊗ A −→ Zm ⊗ (A/F ) −→ 0. Using well-known isomorphisms, we obtain the exact sequence fm

0 −→ (A/F )[m] −→ x1 Zm ⊕ · · · ⊕ xn Zm −→ A/mA. Note that fm (z + F ) is exactly the relation (5.4). For a positive divisor k of the integer m, we finally obtain the commutative diagram fm

0 −−−−→ (A/F )[m] −−−−→ x1 Zm ⊕ · · · ⊕ xn Zm −−−−→ A/mA    ϕm ψm ξm  k  k k 0 −−−−→ (A/F )[k] −−−−→ x1 Zk ⊕ · · · ⊕ xn Zk −−−−→ A/kA , fk

where m ξkm (y) = y, y ∈ (A/F )[m], k   ϕm + mZ)x + · · · + (a + mZ)x (a = (a 1 1 n n 1 + kZ)x1 + · · · + (an + kZ)xn , k ψkm (z

+ mA) = z + kA,

ai ∈ Z, i = 1, . . . , n,

z ∈ A.

The vertical arrows form an inverse spectrum over all the pairs (k, m) such that k is a divisor of m 1 belong to R, where Z ⊂ R ⊂ Q and charR (1) = χ. For inverse limits, we obtain an and both k1 and m ˆ homomorphisms: exact sequence of Zχ -module (Z-module) f

0 −→ (A/F )[χ] −→ x1 Zχ ⊕ · · · ⊕ xn Zχ −→ Aˆχ .

(5.5)

The Z Aˆχ is the completion of A in the “χ-adic” topology with the base of neighborhoods of  χ -module 1 zero mA | m ∈ R . The correspondence C → C[χ] is a functor from the category of torsion groups to the category of Zχ -modules generalizing the functor C → C[m]. 440

Suppose that the group A/F contains a locally cyclic subgroup C of characteristics χ. A generating   1 set (5.1) of the group C, i.e., y = y(m) | m ∈ R , can be considered as an element of the module (A/F )[χ]. This element maps under the homomorphism f in (5.5) to an element α1 x1 + · · · + αn xn , which maps to zero due to the exactness of the sequence (5.5). Thus, every generating set y of the group C corresponds to a relation α1 x1 + · · · + αn xn = 0 holding in the Zχ -module Aˆχ . We now consider an arbitrary direct decomposition: A/F = C1 ⊕ · · · ⊕ Cm ,

(5.6)

where C1 , . . . , Cm are locally cyclic subgroups of characteristics χ1 , . . . , χm , respectively. Choosing generating sets (5.7) y1 , . . . , ym for the locally cyclic groups C1 , . . . , Cn , respectively, we obtain relations α11 x1 + · · · + α1n xn = 0 in Aˆχ , 1

...

(5.8)

αm1 x1 + · · · + αmn xn = 0 in Aˆχm . The coefficients of the ith equality are elements of the ring Zχi . The matrix of the system (5.8) is exactly b
(A) ∈ RM if each of its entries is tensor multiplied by the rational unit. The fact that the sum (5.6) is direct implies that the defect of the matrix is zero. Even without the multiplication - ⊗ Q, the matrix (5.8) gives a QD-situation over columns. This QD-situation gives a quotient divisible group d
(A), as is shown in Sec. 4.2. This completes the construction of d
(A). Assume that a decomposition (5.6) with generating sets (5.7) is chosen. Every generating set yi gives a homomorphism yi0 : Ci → Q/Z by (5.2). We define a homomorphism yi0 : A/F → Q/Z for every i = 1, . . . , m in the following way. Let z = c1 +· · ·+cm ∈ C1 ⊕· · ·⊕Cm = A/F . Then yi0 (z) = yi0 (ci ) ∈ Q/Z. By Lemma 2, we obtain that 0 Zχm , Hom(A/F, Q/Z) = y10 Zχ1 ⊕ · · · ⊕ ym

where χ1 , . . . , χm are characteristics of C1 , . . . , Cm , respectively. Note that the group Hom(A/F, Q/Z) is exactly the group of Pontryagin’s characters for the discrete group A/F . This group of characters is compact in its Z-adic topology, which coincides with the natural topology by Pontryagin in our particular case. For every element xi of the maximal linearly independent set x1 , . . . , xn ∈ A, we define a homomorn xn , where the integer phism x0i : A/F → Q/Z in the following way. Let z + F ∈ A/F and z = a1 x1 +···+a m coefficients a1 , . . . , an are uniquely determined modulo m by the element z + F ∈ A/F . We define ai + Z ∈ Q/Z. (5.9) x0i (z + F ) = m The following lemma can be proved by direct calculation. Lemma 3. The equalities

