Automorphisms and Derivations of Associative Rings
Mathematics and Its Applications (Soviet Series)
Managing Editor: M. HAZEWINKEL CelUrejor Marhematics and Compurer Science. Amsterdam. The Netherlands
Editorial Board: A. A. KIRILLOV, MGV, Moscow. U.S.S.R. Yu. I. MANIN, SreklOl' illsritlite ojMarhematics, Moscow, U.S.S.R. N. N. MOISEEV, Complltillg Celltre, Academy ofSciellces, Moscow, U.S.S.R. S. P. NOVIKOV, Landauinstitllle oj'Theoretical Physics, Moscow, U.S.S.R. M. C. POLYVANOV, Stekloj' Illstitute oj'Mathematics, Moscow, U.S.S.R. Yu. A. ROZANOV, SteklOl' insritllte of Mathematics, MoscoH', U.S.S.R.
Volume 69
Automorphisms and Derivations of Associative Rings by
V. K. Kharchenko Institute/or Mathematics, Novosibirsk, U.S.S.R.
KLUWER ACADEMIC PUBLISHERS DORDRECHT I BOSTON I LONDON
Library of Congress Cataloging-in-Publication Data Knarchenko, V. K. Automorph1sms and derivat10ns of associat1ve rings! by V.K. Kharchenko. p. em. -- (Mathemat1cs and 1tS appl1cations. Sov1et series 69) Translated from the Russ1an. Inc I udes bib Ii ograph 1ca I references and 1ndex. ISBN 0-7923-1382-8 (ac1d free paper) 1. Assoc1at1ve rings. 2. Automorph1sms. I. T1tle. II. Ser1es, Mathematics and its appl1Catlons (Kluwer Academ1c Publ1Shers). Sov1et series; 69. QA251.5.K48 1991 512' .24--dc20 91-25930
ISBN 0-7923-1382-8
Published by Kluwer Academic Publishers, P,O. Box 17,3300 AA Dordrecht, The Netherlands. Kluwer Academic Publishers incorporates the publishing programmes of D. Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press. Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A.
In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, The Netherlands.
Printed on acid-free paper
Translated from the Russian by
L. fuzina
All Rights Reserved © 1991 Kluwer Academic Publishers No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.
In memory of my teachers AKILOV Gleb Pavlovich KARGAPOLOV Mikhail Ivanovich SHIRSHOV Anatolii lllarionovich
SERIES EDITOR'S PREFACE
~t moi, ...• si favait su comment en revenir. je n'y serais point aile.' Jules Verne
One sel'Yice mathematics has rendered the human race. It has put common sense back where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non-
The series is divergent; therefore we may be able to do something with it. O. Heaviside
sense',
Eric T. Bell
Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and nonlinearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered computer science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d 'e\re of this series. This series, Mathematics and Its Applications, started in 1977. Now that over one hundred volumes have appeared it seems opportune to reexamine its scope. At the time I wrote "Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the 'tree' of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algcbraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as 'experimental mathematics', 'CFO', 'completely integrable systems', 'chaos, synergetics and large-scale order', which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics." By and large, all this still applies today. It is still true that at first sight mathematics seems rather fragmented and that to find, see, and exploit the deeper underlying interrelations more effort is needed and so are books that can help mathematicians and scientists do so. Accordingly MIA will continue to try to make such books available. If anything, the description I gave in 1977 is now an understatement. To the examples of interaction areas one should add string theory where Riemann surfaces, algebraic geometry, modular functions, knots, quantum field theory, Kac-Moody algebras, monstrous moonshine (and more) all come together. And to the examples of things which can be usefully applied let me add the topic 'finite geometry'; a combination of words which sounds like it might not even exist, let alone be applicable. And yet it is being applied: to statistics via designs, to radar! sonar detection arrays (via finite projective planes), and to bus connections of VLSI chips (via difference sets). There seems to be no part of (so-called pure) mathematics that is not in immediate danger of being applied. And, accordingly, the applied mathematician needs to be aware of much more. Besides analysis and numerics, the traditional workhorses, he may need all kinds of combinatorics, algebra, probability, and so on. In addition, the applied scientist needs to cope increasingly with the nonlinear world and the
viii
SERIES EDITOR'S PREFACE
extra mathematical sophistication that this requires. For that is where the rewards are. Linear models are honest and a bit sad and depressing: proportional efforts and results. It is in the nonlinear world that infinitesimal inputs may result in macroscopic outputs (or vice versa). To appreciate what I am hinting at: if electronics were linear we would have no fun with transistors and computers; we would have no TV; in fact you would not be reading these lines. There is also no safety in ignoring such outlandish things as nonstandard analysis, superspace and anticommuting integration, p-adic and ultrametric space. All three have applications in both electrical engineering and physics. Once, complex numbers were equally outlandish, but they frequently proved the shortest path between 'real' results. Similarly, the first two topics named have already provided a number of 'wormhole' paths. There is no telling where all this is leading fortunately. Thus the original scope of the series, which for various (sound) reasons now comprises five subseries: white (Japan), yellow (China), red (USSR), blue (Eastern Europe), and green (everything else), still applies. It has been enlarged a bit to include books treating of the tools from one subdiscipline which are used in others. Thus the series still aims at books dealing with: - a central concept which plays an important role in several different mathematical and/ or scientific specialization areas; - new applications of the results and ideas from one area of scientific endeavour into another; - influences which the results, problems and concepts of one field of enquiry have, and have had, on the development of another. Galois theory deals with the action of a finite group of automorphisms on a field. The power and the usefulness of the theory is enormous, and hence, for a long time, there has been great interest in more general Galois theory. This has turned out to be a long and involved quest. Finally, it has led to the modern theory of automorphisms and derivations of associative rings and algebras. A great many technical and conceptual advances were needed to bring the theory to its present rich state. This includes 'rings with generalized identities', 'non-standard algebra' and a powerful logic-algebraic metatheorem. The author is one of the main, currently active contributors to the subject and I am greatly pleased that he has written this first, comprehensive authoritative monograph on the topic. The shonest path between two truths in the real domain passes through the complex domain. I. Hadamard
N ever lend books, for no one ever returns them; the only books I have in my library are books that other folk have lent me. Anatole France
La physique ne nous donne pas seulement I'occasion de re'soudre des problemes ... eUe nous fait pressentir la solution. H. Poincare
The function of an expen is not to be more right than other people, but to be wrong for more sophisticated reason s. David Butler
Amsterdam, August 1991
Michiel Hazewinkel
TABLE OF CONTENTS INTRODUCTION
xi
CHAPTER 1. STRUCTURE OF RINGS 1.1 Baer Radical and Semiprimeness 1.2 Automorphism Groups and Lie Differential Algebras 1.3 Bergman-Isaacs Theorem. Shelter Integrality 1.4 Martindale Ring of Quotients 1.5 The Generalized Centroid of a Semiprime Ring 1.6 Modules over a Generalized Centroid 1.7 Extension of Automorphisms to a Ring of Quotients. Conjugation Modules 1.8 Extension of Derivations to a Ring of Quotients 1.9 The Canonical Sheaf of a Semiprime Ring 1.10 Invariant Sheaves 1.11 The Metatheorem 1.12 Stalks of Canonical and Invariant Sheaves 1.13 Martindale's Theorem 1.14 Quite Primitive Rings 1.15 Rings of Quotients of Quite Primitive Rings
1 6 11 19 24 27 40 50 53 62 65 77 81 87 92
CHAPTER 2. ON ALGEBRAIC INDEPENDENCE OF AUTOMORPHISMS AND DERIVATIONS 2.0 Trivial Algebraic Dependences 2.1 The Process of Reducing Polynomials 2.2 Linear Differential Identities with Automorphisms 2.3 Multilinear Differential Identities with Automorphisms 2.4 Differential Identities of Prime Rings 2.5 Differential Identities of Semiprime Rings 2.6 Essential Identities 2.7 Some Applications: Galois Extentions of PI-Rings; Algebraic Automorphisms and Derivations; Associative Envelopes of Lie-Algebras of Derivations CHAPTER 3. THE GALOIS THEORY OF PRIME RINGS (THE CASE OF AUTOMORPHISMS) 3.1 Basic Notions 3.2 Some Properties of Finite Groups of Outer Automorphisms 3.3 Centralizers of Finite-Dimensional Algebras 3.4 Trace Forms 3.5 Galois Groups 3.6 Maschke Groups. Prime Dimensions 3.7 Bimodule Properties of Fixed Rings 3.8 Ring of Quotients of a Fixed Ring 3.9 Galois Subrings for M-Groups
96 98 103 109 112 122 126 133
142 144 146 157 160 163 170 174 176
x
3.10 3.11
TABLE OF CONTENTS
Correspondence Theorems Extension of Isomorphisms
CHAPTER 4. THE GALOIS THEORY OF PRIME RINGS (THE CASE OF DERIVATIONS) 4.1 Duality for Derivations in the Multiplication Algebra 4.2 Transformation of Differential Forms 4.3 Universal Constants 4.4 Shirshov Finiteness 4.5 The Correspondence Theorem 4.6 Extension of Derivations CHAPTER 5. THE GALOIS THEORY OF SEMIPRIME RINGS 5.1 Essential Trace Forms 5.2 Intermediate Subrings 5.3 The Correspondence Theorem for Derivations 5.4 Basic Notions of the Galois Theory of Semiprime Rings (the case of automorphisms) 5.5 Stalks of an Invariant Sheaf for a Regular Group. Homogenous Idempotents 5.6 Principal Trace Forms 5.7 Galois Groups 5.8 Galois Subrings for Regular Closed Groups 5.9 Correspondence and Extension Theorems 5.10 Shirshov Finiteness. The Structure of Bimodules CHAPTER 6. APPLICATIONS 6.1 Free Algebras 6.2 Noncommutative Invariants 6.3 Relations of a Ring with Fixed Rings A. Radicals of Algebras B. Units, Semisimple Artinian Rings, Essential Onesided Ideals C. Primitive Rings D. Quite Primitive Rings E. Goldie Rings F. Noetherian Rings G. Simple and Sub directly Indecomposable Rings H. Prime Ideals. Montgomery Equivalence I. Modular Lattices J. The Maximal Ring of Quotients 6.4 Relations of a Semiprime Ring with Ring of Constants 6.5 Hopf Algebras
190 193
202 206 209 213
216 221 228 231 234
236 241
253 25'i 258 265 266
270 280 292 293
298 300
306
309 313
315 318 327
336 345
355
REFERENCES
371
INDEX
383
INTRODUCTION Starting from the Bergman-Isaacs work on regular actions of finite groups, a great number of papers have been devoted to studying the automorphisms of associative rings. For some time the central aim of these studies was to find out and investigate those ring properties which are preserved under passage to a fixed ring under finite group actions and, conversely, from the fixed ring to the initial ring. The methods which emerged in these studies proved their value for investigating arbitrary automorphisms and derivations, basically from the point of view of their algebraic dependences. These methods proved so effective that they made it possible to prove Galois correspondence theorems in the class of semiprime rings both for automorphism groups and for Lie algebras of derivations. A rapid development of the theory was facilitated by the data accumulated by that time on the structural theory of rings with generalized identities (Amitsur, Martindale theorems), and it profited greatly from the concepts formed in noncommutative Galois theory as it was transferred from skew fields to complete rings of continuous linear transformations of spaces in work by E.Noether, N.Jacobson, G.Cartan, T.Nakayama, G.Azumaya, J.Diedonne, A.Rosenberg and D.Zelinsky. In the process of studying automorphisms, involutions and derivations the authors started noting that the basic obstacles in proving the theorems disappeared as soon as one considers prime rings. However, for these problems, it turned out that the direct reduction from semiprime rings to prime rings was impossible and one had to do things directly at the semiprime level, which quite frequently resulted in monotonous considerations. This effect was investigated in an interesting paper by K.I.Beidar and A.V.Mikhalev [16]. The essence of their result (from now on called "a metatheorem" ) is that all theorems written in terms of Horn formulas of the elementary language can be transferring from the class of prime rings to that of orthogonally complete semiprime rings. If we add to what was said above that in the class of prime rings the formulas of the elementary language are equivalent to Hom ones, then the "applied" value of the metatheorem "becomes evident. At approximately the same time Burris and Werner [25] obtained similar results on sheaves of algebraic systems independent of applications in the theory of rings. Additionally, in studies by E.I.Gordon and V.A.Lyubetzky [100] based on non-standard analysis, similar problems were considered. Within these frameworks a semiprime orthogonal complete ring can be viewed as a non-standard prime ring. The formulation and proof of the metatheorem presented in this book take, more or less, into account the background intuition of all three of the approaches just mentioned. The material is arranged in the following way. The first chapter is an introductory part presenting the current state of the art of the structural theory of rings for the Baer radical. We, naturally, tried to keep the xi
xii
INTRODUCTION
presentation as close to the basic subject as possible and as early as in the third paragraph one can find the Bergman-Isaacs theorem on the nilpotency of any ring having a regular finite group of automorphisms (under natural limitation on the characteristics), as well as the Quinn theorem on the integrality of a ring over the ring of invariants. These theorems together with Martindale theorem and the metatheorem are the key results of the chapter. In the second chapter problems of algebraic dependences of au tom orphisms and derivations are considered. The results presented show, essentially, the algebraic independence of automorphisms and derivations of semiprime rings. To grasp this idea certain refinements are necessary since some dependences of a certain type (trivial ones) still exist. For instance, the sum of any two derivations is equal to a suitable third one. In a similar way, the product of any two automorphisms will be an automorphism. To be more exact, the results of the second chapter show that all the algebraic dependences between automorphisms and derivations are consequences of the simplest ones, i.e., of those resulting from the algebraic structure of a set of derivations and automorphisms, as well as of those relations J1 = r a - 1 a' g = r a 1 _ 1 a
which
determine
the
inner
derivations
and
automorphisms for a Martindale ring of quotients. Associated with the above refinement, there arises a problem of obtaining criteria allowing one to distinguish between trivial generalized polynomials with automorphisms and derivations from non-trivial ones. In the sixth section we give such a criterion for the multilinear case in the spirit of the L.Rowen's approach and prove that if the ideal generated by the values of generalized monomials of an identity with automorphisms contains a unit, then the ring satisfies a usual pulynomial identity. This statement is also valid for differential identities with automorphisms in the case when the ring has no additive torsion. The seventh chapter presents certain immediate corollaries of independence. Let us cite two of them: any algebraic derivation of semi prime ring with zero characteristic is an inner one for a Martindale ring of quotients; if the fixed ring of a finite group G of automorphisms of the ring R satisfies a polynomial identity and R has no additive torsion, then R also satisfies a polynomial identity. In the next two chapters the Galois theory of prime rings is developed and then, in the fifth chapter, using the metatheorem, it is transferred to semiprime rings. The results presented in these chapters are, basically, due to the author; however, the formulations used and the proofs incorporate the ideas from a paper by S.Montgomery and D.Passman [117], as well as the results of studies presented in a series of works by J.Goursaud, J.Osterburg, J.Pascaud and J.Valette [50, 51, 52]. In transferring the results to semlpnme rings, the ideas by A.V.Yakovlev [155] concerning the Galois theory of sheaves of sets also proved useful.
INTRODUCTION
xiii
In any event there arises the question of relating the theory developed in this book with the classical Galois theory of fields, sfields and rings of continuous transformations. The fact that the classical results follow from those cited in the book is not enough information, since any true statement follows from other true statement (as well as from a false one). Therefore, from our viewpoint, the problem should be posed at another level, i.e., to what extent is acquaintance with the general results useful for understanding the classical ones? As to the Galois theory of fields, the answer is unambiguous; namely, it is the ABC of modern algebra and it is, therefore, doubtful that anyone who wishes to get acquainted with it, will start with studying noncommuta-tive rings. In the case of the automorphisms of skew fields a direct construction of the Galois theory is much simpler, so that it would be reasonable to get acquainted with this theory by means of the seventh chapter of the monograph by N.Jacobson [65], which, however, does not cover the case of derivations given in the fourth chapter of the present monograph. The latter results for skew fields were obtained by M.Weisfeld [153]. As to the case of rings of continuous linear transformations of vector spaces over skew fields (or simple artinian rings), the classical proof is based on the fact that the automorphisms turn out to be conjugate under semilinear transformations and the problems are, therefore, reduced to a great extent to skew fields. In the general case there is no description of automorphisms, so, as applied to rings of transformations, the new proofs reveal other reasons for the Galois correspondence theorems. It is a posteriori obvious that for a ring of linear transformations the correspondence described in 3.10.6 reduces to the classical one, and thus the BM and RF conditions, as well as the centralizer condition, can be viewed as an internal characterization of the homogenous rings of endomorphisms. Therefore, the Galois theory of prime rings sheds additional light on rings of continuous linear transformations as well. Of much greater importance, however, is the application of this theory to other classes of rings which have not been studied in terms of the Galois theory so far. For instance, the correspondence theorem for domains is rather far from the analogous theorem for skew fields; there exist domains which cannot be embedded into skew fields at all. The application of this theory to free algebras also proved unexpected, since there is nothing of this kind for rings of polynomials. This application leads into the concluding chapter of the book which is mainly intended for the reader trained in the theory of rings. The chapter treats the A.N.Koryukin theorems on noncommutative invariants of linear groups which gave a complete solution of an analogue of Hilbert's 14-th problem for a ring of noncommutative polynomials. The chapter considers a wide range of problems on the transfer of the properties from a ring of invariants (a ring of constants) to the initial ring and conversely, it clarifies the relations between the spectra of prime ideals of a ring and those of a fixed ring (the Montgomery equivalence), it proves the Fisher and GrzesczukPuczylowsky's theorems on finite groups acting on modular lattices, and it
xiv
INTRODUCTION
gives a proof of the Goursaud-Pascaud-Valette and Piers Dos Santos theorems on maximal rings of quotients. The last section presents a general concept of Hopf algebra actions embracing both the case of automorphisms and that of derivations. The well-known Kostant-Sweedler theorem (Theorem 6.5.1) demonstrates that the study of the action of cocommutative Hopf algebras over algebraically closed fields of characteristic zero reduces to those of actions of the corresponding groups and Lie algebras. Therefore. it is the study of noncommutative Hopf algebras that is of prime interest. In the last section we consider some approaches to studying skew derivations from this viewpoint. The author wishes to express his gratitude to the participants of Shirshov seminars on the ring theory held at the Institute of Mathematics of the Siberian Branch of the USSR Academy of Sciences for their interest in his work.