0 x0i = α1i y10 + α2i y20 + · · · + αmi ym

ˆ hold true in the Z-module

0 Zχm Hom(A/F, Q/Z) = y10 Zχ1 ⊕ · · · ⊕ ym for every i = 1, . . . , n, where the coefficients αij are entries of the matrix (5.8). 0 . Lemma 3 shows that the QD-situation is realized exactly for the elements x01 , . . . , x0n and y10 , . . . , ym Considering the topological group Hom(A/F, Q/Z) algebraically, i.e., forgetting the topology, we obtain the quotient divisible group d
(A) as the pure hull of the elements x01 , . . . , x0n in the additive group of Hom(A/F, Q/Z), d
(A) = x01 , . . . , x0n ∗ , plus possibly a torsion-free divisible group of finite rank, as is shown in Sec. 4.2.

441

On the other hand, let an arbitrary quotient divisible group A∗ be given with a basis x∗1 , . . . , x∗n . This means that the set x∗1 , . . . , x∗n is linearly independent over Z and the quotient A∗ / x∗1 , . . . , x∗n is a torsion divisible group. The following simple but interesting lemma is useful. Lemma 4. Let G be an arbitrary group such that the torsion part t(G) of G is reduced and let z1 , . . . , zk ∈ G be a maximal linearly independent over Z set of elements. The group G is quotient divisible with the basis ˆ z1 , . . . , zk if and only if the images z10 , . . . , zk0 of the elements z1 , . . . , zk under the Z-adic completion G → G 0 0 ˆ i.e., z , . . . , z ˆ = G. ˆ ˆ over the ring Z, generate G 1

k Z

A∗

By Lemma 4, the group can be represented as a pure subgroup of a group M ⊕ D, where M is the ˆ namely M = Aˆ∗ , and D is a torsion-free divisible additive group of a finitely presented module over Z, group, which coincides with the first Ulm’s subgroup of the group A∗ , as shown in [13]: A∗ = x∗1 , . . . , x∗n ∗ ⊂ M ⊕ D. The projection M ⊕ D → M restricted on A∗ is the Z-adic completion of the group A∗ . We recall ˆ that every finitely presented Z-module is of the form M ∼ = y1 Zχ1 ⊕ · · · ⊕ ym Zχm for some characteristics χ1 , . . . , χm (for details, see [7]). By Lemma 2, HomZˆ (M, Q/Z) ∼ = C1 ⊕ · · · ⊕ Cm , where C1 , . . . , Cm are locally cyclic groups of characteristics χ1 , . . . , χm , respectively. Now we define the group A = d(A∗ ) as a subgroup F = x1 Z ⊕ · · · ⊕ xn Z ⊂ A ⊂ x1 Q ⊕ · · · ⊕ xn Q = V + Z be arbitrary elements of Q/Z. We denote a1 x1 + · · · + an xn ∈ V. z = γ1 x1 + · · · + γn xn = m The element z is defined not uniquely but uniquely up to F , i.e., z +F is defined uniquely as an element of V /F , and the subgroup z, x1 , . . . , xn of the group V does not depend on the choice of integer coefficients a1 , . . . , an either. Finally, we define the group A = d(A∗ ) by the set of generators

in the following way. Let γ1 =

a1 m

+ Z, . . . , γn =

an m

A = f (x01 )x1 + · · · + f (x0n )xn | f ∈ HomZˆ (M, Q/Z) . Since the group A/F is a finite direct sum of locally cyclic groups, it follows from Lemma 2 that (5.10) HomZˆ (Hom(A/F, Q/Z), Q/Z) ∼ = A/F. ˆ The Z-module M = Hom(A/F, Q/Z) = Aˆ∗ contains the elements x01 , . . . , x0n which are simultaneously the homomorphisms A/F → Q/Z defined in (5.9) and the images of the basic elements x∗1 , . . . , x∗n ∈ A∗ under ˆ homomorphism f : M → Q/Z corresponds the Z-adic completion A∗ → Aˆ∗ . A Z-module 
to an  element 0 0 f (x )x + · · · + f (x )x + F ∈ A/F under the isomorphism (5.10). This shows that d d (A) = A and n n  1 ∗1 
∗ ∗ ∗ ∗ ∗ d d(A ) = A for dual groups A and A with dual bases x1 , . . . , xn ∈ A and x1 , . . . , xn ∈ A . We summarize this in the following theorem. Theorem 6. Any finite sequence (possibly with repetitions) of elements x01 , . . . , x0n of any finitely presented ˆ ˆ M = x0 , . . . , x0 ˆ , represents two Abelian groups Z-module M that generates this module over the ring Z, n Z 1 A and A∗ with sets of elements x1 , . . . , xn and x∗1 , . . . , x∗n in the following way. The torsion-free group A is disposed between a free group and a divisible group, x1 Z ⊕ · · · ⊕ xn Z ⊂ A ⊂ x1 Q ⊕ · · · ⊕ xn Q, ˆ and it is generated by all the elements of the form f (x01 )x1 + · · · + f (x0n )xn , where f runs all the Z-module homomorphisms f : M → Q/Z, i.e., A = f (x01 )x1 + . . . + f (x0n )xn | f ∈ HomZˆ (M, Q/Z) . 442