INTRODUCTORY PART CHAPTER 1. STRUCTURE OF RINGS 1.1 Baer Radical and Semiprimeness We shall start constructing Baer radicals with a simple remark that any abelian group can be turned into a ring by way of defining multiplication by means of a formula xy = O. Such a ring is called a zero-multiplication ring and is trivial in terms of the theory of rings. Now, starting from an arbitrary ring R, we can try to "kill off" the trivial part in it. If the ring R contains a nonzero ideal
II
with a zero multiplication,
Ii= 0, then
it
would be natural to go over to considering a factor-ring R / II' since the whole ring R is presented as an extension of the "trivial" ring II by R / II. The same approach can be used for the ring
try to find in it a non-zero ideal
II /
12
R / II' i.e., we can
with a trivial multiplication, and
focus then our attention on a factor-ring R / 12 == (R / II) / (I2 / II). A continuation of this process is equivalent to finding in the ring R an ascending chain of ideals (O)C Ilc 1iC •. C InC •.
such
that
the
factors
In / In _ I
have
a
zero
multiplications,
i.e.,
2 I nl:: In _ 1. It goes without saying that such a chain can prove infinite. In
this case we can introduce into the consideration an ideal I", = n In which is, as has been shown above, obtained by extending the "trivial" rings (using the "trivial" ones), and nothing can prevent us from using the same approach for killing off "trivial" parts in the ring R I I", . Therefore, we get a transfinite chain of the ideals of the ring R.
V. K. Kharchenko, Automorphisms and Derivations of Associative Rings © Kluwer Academic Publishers 1991
2
AU1OMORPIDSMS AND DERIVATIONS
where
r
a+ I!:; I a , and
the equality
Ia
=
U
f3 < a
I
f3 holding
for
limit
ordinals a. Obviously, in its cardinality the length of such a chain is not greater than that of the set of elements of the ring R, so that we can consider a union B (R) = U I a' An ideal B (R ) is called a Baer radical (or a lower nil-radical) of the ring R. By construction, we can no longer "kill off' any trivial part of the ring R / B(R), since this ring contains no nonzero ideals with a zero multiplication. 1.1.1 Definition. A ring is called semiprime if all its nonzero ideals have a nonzero multiplication, i.e., for an ideal I the equality I2= 0 implies I = 0 . 1.1.2 Definition. An ideal I of the ring R is called semiprime if the factor-ring R / I is semiprime. We see, therefore, that the Baer radical of a ring is a semiprime ideal. From the viewpoint of the construction process above, it is also possible (as a limiting case) that the chain being constructed reaches the whole ring R in a finite number of steps. It is easily seen that in this case there exists a natural number n, such that the product of any n elements of the ring R equals zero, i.e., R n= 0 . A ring with such a property is called nilpotent and, accordingly, the ideal I is called nilpotent if for a certain natural n we have I n= 0 . 1.1.3 Note. A semiprime ring contains no nilpotent ideals since one of the powers of a nilpotent ideal has a zero multiplication. Accordingly, for a semiprime ideal J and and ideal I the condition In!:; J implies I!:;J.
The above definition of a Baer radical of a ring clearly demonstrates that the radical is "composed" of rings with a zero multiplication, its form, however, depending on a certain arbitrariness of the choice of the ideals Ia, which is a disadvantage. It is, therefore, useful to prove the following theorem. 1.1.4 Theorem.(a) The Baer radical of a ring can be obtained as a union of the transfinite chain of ideals (0) = Ii> !:; Nz !:: _. !:: N a r;;, .wherein the ideal N a + 1 is chosen in such a way that the sum of all the
3
CHAPTER 1
nilpotent ideals of the ring
R I Na is
Na = UNa holds for the limit ordinals
p< a
Na+ 11 Na and
the
equality
a.
I'
(b) The Baer radical of a ring is the smallest semiprime ideal. Proof.(a) Let N be a union of the chain given in the formulation of
the theorem. Using transfinite induction, let us prove that I a!:: Na for all a. Because the ideal with a zero multiplication is nilpotent, II!:: N1 . If a is a limit ordinal and While
a
if
I~ !;:;
is
/3 < a,
I p!;:; N P for all
not
a
I a-I !:: Na-I' so that
limit
I~ ~
then UIa
= UI P = Na ·
ordinal
and
I a-I
Na-I is a nilpotent ideal of
!:;
I
1 !:: N
a-
a-
l'
the n
RI Na _ 1, and, hence, Ia!:;;;Na · Therefore, B(R)~N. Let us, conversely, assume that N CZB(R) and choose the smallest a, such that Na CZ B(R). In this case a cannot be a finite ordinal, i.e., there exists an ordinal a - I for which N a-I !;:; B ( R). Because the ideal Na is, by definition, equal to the sum of all the ideals of the ring R which are nilpotent modulo N a-I' then for a certain ideal N a nilpotent modulo N a-I
(i.e.,
such
that
N
CZ
B ( R)
holds. The condition
N n!:;;; Na-I!:; B ( R), however, implies that the ideal N + B ( R ) is nilpotent modulo B (R), which contradicts the semiprimitivity of the factorring R /B(R).
(b) Let J be an arbitrary semiprime ideal. Since II
s::
J.
If
Ia
~ J,
then
2
I a + 1~ Ia
~ J,
Ii= (O)!:;;;
I
a =U I
induction,
P~
B (R) r;;;.J.
J.
then
and, therefore,
Finally, for a finite ordinal a we see that the conditions yield
J,
I p!:; J(/3 < a)
Therefore, according to the principle of transfinite
The theorem is proved.
1.1.5 Definition. A ring R is called prime if the product of any two of its nonzero ideals is different from zero. Accordingly, the ideal I is called prime if the factor-ring R/ I is prime. Therefore, the ring R is prime iff (0) is a prime ideal. The following characterization of prime and semiprime ideals in terms of the ring elements often proves to be useful. 1.1.6 Lemma.(a)The ideal I of the ring R is prime iff for any a, be R the inclusion aRb E I implies a E I or be I.ln particular,
AUTOMORPIDSMS AND DERIVATIONS
4
the ring R is prime iff for any nonzero a, b there exists ~ R, such that axb:;tO. (b )The ideal I of the ring R is semzpnme iff for any element a E R Ihe inclusion aRa E I implies a E R. In particular, the ring R is x E R, such semiprime iff for any nonzero element a E R there exists that axa:;t O. Proof. (a) Let I be a prime ideal and , and suppose that b ~ I. Then the ideal (b) = bZ + bR + Rb + RbR which is generated by the element b is not contained in I, but the product of the ideals (aR + RaR) . (b) is contained in I, which implies that aR + RaR !:: I. Let (a) be an ideal generated by the element a, then (a)· (a)!:: aR + RaR and,by the definition of a prime ideal, (a) ~ I i.e., a E I, which is the required proof. Conversely, if condition (a) is valid for the ideal I, and the product PQ of the ideals P and Q is contained in I, then for any pEP, q E Q we have pRq!:: PRQ !:: PQ !:: I • If now q ~ I, then p E I and, since p is an arbitrary element of the ideal P, then P {; I. If there exists no element q E Q with the properties q ~ I, then Q ~ I, which is the required proof. (b) If I is a semiprime ideal, aRa!:: I, then (a)3!:: I. That implies that the ideal (a) + I I I is nilpotent in the factor-ring R I I, i.e., (a) !:: I and, in particular, a E I.Let condition (b) of the lemma be fulfilled and p2!:: I, where P is an ideal of the ring R. Then for any PEP we have
pRp!:: PRP !:: p2!:: I, i.e.,
proof. rings. rings
p E I, which is the required
1.1.7 Lemma. Any semiprime ring is a subdirect product of prime It should be recalled that a ring R is called a subdirect product of a E A, if there exists an imbedding 1r of the ring R to the
Ra,
direct product
IT R a ,
such
that
its
composition
with
the
projections
Pa:Il Ra ~ R a , results in epimorphisms (it is implied here that the subdirect product is not uniquely defined by the rings R a , a E A). If we consider the kernels Ia of the compositions 1rpa, then we can easily see that the condition of decomposability of the ring R into a subdirect product of the rings R a , a E A is equivalent to the existence of a set of the ideals I a , such that
na
Ia = 0 and
R
I Ia == Ra' Under these conditions the ring
is also said to be approximated by the rings Ra' Proof of lemma 1.1.7. Let a = c30 be a nonzero element of the ring R. According to Lemma 1.1.6, we can find an element Xl E R, such that al = aXI a :;t O. Using the element a l we find an element -X2 such that R
5
CHAPTER I
= a l x 2a I -:F. o.