Arbitrary elements d1 , . . . , dn are chosen in a torsion-free divisible group D so that the set of elements = x01 + d1 , . . . , x∗n = x0n + dn of the group M ⊕ D is linearly independent over Z. Then the quotient divisible group A∗ is the pure hull of these elements in the group M ⊕ D: x∗1

A∗ = x∗1 , . . . , x∗n ∗ . For every torsion-free, finite-rank group A with a maximal linearly independent set of elements ˆ x1 , . . . , xn (quotient divisible group A∗ with a basis x∗1 , . . . , x∗n ) there exists a finitely presented Z-mod0 0 ∗ ule M with a set of generators x1 , . . . , xn such that the group A (the group A ) is presented in such a way. Correspondences A∗ = d
(A) and A = d(A∗ ) are functors that give the duality between the categories QD and QT F introduced in [13]. ˆ Since every finitely generated submodule of a finitely presented Z-module is itself a finitely presented ˆ Z-module (see [8]), we obtain the following version of Theorem 6. ˆ Corollary 2. Every finite sequence (repetition is possible) of every finitely presented Z-module determines a mutually dual pair of Abelian groups: a torsion-free, finite-rank group and a quotient divisible group. The following proposition takes place in the notations of Theorem 6. Corollary 3. The following three statements are equivalent. (1) M = x01 , . . . , x0k Zˆ ⊕ x0k+1 , . . . , x0n Zˆ . (2) A = B ⊕ C, x1 , . . . , xk ∈ B, and xk+1 , . . . , xn ∈ C. (3) A∗ = B ∗ ⊕ C ∗ , x∗1 , . . . , x∗k ∈ B, and x∗k+1 , . . . , x∗n ∈ C. Now we consider in detail the simplest case of Corollary 2, where only one element x0 of a finitely ˆ presented Z-module M is given. For every prime number p, let mp be the least nonnegative integer such that the element pmp x0 is divisible by every power of p in the module M . If such a number mp does not exist, then we define mp = ∞. The characteristic (mp ) is said to be the co-characteristic of the element x0 , cochar(x0 ) = (mp ) = χ. It is easy to see that ˆ 0 = Zχ x0 ∼ N = x0 Zˆ = Zx = Zχ . Then the group A is torsion-free of rank 1 with maximal linearly independent set x. Herewith A = Rχ x ∼ = Rχ , where Z ⊂ Rχ ⊂ Q and the characteristic of the unit in Rχ is equal to χ. The group A∗ is quotient divisible of rank 1 with basis x∗ . Two cases are possible. If the characteristic χ belongs to a nonzero type, then A∗ = x0 ∗ ⊂ N and x∗ = x0 . In this case, the group A∗ is = 1 ∗ ⊂ Zχ . If the characteristic χ = (mp ) belongs to the zero type, then it isomorphic to the group Rχ
determines the integer m = pmp . Then Zχ = Zm . Let Rχ = Zm ⊕ Q. In this case, x∗ = x0 + d, and we p

obtain A∗ = Zm + Qd ∼ = Rχ . In both cases, we use the notation A∗ = Rχ x∗ . ˆ is called linearly Definition 5. A finite sequence of elements x1 , . . . , xn of a module over the ring Z ˆ if the equality independent over Z ˆ α1 x1 + · · · + αn xn = 0, α1 , . . . , αn ∈ Z, implies α1 x1 = · · · = αn xn = 0. We emphasize the difference between this definition and the definition of the linear independence ˆ but not over Z used in this paper. For example, the sequence of zeros is linearly independent over Z over Z. ˆ Corollary 4. If a finite sequence of elements of a finitely presented Z-module is linearly independent ∗ over Z, then the corresponding quotient divisible group A is reduced and the torsion-free, finite-rank group A is co-reduced, i.e., it does not contain free direct summands. 443