Continuing this process. we can construct a countable sequence of nonzero elements 80. a p .• an.··. a n+ 1= anx n+ I an for certain elements xl' X:2~. x n~. of the ring R. Let us consider a set M of all the ideals of the ring R containing no elements of the sequence constructed and ordered by inclusion. This set is not empty since it contains the zero ideal as an element. Moreover. the set is directed. i.e.. according to the Zorn lemma. the set M contains maximal elements. Let P a be one of them. In this case the ideal P a does not intersect the sequence 80. al~.' a n.... but any ideal strictly containing P a has ~
a 0*
nonempty
n
aE R
intersection
with
this
sequence.
Since
a
EPa.
then
P a = 0 and it remains to show that P a is a prime ideal.
Let
A
= A + P a::>
and
be ideals of the ring
B
R
not contained in
P a. Then
=B
+ P a::> P a and. since P a is maximal in the set M, the ideals Al and BI do not belong to M, i.e., there can be found natural n, m, such that an E ~, am E B I . Let, for instance, n ~ m. In ~
P a.
BI
this case, since a k + I = a k x k + 1 a k case we get:
E
(a JJ ' then
an
E
(a zn)
!:; B 1 ·
In this
Therefore, the ideal AB + Pais not contained in P a and, hence, AB cz. P a' which is the required proof. The reader could ask a natural question as to how far the Baer radical can be away from being nilpotent because of the transfinite process of its construction. To a certain extent this question can be answered by the following theorem.
1.1.8 Theorem The Baer radical of a ring is locally nilpotent, i.e .• any finite set of its elements generates a nilpotent subring. Proof. Let us consider the inductive construction. We have already noticed that at any finite step nilpotent (so afortiori locally nilpotent) ideals are obtained. Let a be a limit ordinal, that all the ideals I a,
I
P
I
a=
U
P<
a
I a
be locally nilpotent. If a1,.., a n are elements from
then there exist transfinite numbers
and such that numbers
a 1 E I PI'··' an E I
/31"·' /3 n'
and let us suppose
f'
/31'·'" /3 n which Pn· Let /3 be the
then. since the ideals
(I
r)
form
are less than a largest from
the
a chain, we get
AU1OMORPHISMS AND DERIVATIONS
6
at' ~, ... , an E I P and, due to the local nilpotency of I p these elements generate a nilpotent subring. If a is not a limit ordinal, a = f3 + 1 and the ideal I p is locally nilpotent, then any finite set
8} ,..., 8
generate a subring S, such that
n
of the elements of the ideal
2 S I: I
p' However, the subring
generated by a finite set of elements (8 i 8 j I 1 ~ 1., j ~ n) and, hence, S2 and, consequently, also s, are nilpotent. The theorem is proved. No further details pertaining to the Baer radical are going to be discussed here; an interested reader is referred to the thorough study by V.A.Andrunakievich and Yu.M.Ryabukhin "Radicals of Algebras and Structural Theory" (Moscow, "Nauka", 1979).
1.2 Automorphism Groups and Lie Differential Algebras
1.2.1. An automorphism of the ring R is an isomorphism onto itself, i.e., the one-to-one correspondence g: R ~ R, which preserves the operations
One can easily see that a composition of two automorphisms as well as the mapping which is inverse to an automorphism will be an automorphism. This fact implies that a set of all automorphisms forms a group designated by Aut R. In the case of non-commutative rings of importance is the notion of inner automorphism. Let R be a ring with a unit and b be its invertible element, i.e., bb-} = b- Ib = 1 for a certain b- 1 ERIn this case the mapping
-
-}
b:x~b
xb
is an automorphism of ring R which is called inner. A set of all inner automorphisms Int R forms an invariant subgroup in Aut R, since g
where b. de R.
g
E
Aut R.
-1 -
b g =b
g
CHAPTER 1
7
Let us denote through If a group of all invertible elements of the ring R. Then the mapping b-+ £ will be a homomorphism of this group onto the group Int R, the kernel of this homomorphism consisting of all central invertible elements. An element z of the ring R is called central if it commutes with all the elements of the ring R: zx = xz for all x E R. The set Z( R) of all central elements is called a center of the ring R. Therefore, we have an exact sequence of the homomorphisms of the groups (1 ) -+ Z( R)* -+ It -+ IutR.
1.2.2. A derivation of the ring R is a mapping obeys the following properties:
JL: R -+ R which
A set of all derivations of the ring R is denoted through Der R . This set is closed relative to the commutator operation: if JL, v are derivations, then [ JL, v]
c!f
JLv-vJL is also a derivation: (xy )
[p, v ]
_(xV y
+
= «xy)
xyV)J.l.
J.l. v ) -
( (xy)
v
)
J.l.
J.l.
= (x y
= xJ.l.V y+ xJ.l. yV+ xV yll+
+ xy
J.l. v ) -
xyllV_
xVJ.l. y _ xV yJ.l._ xJ.l. yV_ xyVJ.l.= }J.l.. v] y+ x) J.l.. v].
Therefore, Der R is a Lie ring. This ring is simultaneously a right module over the center of the ring R, provided the multiplication by the central elements is determined by the formula x< J.l.z) c!f xJ.l.· z. Indeed,
At the same time, one cannot claim that Der R is an algebra over the ring z( R) since the definition of an algebra includes the identity [ JLz, v]
while in the ring
= [JL. Der R
v] z ,
the following identity holds instead:
[JLz, v] = [JL, v] z + JL' z v
This peculiarity results in the following definition.
(1)
8
AU1OMORPlllSMS AND DERIVAnONS
1.2.3.Definition.Let Z be a commutative ring, and D be a certain Lie ring which is simultaneously a right module over z, and let us assume that there is a certain homomorphism of Lie rings given: u: D ~ Der R . Then D is called a differential Lie-z-algebra (or, for shott, a Lie- dalgebra), if identity (I) holds in D, where the zV value is determined as z u(v) . Therefore, we come to the conclusion that the set Der R forms a differential Lie Z( R) -algebra, where the homomorphism u: Der R ~ Der Z is determined as a restriction of the derivations onto the center. It should, naturally, be noted that all the derivations transform central elements into central ones: if zx = xz for all x, then z Jl x + zx Jl = x Jl z + xz Jl, which results in z Jl x+ xz Jl, i.e., z Jl e Z(R) • Let us now assume that the ring R has a simple characteristic p> 0, pR = O. In this case the p-th power of the derivation appears to be another derivation (xy)
JlP
=Lclx Jl
k
y
Jl
P -
k
P
= x Jl y+ xy
JlP
,
c';
since the binomial coefficients are divisible by p, and, hence, a set of derivations forms a restricted Lie ring. It should be recalled that a Lie ring is called restricted if a unary operation jJ.~ jJ.[p] is given on it, which is related to addition and Lie multiplication with the following identities [jJ., v
( jJ.
where
[ p]
] = [ •. [ jJ., v]v] .• v],
+ v)[ p]=
w( x, y)
is
(2)
jJ. p + v p + w( jJ., v),
a commutator presentation
of the
noncommutative
associative polynomial (x + y) p - x p - yp. Well known is the fact that such a presentation exists. For instance, at p = 3: (x
+ y)3 - ~ - .0 = 2- y + xyx + y2- + xy. + + Y 2x ;: [ f?c, y ]y] + [[y, x]x] (mod 3 )
yxy
The Lie algebra over a commutative ring Z is called a restricted Lie algebra (or sometimes a Lie p-algebra) if it is a restricted Lie ring and the operation jJ.~ jJ.[p] is related to the module structure through the
CHAPTER 1
9
formula
In the ring of derivations the operation of bringing to the power p does not obey the above identity, and the following identity holds instead: (p.c)
p
= p. p c p + p.. ( •. «c J.l c) J.l c) J.l .• ) J.l c.
(3)
We thus naturally come to the following definition. 1.2.4.
Definition.
A
differential Lie Z -algebra D is called is given on it, which transforms it into a restricted Lie ring and is associated with the module structure through the formula restricted if the operation p.-+
p.[p]
p-l I
(p.c)
[ p]
.J p] p r J.l J.l J.l J.l ' =1-'" c +p.·( •. «c c) ) c .• ) c.
(4 )
It is clear that if D acts trivially on z (Le., if u =0), then this definition turns into that of a Lie p-algebra over z. Hence, if the ring R has a simple characteristic, then the set of derivations forms a restricted differential Lie z( R) -algebra. Among derivations, as well as among automorphisms, one can single out an inner part. Let b be an element of ring R, in which case the mapping adb, which operates by the formula df ad b: x-+[ x, b] = xb- bx,
will be a derivation, since [xy, b]
= [ x, b] y + x[ y,
b].
This derivation is called an inner one. Let us consider R as a (restricted) Lie ring with a multiplication operation [xy] -+ xy- yx (and the operation .J p] = x P at p> 0; the addition remaining the same). This Lie ring is called adjoined to R and is (- )
designated by R . In this case the mapping
x -+ adx
sets a homomorphism of the
(restricted) Lie rings ad: R( - ) -+ Der R, which can be easily seen to preserve the z( R) -module structure:
10
AUTOMORPlllSMS AND DERIVAnONS
[ad b, ad r] = ad [b, r] ;ad (bz) = (ad b) z .
The kernel of this homomorphism coincides with the center of the ring R, so that we have an exact sequence of the homomorphisms of Lie rings
It should be also remarked that [adb,,Ill =adb
J.I.
and, hence, the image adR forms an ideal of the (restricted) Lie ring Der R , which makes it possible to consider the outer part of the set of derivations as a factor-ring as well: Dext(R) = Der R
I
adR.
1.2.5. Adjoined Lie a-algebras and universal envelopings. By analogy with adjoined Lie rings, one can also determine adjoined Lie aalgebras. Let A be an associative ring with a unit, z be its subring, and let us suppose that
i - ).
Z ( -)
is an ideal with a zero multiplication in the Lie ring
This implies that
Z
is a commutative subring in
A,
which is
. . ' mvarlant reIative to aII'mner d" envatlOns: az - za = Z aE Z • V"Iewmg A( - ) 'ht modl . (assoCIatIve .. ). as a ng u e over the commutatIve nng Z, we see A(-) convening into a (restricted) differential Lie z-algebra, where the action of (- ) A
on
Z
is governed by the formula z a = a z - z a. This algebra will
a-
hereafter be designated by A(;) and called an adjoined (restricted) Lie algebra. The ring A will be referred to as an associative enveloping of the (restricted) differential Lie a-algebra D if there exists a Z-linear homomorphism of the (restricted) Lie rings ~ A: D ~ A z
(- )
' such that the
action of dE D on Z coincides with that of ~ i d ), and the set ~ A( D) generates A as an associative algebra with a unit. The associative enveloping A will be called a universal enveloping for D if for any associative enveloping B there exists a Z -linear homomorphism of the rings qJ: A~ B, such that ~ AqJ=~ B. From the theory of categories one can easily deduce that the universal enveloping for D exists in all cases. It is much more difficult to elucidate under what conditions the homomorphism ~ A proves to be an embedding into the universal enveloping. If D is presented by the derivations of ring R with a unit and
CHAPTER 1
11
center z, i.e., D s:; Der R, then D is embedded into the ring of endomorphisms of the abelian group < R, + >, in which case the subring
x ~ xz).
( -)
It is clear that ~I'l>: D ~c1> (D) z
will be an
embedding and, hence, the universal homomorphism ~ A will be also an embedding. In a general case, if D is a free module over Z, one can prove that ~ A is an embedding, which fact will be proved below only for the case when Z is a field (see 6.1.12) and we shall see that any Lie a-algebra over a field can be presented by the derivations of a free algebra. 1.2.6. By way of concluding this paragraph we should note that the automorphisms of the ring R act in a natural way on its derivations: if gE Aur R,
derivation (xy)
J.lE
g- 1 J1.g
Der R,
then
- 1 J1. =(x g yg
the 1
superposition
g- 1 J.lg
C;;
J.l g
will
be
a
- 1 - 1 J1. g - 1 J1. - 1 J1. + x g yg ) = xg g y+ xyg g.
1.3 Bergman-Isaacs Theorem, Shelter Integrality
In this paragraph we shall prove some results by Bergman and Isaacs on fixed rings of finite groups of automorphisms, as well as the Queen's theorem. 1.3.1. Definition. Let G be a group of automorphism of the ring R. A fixed ring of group G is a subring RG=(r E RI V gE G, r g = r} of all elements which are fixed relative G. Our nearest aim is to find out the conditions under which the ring RG proves to be sufficiently "small" (zero or nilpotent), which makes further studies of invariants of finite groups possible. 1.3.2. Example. Let F be a field with a characteristic p 'i:. 0 and an element ro 'i:. 0, 1 of a finite multiplicative order ro n= 1. Let F < x, y > be a ring of all noncommutative polynomials of all variables without free terms. Let us denote by R a ring of all 2 x 2 matrices with the elements from F < x, y >. The matrices
12
AUTOMORPlllSMS AND DERIVAnONS
though
not
belonging
-1
to
R,
do
determine
the
of the ring R. The order of a group
g i: M ~ Ai MA i
npz.
automorphisms G generated
by
Direct calculations also show that RG = {O}. (It these automorphisms is is sufficient to remark that the matrix which is commutative with ~ and
~
(~~),
has the form
while the matrix of this form commutes with
~
only if f = 0.) This example shows that under most general conditions there is no tight association between the rings Rand R G. Nonetheless, when in R there is no additive n -torsion, where n is the order of the group, such associations can be established. Let x = (gl'-' gnJ be a finite set (not obligatory a group) of automorphisms of the ring R. Let us denote by
1.3.3.
Definitions.
U( X) a set of all sequences
Let
r be
a
u=(ug1'_"u g )
mapping
Lemma. Let
(ug1'''''u g) from
XE R.
= (u g) g U( X)
E
x of the elements of the ring
in R,
which
sets
the
R.
sequences
in correspondence with the element u g1 +U <7z + ... +U gn.
1.3.4.
for all
U
from
V( X)
be
a
subset
of
the
sequences
u( X), such that
In this case the left ideal W( X) is nilpotent to the power n
not greater than
IT (C~ + I ). k=
0
Proof. For any au=(aug)EU(X)
a
E
Rand
U E
and
V( x) = (u E U( x): r(u. a) =0 for all
U( x)
let us determine the products Then
u.a=(ugag)EU(X). aE R}. For any
Y~ X
a set v(Y) with a subset of sequences (Ug)E vex), for which gE X\ Y.
Let u,vEV(Y),gEY.Wehave
let us identify ug=O
at any
13
CHAPTER 1 -1
g UgV-U'(vg
)EV(Y\{g}).
Indeed, this element lies in V( Y), and its g-component is UgV g -
g- \ g
ugl...Vg
)
=0.
If we apply -r to this element, then the second tenn disappears by the definition of V( X), and we get U g( 't'v) E w( Y\ ( g ) ), obtaining, after summing it up over all gE Y, ('t' u)('t'v) E
Ltv(Y\ (g)) . gE Y
Hence, (W(Y)/!:;;
LW(Y\(g)).
(5)
g E Y
Let now Y run through elements. Then we get (Ltv(y»k+ls:
I yl = m
all
C ~= k
of the subsets
x of
m
Ltv(Z)R#.
I zl = m -
1
(Here R# is a ring obtained from R by joining the unit). In order to visualize it, the left part should be presented as a sum of products
JJ. According
to the Dirichlet principle, one of the factors is encountered twice, i.e., each product can be presented as A't'V(Y)B't'V(Y)C s: (R #'t'V(Y»(R #-r(V(Y»R #= ('t'V(Y»2 R #, i.e., according to (5), it is contained in the left part. w( YO)' .•. W( Y
n
n(C~+I)
By induction over m we now get (W(X»1=0 is proved.
= O.
The lemma
1.3.5. Proposition. Let G be a finite group of automorphisms of the ring
R, n be
its
order, h=l+
n
II (C n1 + 1).
1=0
d
In
this
case for
natural d the following inclusion is valid: (nR)h s:; nR#t(R)d R, where
any
14
AUTOMORPlllSMS AND DERIVATIONS
t( r) =
Lr
g is
a trace of element r.
gE G
Proof. Let
L= ( 1 E R, 1 t ( R) = O}
t( R). If 1E L, then
L lx g = 0,
be a left annihilator of all traces
which implies that
U
= (1, 1,_., 1) E V( G)
gE G h - 1
and, hence, n1='l"(u)E'l"V(G). According to Lemma 1.3.4, (n1) =(O). Let now J be an arbitrary G- invariant left ideal of the ring R, RJ
~ J, J g = J
R = R / Jt(R)R
for
all
g
E
G.