Corollary 5. The following three statements are equivalent. ˆ ˆ (1) The sequence of elements x01 , . . . , x0n of a finitely presented Z-module is linearly independent over Z and the co-characteristics of these elements are equal to χ1 , . . . , χn , respectively. (2) A = Rχ1 x1 ⊕ · · · ⊕ Rχn xn . (3) A∗ = Rχ1 x∗1 ⊕ · · · ⊕ Rχn x∗n . Finally, we distinguish two extreme cases of Corollary 5, using the same notation. Corollary 6. If χ1 = · · · = χn = (∞, ∞, . . . ), then A is a divisible torsion-free group of rank n and A∗ is a free group of rank n. Corollary 7. If x01 = · · · = x0n = 0, then A is a free group of rank n and A∗ is a divisible torsion-free group of rank n. REFERENCES 1. U. Albrecht, H. P. Goeters, and W. Wickless, “The flat dimension of mixed Abelian groups as E-modules,” Rocky Mountain J. Math., 25, 569–590 (1995). 2. U. Albrecht and J. Hausen, “Mixed Abelian groups with the summand intersection property,” in: D. M. Arnold (ed.), Abelian Groups and Modules. Proc. Int. Conf. Colorado Springs, CO, USA, August 7–12, 1995, Lect. Notes Pure Appl. Math., Vol. 182, Marcel Dekker, New York (1996), pp. 123–132. 3. R. Beaumont and R. Pierce, “Torsion-free rings,” Illinois J. Math., 5, 61–98 (1961). ¨ 4. D. Derry, “Uber eine Klasse von abelschen Gruppen,” Proc. London Math. Soc., 43, 490–506 (1937). 5. S. Files and W. Wickless, “The Baer–Kaplansky theorem for a class of global mixed Abelian groups,” Rocky Mountain J. Math., 26, 593–613 (1996). 6. A. A. Fomin, “Abelian groups with one τ -adic relation,” Algebra Logika, 28, 83–104 (1989). 7. A. A. Fomin, “The category of quasi-homomorphisms of Abelian torsion free groups of finite rank,” in: Algebra, Proc. Int. Conf. Memory A. I. Mal’cev, Novosibirsk/USSR 1989, Pt. 1, Contemp. Math., Vol. 131, Amer. Math. Soc., Providence (1992), pp. 91–111. 8. A. A. Fomin, “Finitely presented modules over the ring of universal numbers,” in: R. G¨ obel (ed.), Abelian Group Theory and Related Topics. Conf., August 1–7, 1993, Oberwolfach, Germany, Contemp. Math., Vol. 171, Amer. Math. Soc., Providence (1994), pp. 109–120. 9. A. A. Fomin, “Some mixed Abelian groups as modules over the ring of pseudo-rational numbers,” in: Abelian Groups and Modules, Trends in Mathematics, Birkh¨ auser, Basel (1999), pp. 87–100. 10. A. A. Fomin, “Quotient divisible mixed groups,” in: A. V. Kelarev (ed.), Abelian Groups, Rings and Modules. Proc. AGRAM 2000 Conf., Perth, Australia, July 9–15, 2000, Contemp. Math., Vol. 273, Amer. Math. Soc., Providence (2001), pp. 117–128. 11. A. Fomin and O. Mutzbauer, “Torsion-free Abelian α-irreducible groups of finite rank,” Comm. Algebra, 22, 3741–3754 (1994). 12. A. A. Fomin and W. Wickless, “Categories of mixed and torsion-free finite rank Abelian groups,” in: Facchini A. (ed.), Abelian Groups and Modules. Proc. Padova Conf., Padova, Italy, June 23–July 1, 1994, Math. Appl., Dordr., Vol. 343, Kluwer Academic, Dordrecht (1995), pp. 185–192. 13. A. A. Fomin and W. Wickless, “Quotient divisible Abelian groups,” Proc. Amer. Math. Soc., 126, 45–52 (1998). 14. A. A. Fomin and W. Wickless, “Self-small mixed Abelian groups G with G/T (G) finite rank divisible,” Comm. Algebra, 26, 3563–3580 (1998). 15. L. Fuchs, Infinite Abelian Groups, Academic Press, New York (1970, 1973). 16. S. Glaz and W. Wickless, “Regular and principal projective endomorphism rings of mixed Abelian groups,” Comm. Algebra, 22, 1161–1176 (1994). 17. B. Jonsson, “On direct decompositions of torsion-free Abelian groups,” Math. Scand., 7, 230–235 (1957). 444

18. A. G. Kurosh, “Primitive torsionsfreie abelsche Gruppen vom endlichen Range,” Ann. Math., 38, 175–203 (1937). 19. A. I. Maltsev, “Torsion-free Abelian groups of finite rank,” Mat. Sb., 4, 45–68 (1938). 20. L. S. Pontryagin, “The theory of topological commutative groups,” Ann. Math., 35, 361–388 (1934). 21. W. J. Wickless, “A functor from mixed groups to torsion-free groups,” in: R. G¨ obel (ed.), Abelian Group Theory and Related Topics. Conf., August 1–7, 1993, Oberwolfach, Germany, Contemp. Math., Vol. 171, Amer. Math. Soc., Providence (1994), pp. 407–417. A. A. Fomin Moscow Pedagogical State University E-mail: [email protected]

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