Let
#. Since the two-sided ideal
. then the actIOn transferred onto
us
consider
T= Jt(R)R #
a
factor-ring
is
G-invariant,
0f
t he group G: (r + T) g = r g + T WI'11 be natura11 y R. For the element r= r+ T E R we have t( r)=t(r)+ T=t( r) and hence, Jt(R)=O, i.e. J is contained in the left -
annihilator of all the traces of the ring R. Therefore, (nJ) -
the ring R this equality implies that (nJ) the right by nJ, we see that
h- 1
~
h- 1
-
= (0
J. For
_#
Jt( R) R . Multiplying from
(6) Stemming from this conclusion, let us carry out induction over d. Let
Raising both parts to the power h and using inclusion (6) for the ideal
JQ =
Jt( R)
d-I
J, we get
Setting J = R, we come to the statement of the proposition. 1.3.6. Theorem Let G be a finite group of the automorphisms of the ring R, n be its order. In this case, if R has no additive n -torsion, the d-power of the nilpotency of the fIXed ring implies that of the ring R of the power not greater than n d. In particular, if R G= (O), then Rh= (O),
where
15
CHAPTER 1
n h=l+ II(d+l). i= 0
The proof follows immediately from proposition 1.3.S. The estimate of nilpotency in this theorem seems to be extremely rough. For instance, the degree of nilpotency for solvable groups is known not to exceed nd. A detailed study, however, has not yet been carried out, and it would be of interest first to find the boundaries for estimating nilpotency from below, using a series of examples. The theorem obtained above is an extremely important instrument for studying invariants of finite groups in the case when the ring has no additive n-torsion (where n· is the order of the group). In this paragraph we shall consider only one application. 1.3.7. Corollary. Let G be a finite group of the automorphisms of a semiprime ring R which has no additive n -torsion, where n is the order of the group G. In this case the fued ring R G is also semi-prime. Proof. Let I be a non-zero ideal of the ring R G, such that I2= (0). Let us consider a right ideal nIR in R. If ns= n Limrm is a m
fixed element of this ideal, then
m
m
If we apply theorem 1.3.6 to the ring nIR. we get (nIR) G is a ring with a zero multiplication and. hence. nIR is a nilpotent ring of the power h 2 . Since the ring R has no additive n-torsion, the right ideal IR will also be nilpotent. And. finally, the two-sided ideal J generated by IR will also be nilpotent:
which does not agree with the semiprimeness of the ring R. The corollary is proved. If the initial ring R is commutative, then all the coefficients of the polynomial
II(x - a g) at gE G
aE R
will be fixed relative the action of the
finite group G. Opening the brackets, we find an integer polynomial over R G, a root of which is a. i.e.. R is an integer extension of R G of the degree I GI. This fact has an analog for non-commutative rings as well (under
AU1OMORPIDSMS AND DERIVATIONS
16
natural limitations on discovered by D.Quinn.
has recently
been
1.3.S. Definitions. Let T be a subring of the ring R. integer T-polynomial of the degree n will be expressed as
The
xn +
L.
the
additive structure),
which
t ~'1 xt ~'2 x .• xt ik
~
k~
n
where t i j are either elements from T or formal units; in both cases in every term under the sign of the sum there is at least one factor from T (if T contains the unit, the latter amendment is excessive). The ring R is called Shelter integer over T (of the degree n) if for every aE R there exists an integer T-polynomial f( x) (of the degree n), such that f(a) = O. A linearized variation of the above notion is somewhat deeper and more useful. A quasi-integer T-poiynomial of the degree n is an expression of the following kind: x 1x r · Xn +
L n t i1 x j1 t i2 x j2··
k~
x j k- 1 t ik'
i, j
where, as above, t
ij
is either an element from T or the unit, in which
cases not all t i j under the sign of the sum are equal to the formal unit. The ring R is termed quite integer over T of the degree n if for every set al'." a n of elements from R there exists a quasi-integer Tpolynomial f( xl'." xJ of the degree n, such that f(a 1,.., aJ = O. 1.3.9. Theorem (D.Quinn). Let G be a finite group of automorphisms of the ring R. If 1GI R = R; 1GI = 0 ~ x = 0, then the ring R is quite integer over R G of a certain degree m. The proof of this theorem is based on the following statement obtained by R.Pare and W.Schelter [131]. R.
1.3.10. Theorem. Let a group G of an order n act on a ring In the ring of n x n matrices Rn let us consider a sub ring of gl
g
diagonal matrices T=(diag(a , .• ,a ')1 aE R)=R where (gp.' gJ= G. In this case the ring Rn will be quite integer over T of a certain degree m. Proof. For every k = 1, 2~., n we shall denote by R k a subring in
CHAPTER 1
17
R n , which consists of all the matrices the last columns of which are zero. By induction over quite integer over T of a certain degree m. Let k = 1 and a 1= diag( rl'0, •. O)E ~. tE T, such that t = diad( rl'.')' Then for
k
n - k rows and n - k we will show R to be
Let us find an element any ~E ~ we have
(a1 - t) ~ = 0, and, hence, at a 2 = ta 2 which is the required proof. Let k> 1. Let us present an arbitrary matrix aE Rk as
a=
[
0] * * ° °°° a' r"
(7)
where a' is the (k - 1) x (k - 1) matrix, k - 1. Let us set
(a'O)
iI= 00
at
=
~
E R k _ t'
Then for all the elements at,
r'i
iff the columns
and
r'i
~E Rk
we shall introduce the relation
corresponding to the matrices at and
r;.
~ in presentation (7) coincide, r~' = relation obeys the following properties. (1). Let at'~, aE Rk and tE
tat
r" is a column of the height
It should be remarked that this
T. If
at
=~
=11 "2 and
then
11 at
t E T,
such that
5Ot~.
(2). If at, a E R k' then there exists an element
(8) Indeed, one can find
a- t=
and, hence,
t E T,
"0]°
[ ** ~ o *
°
such that
AU1OMORPIDSMS AND DERIVATIONS
18
Let al ,"., ambe arbitrary elements from R k , where the number m equals the degree of integrality of Rk _ lover T. In this case we get the following dependence:
at"" am + Lt il if' j'" a j m-\ t ik = 0 . \ Let us multiply the above equality from the right by an arbitrary element aE R k • Then, due to the flrst property of relation ==, a multiple application of relation (8) enables one to eliminate the wavy lines over the letters. Since relation (8) is homogenous both over a l and the totality of the variables comprising it, the senior form is transformed into a quasi-integer polynomial with the senior term a l ." a m,a, while the remaining terms are transformed into aT-polynomial f of degree m + 1, such that f( a l ,"., a m,a ) == O. This fact implies that there is an element tE T, such that the difference s = f( ai' ". ) - t has the form
S~(: ~ ~J
(9)
In line with the prOpOSItIOn on induction for any elements
sl '".,
S
m
of type (9), there exists a quasi-integer polynomial q> of degree m, such that the
rnatti.
q( s l' -., s,.)
has
the
q>(sl'"" sm> qJ(sl',"., sm') a=O, where
other elements s'l ,."'
S
fmm:
(:
~ ~J
and,
hence,
qJ is a quasi-integer polynomial for
m' of type (9). If we now assume that the elements
(i) (i) . 2 are constructed by the arbitrary sets a(i) l '"., am , a , (1 s:; ~s:; m) in the same way as the element s by the set a l ,"., a m, a, then we will get S i's i
(i)
(i)
(i)
the required dependence between the elements a l ,"., am, a
, (1 s:; i s:; 2 m) ,
2
a of the degree 2( m+ 1) + 1. The theorem is proved. 1.3.11.Corollary. The ring of matrices Rn is Shelter integer over
the ring of scalar matrices (diag(r,.", r)\ rE R}. Proof. It suffices to assume that in the preceding theorem the group acts trivially. We are now ready to prove Theorem 1.3.9. Let us consider a
19
CHAPTER 1
I ; II,
formal n x n matrix e= with all its points occupied by ; , where n is the order of the group G. Since nR = R and nx = 0 ~ x = 0 in the ring R, then the multiplication of matrices from Rn by e is defined, in which case e2 = e both as a matrix and an operator. Let al'." am be arbitrary elements from Rn' Let us consider an integer dependence of the elements ea1e, .• , eame over
.
T={d~ag(r
gl
, .• , r
gn..
II rE R):
Multiplying this equality from the right and from the left bye, we see that the ring eRne is quite integer over the subring eTe of a certain degree m. The ring eR n e, however, consists of the matrices, all the coefficients of which are the same, while the ring eTe consists of the matrices, all the coefficients of which are fixed relative G and equal to each other. Bearing in mind the fact that the product of the matrices I a 11·11 bll has the form I nabll, and the possibility of cancelling by n, we come to the conclusion that R is quite integer over R G. The theorem is proved.
1.4. Martindale Ring of Quotients Let R be a semiprime ring. Let us denote by F a set of all (twosided) ideals of the ring R which have zero annihilators in R. It should be recalled that the right annihilator r R (A) of a set A in R is a totality of all XE R, such that Ax= O. Accordingly, the left annihilator 1 R(A) is a set of all XE R, such that xA= O. The intersection ann R A= r R(A) ("\ 1 R(A) is called an annihilator of A in R. The lemma below demonstrates that all the three notions coincide for ideals of a semi prime ring.
1.4.1.Lemma.
Let I be an ideal of a semiprime ring R. Then
ann R(I) = r R(I) = 1 R(I).
Let us first make the following remark.
1.4.2.Remark. A semiprime ring has no nonzero nilpotent onesided ideals. Indeed, let, for instance, B be the right nilpotent ideal, B n=O. Let us consider an ideal B + RB generated by the set B. Then, allowing for
20
AUfOMORPIDSMS AND DERIVATIONS
BR s:: B, we have
(B +RB )n
= 0,
i.e., due to the semipriness,
B
+ RB =0
and, moreover, B=O. Proof (of the lemma). It is evident that it would suffice to establish the second equality. Let Ix=O. Then (XI)2= xIxI=O, i.e., xI is a nilpotent right ideal. According to the remark, xI = O. Thus, r R(I) s:: 1 R(I). The inverse inclusion is proved in an analogous way. The lemma is proved. Lemma 1.4.1 shows, in particular, that the set IF is multiplicatively closed, i.e., together with the ideals II' 12. it contains their product II I 2 , because the equality
II I2 x
=0
yields
12 x s:: ann R II = 0,
i.e.,
I2 x
=0
and
x.e ann R I2 = O.
1.4.3. Lemma. Any ideal I of a semiprime ring R has a zero intersection with its annihilator ann I. The direct sum I+ annI belongs to IF. Proof. The intersection III annI has a zero multiplication and, hence, equals zero. If (I + ann I)x =0, then Ix = 0 and, hence, x belongs to the ideal A= ann I. On the other hand, Ax =0 and, therefore, x also belongs to the annihilator of the ideal A, i.e., xE All annA= 0, which is the required proof. 1.4.4. Definition. An ideal of the ring R is called essential if it has a zero intersection with any nonzero ideal of the ring R. 1.4.5. Lemma. The ideal I of a semiprime ring R belongs to IF iff it is essential. Proof. If I E IF, then O;t I J r;;, III J. Inversely, according to the previous lemma, III annI=O and, hence, annI=O if I is essential, which is the required proof.
group
1.4.6. For every IE IF let us denote by Hom ( R I, R) an abelian of all homomorphisms of the left R -module I to R. When
considering the union v of all these abelian groups, let us introduce the equivalence relation IE IF ~,
fIJI'" flJ2
V=
v
IE F
Hom( R I, R),
iff there exists an ideal
lying in the intersection of the domains of the definitions of fIJI and
such that flJI(a)= flJ 2(a) for all
introduce ring operations: if assume that
aE I.
On the factor-set
vI'" let us
flJI(a)E Hom(Il'R),flJ2E Hom(I 2 ,R),
then we
21
CHAPTER 1
in which case
1.4.7. Lemma. The set is a ring. Proof. We should, first
V
of all,
operations. Let ~ 1 "'
~ 2 "'
that of
r 1E F ,
of
r2
~1
~2
on the ideal
on
the
r 2 E F.
ideal
I'1 E F, and, hence,
homomorphism homomorphism
is
I'" with the above operations introduced the
correctness
of
the
and let the action of
this
case
~ 1 ±
determined on the left
of
check
r2
I'1 {;;;
~ 1 ~ 2 ""
of
I'1
and
Moreover, the
since
modules
r2 "
the
is the hence, laws
of
1.4.8. The constructed ring V I", is called a left Martindale ring of
quotients, and will be denoted by R F . The reader familiar with the notion of a direct limit can easily see that RF =
lim Ie F
)
Hom R ( R I, R).
If a is a certain element of the ring R, then we can determine a
homomorphism
aE
Hom( R R, R)
of the left R - modules
by
the
formula
a(x ) = xa. The mapping i which puts the class of the equivalence defined by the homomorphism a in V I "" into correspondence to the element a, will be an embedding of the ring R into the ring R F. Indeed, if a '" 0 ,
then, by the definition, a (I) = O. for a suitable IE F. The latter equality, however, implies that Ia=O, i.e., a=O. Consequently, if al'~ER, then
'"
a 1 ± a 2 (x) = x(a 1 ± ~) = xa 1 ± xa 2 =
and, hence, i
a1(x) ± ll2( x),
preserves the ring operations. Let us now identify R with its
22
AU1OMORPlllSMS AND DERIVATIONS
image i(R) and assume R to be a subring of R F .
1.4.9. Theorem. The ring (1) For
any element
R F satisfies the following properties: R F . there exists an ideal
aE
IaE IF, such
that I a a !:: R . (2)
If
aE
RF and Ia=O for a certain
IE IF, then a=O.
(3) If IE IF and ~: I ~ R is a homomorphism of left modules, then there exists an element rE R F' such that ~(i) = ir
Rfor
all iE I. (4) Let w be an (R, R) -subbimodule in RF and ~:w ~ RF be a homomorphism of left R-modules. If W contains the ideal I of the ring R, such that ~(I) ~ R and ann R I =1 R ~ ), then there is an element rE R F , such
that
~(b)
= br for
any
bE Wand
1r =0 for
any
1E 1 R (W).
Properties (1 )-(3) result immediately from the definition. Moreover, these properties define the ring of quotients R F uniquely, to the accuracy of an isomorphism over R. Let us show the way properties (1)-(3) yield property (4).
1.4.10.
Lemma. If W is a nonzero submodule of the left (right) R-module R F' then w2 ;t: O. In particular, R F is a semiprime ring. Indeed, if O;tve wand Iv is an essential ideal of the ring R, such 2 2 that Ivv~ R, then O;t:( Ivv) ~w . Therefore, the left and right annihilators of the (R, R ) -submodule W in
RF coincide, since the equality
1W=0 implies
(W1l=0
and, vice
versa, W1=0 implies (lW)2=0. This peculiarity makes it possible to substitute right or two-sided annihilators for left ones in the definition of property (4). Let W obey condition (4). Let us denote by L an annihilator of the ideal I in R. Let us also ex tend ~ onto L+ I, assuming qi.. L)= O. Since the annihilator of the ideal L+ I equals zero, then, according to property (3), we find an element
rER F , such that
Lr=O,tr=qi..t), where
tEI.
Then, if v E w, a E IIv, then av E I and, hence, aqi..v) = ~(av) = avr. Therefore, I( qi.. v) - vr) = O. Besides, according to the condition, L· W = 0, i.e., Il . lW
L(qi..v)- vr)=O, =
0
and,
which
hence,
I1
implies '
1
~ L.
qi..v)= vr. Consequently,
If
1W=O,
I 1 1r = 0,
then which,
23
CHAPTER 1
according to property (2), affords l r =0, which is the required proof.
1.4.11. By analogy with quotients. IfF
=
lim
one can define the right ring of
RF
Hom(I R' R).
Ie P
The difference in the determination of the operations is in the only that
fact
the
product
(qJl ~) (x) = qJI (~( x».
,
in
R 0
is
determined
by
the
formula
This formula yields the correspondence
a -+
a,
where a( x) -+ ax is an embedding of R in R F' In R F let us consider a subring Ta
E
ex R)
which consists of all elements a for which there exists ideals F, such that aT a !;;;; R. In this case the mapping a -+ ax, x ETa
determines a certain element athen the mapping
x -+ xb,
E
R;. Analogously, if b
X E
E
R;
I determines the element
£
and E
Ib
R F'
E
R,
It is
evident that the mappings a -+ a, b -+ £ are mutually inverse isomorphisms, and the rings
ex
Q(R)
and
0
o
Q (R) = (r E Rp I E IE F, Ir!;:; R
can be identifies.
The ring R) is called a two-sided Martindale ring of quotients (or a symmetrical ring of quotients).
1.4.12. Proposition. Let the ring R be expanded into a direct sum of ideals R = T ED 5. Then R F = T F ED 5 F Proof. It should be remarked that the rings T F and 5 F are naturally embedded into R F: if t E T F and It!:: T, where I is an essential ideal of T, then TIT + 5 is an essential ideal of R and the mapping a + s -+ at defined on this ideal determines the element t E R F which is naturally identical to the element t. Since 5T,. = and T5,. = by construction, then we have
°
°
1/3 + T) a',. n 5,.) = 0, i.e., the intersection a',. n 5,) equals zero.
then TIq!;;;; T , 5Iq !;;;; 5, in which case TI and 5I are essential ideals in T and 5, respectively. Hence, right multiplications by q determine the elements t E T F' s E 5 F . If
q E RF
and
Iq!;;;; R, I E F,
Let us show that q = t + s. Indeed, if a E TI, then aq = at, as = 0, if a E 5I, then aq = as, at = and, hence, for a E TI + 5I the equality aq = at + as is valid, i.e., RI (q - t - s) = 0, and property (2) of the
°
AUTOMORPHISMS AND DERIVATIONS
24
ring
RF yields q =
Let (TIT
+
us
S) t I:;; T
s + t.
show
that
T
r . Sr
= O.
Indeed,
if t
E T ,. ,
then
for a certain essential ideal I of the ring T, since by the
construction of SF property (2) we get
we have TS F = 0,
ts,.
then (TIT + S ) ts,. = 0,
i.e.,
by
= 0.
Therefore, the subrings T F' SF they present the whole ring
R F'
annihilate each other and as a sum i.e., they are ideals and we h a v e
RF = TF $ SF'
For the ring of quotients Q the considerations are the same, the only requirement is to check when constructing t if t E Q (R ) when t E C( T) . The proposition is proved. 1.4.13. Proposition. If R is a semzprtme ring and I E F then the left Martindale ring of quotients of I can naturally be identified with R F ' in which case C(I) = Q(R). Proof. If A is an essential ideal of the ring I, then IAI ElF, in which case IAI I:;; A and, hence, if q: A -t I is a homomorphism of the left I-modules, then the restriction of q on IAI will be a homomorphism of the left R-modules, and we thus get the embedding I F I:;; R F .
The inverse inclusion is also evident: if J is an essential ideal of R, then JI J is simultaneously an essential ideal of the ring R and the ring I. In this case the image of the restriction ip of the homomorphism qJ: J -t R onto JI J is contained in IqJ (J) I:;; I, which allows one to identify qJ / "" with the element ip / ""
E
IF' The equality
C(I) = Q(R)
is obvious under
such an identification. The proposition is proved.
1.5
The Generalized Centroid of a Semiprime Ring
1.5.1. Definition. A generalized centroid of a semi prime is a center C of its Martindale ring of quotients R F' Is is evident center of a ring is a commutative subring. It should be recalled centralizer of the set S in the ring R is a ZA(S) =
(aE
A'rf
SE
S(as = sa)).
ring R that the that the subring
CHAPTER 1
25
1.5.2. Lemma. The centralizer of any ideal RF
I E
F in the ring
is equal to the generalized centroid of the ring R.
Proof. As identity [xz, y] the centralizer of the above identity,
usual, let us denote [ x, y] = xy - yx. In this case the = x[ z, y] is valid. Let an element a belong to I, i.e., [I, a] = O. If r E R F , then, in conformity with we have
[ x, y] z
IIr[r, a]\::[IIrr, a]+[IT r , a]r=O
Herefrom, by property (2) of the ring of quotients (see 1.4.9), we get [ r, a] = 0, i.e., a E C. The lemma is proved.
1.5.3. Definition. A ring is called regular if for any element x there exists an element x', such that xx = x .
x
1.5.4. Lemma. The generalized centroid of a semiprime ring is a regular ring. Proof. Let a be a central element of a Martindale ring of quotients. Let us show that the mapping qr. x~ ~ xa, where x runs through the set I a / ' I a'
is correctly determined. Let
Therefore,
qJ
xa2 = O.
Then
is a homomorphism of the left R-module
(xa)R F(xa) = O. J = Ia 2
n Ia
E
F
to R, i.e., by property (3) of the ring of quotients (see theorem 1.4.9), there is an element a 1 E R such that x~ a 1 = xa for all
x
E J.
According
2
to property (2) of the ring of quotients we have a a 1 = a. Let us show that the element a 1 lies in the center. Let x be an arbitrary element of the ring of quotients. Then P,f(a 2 a 1)2 l = ~,a2l=O, Multiplying
the
obtained
equality
from
and, the
hence,
left
by
a4 [x, ai]=O.
ai,
we
get
2 O=a~fall= P,fa1l.
The lemma is proved. As any regular ring, a generalized centroid has a sufficient number of
idempotents. Indeed, if ~a' = a, then (aa,)2 = ~a' . a = aa', i.e., e = aa' is an idempotent, in which case ea = a. The latter condition implies, in particular, that the c-module generated by the element a is generated by the idempotent e, i.e., eC = aC. In the set E of all the central idempotents the following relation of a partial order naturally arises:
26
AUTOMORPHISMS AND DERIVATIONS
e l :::; 1.5.5.
~
df
<=> e l e 2 = e r
idempotent e(S) = e 1.5.6.
The support
Definition. E
Lemma.
this case the equality
of the set
C, such that se = s for all s Any set
S ~ RF
S
~ RF
is the least
E S.
has a support. e(S) = e
xRS = 0 for an element
x E
E
C. In
R F (as well as the
equality SRx = 0) is equivalent to x e 1$) = O. Proof. Let V be a two-sided R-submodule in
R F , generated by the set S. In this case I = V n R is a two-sided ideal of the ring R. Let us prove that its annihilator in the ring R coincides with the annihilator of V in R. Let Ix = O. If v E v, then (see 1.4.9) Iv' v ERn V = I and, hence, Ivvx = O. By property (2), we get vx = O. By property (4) of the ring of quotients, for the identical mapping cp: V ~ V, there exists an element e E R F' such that ve = v for all
v E V (in particular, se = s for all s E S) and e annihilates the annihilator L of the set V in the ring R F' This fact implies that for any 1 E L, v E V, X E RF
the
following
equalities
are
valid:
+ v)[ x, e] = 0, (1 + v)(e 2 - e) = O. Since the annihilator of the sum L + V has a zero multiplication, then, by lemma 1.4.10, the element e is a
(1
central idempotent. Further on, if el is a central idempotent, such that se l = s for all s E S,
then, in accordance with the centrality, vel = v for all v
E
V
and,
hence, 1 - e l E L, i.e., 0 = (l - e l ) e = e - e l e, which implies e l ~ e by the definition. Let, finally, SRx = O. Then Rx lies in the annihilator of V and, hence, eRx = 0, which affords Rex = 0 and ex = O. The lemma is proved. 1.5.7.
Lemma.
If 0:;; e l :::; e(S) then e l S :;; O.
Proof. If e l S = 0, then the idempotent f = 1 - e l obeys the equality for all s E S. Therefore, f ~ e(s) ~ e l , i.e., e l = fe l = 0, which is a contradiction. The essence of proportionality over C in terms of the initial ring R is well demonstrated by the following lemma. sf
=s
1.5.8.Lemma.
The
elements
a, bE R are proportional over
C,
27
CHAPTER 1
i.e., a XE
= cb for
a certain
C E C
iff
e(a) ~ e(b) and
axb = bxa for all
R
This lemma will later be singled out from the general statement 1.7.14, and meanwhile the reader is offered the idea to prove it as an exercise. 1.5.9. Proposition. A generalized centroid of a prime ring is a field. Inversely, if a generated centroid of a semiprime ring is a field, then the ring is prime. Proof. Let R be a prime ring. As the ring C is regular, it is sufficient to prove that it has no zero divisors. If <1.. ~ = 0, <1.' ~ E C, then, choosing nonzero ideals II' I2 in R, such that <1. II ~ R, C2 I 2 !:: R we see that the product of the two-sided ideals
(<1. I 1)( ~I2) is equal to
zero, which affords <1. = 0 or ~ = O. Inversely, if aRb = 0, then by lemma 1.5.6 we have e (a)b = O. From here we get e (a)Rb = 0, i.e., in C there are zero divisors. The proposition is proved.
1.6 Modules Over a Generalized Centroid In this paragraph we in fact shall be speaking about modules over an arbitrary regular self-injective commutative ring, though it becomes evident only at the very end. The approach for presenting such a classical material is clear to be quite different. We prefer a topological approach by an important reason that all module homomorphisms, ring automorphisms and derivations, as well as termal unary operations prove to be continuous and, hence, the problem of transferring many properties onto an injective hull (or, in our terms, onto a topological closure) proves to be absolutely evident. In the case of a prime ring all the modules considered here are converted into linear spaces, and the topology becomes discrete. Therefore, the reader mainly interested in prime rings can omit this paragraph. 1.6.1. Let R be a semiprime ring, C be its generalized centroid. Let us determine the topology in an arbitrary c-module M with the help of idempotents from C. It should be recalled that a partially ordered set A is called directed if for any two elements there exist their upper boundary. The set E of all the central idempotents is partially ordered with respect to the order
28
AuroMORPmSMS AND DERIVATIONS
introduced above (see 1.5.4). In this case any subset El ~ E has an exact upper boundary sup ~ which, as can be easily seen, is equal to the support e( Et )
of the set
E1 .
1.6.2. Definitions. Let A be a directed partially ordered set. Let us call an element m of module M over ring C a limit of the family (m a e M, a e A), if there exists a directed family of idempotents such that
(ea,aeA),
ae
for
eaSe p
asp; sup(e a } =1,
and for all
A the equalities
(10)
me a = maea
are valid. In this case we shall write m = lim ma' A
Accordingly, the set T !:; M is called closed family of elements from T lies in T. A closure of closed set containing the given one. Therefore, the determines a certain topology on the C-module M. Let now qJ: 11!1 --+ M be a certain mapping of shall call qJ quite continuous if the equality ~ m)
= lim qJ(m a)' A
if any limit of the the set is the least operation of closure the C-modules. We m = lim ma yields A
It should be remarked that any quite continuous mapping
is continuous. Indeed, in this case the total preimage of a closed set is closed and, hence, qJ is a continuous mapping. It is evident that if qJ preserves the operators of multiplication by the idempotents from c: qJ(me) = ~ m) e, then qJ will be quite continuous. We, in particular, can conclude that any module homomorphism is continuous, while an isomorphism is a homeomorphism.
1.6.3.Proposition.The operators of left and right multiplication, those of the shift [Ja(x) = x + a, as well as all the derivations and RF are quite continuous. Moreover, if automorphisms of a ring r
= lim A
r a, S
= lim A
Sa
then r ± s
= lim(r a ± sa), A
r·
S
= lim(r a saL A
Proof. The operators of the left and right multiplications are quite continuous, since they are module homomorphisms. If r = lim r a , then A
(r
+
a)e a = re a
+ ae a
= raea+
aea=(ra+ a)e a ,
29
CHAPTER 1
i.e.,
r + a = lim (r a + a). A
In order to prove the continuity of derivations. let us first remark that the central idempotents are constants for any derivation. If if = e. then 2e e J.L= e J.L. Multiplying bye. we get eJ.L= 2ee J.L= eeJ.L and. hence. ee J.L= O' • t.e., e J.L= 0 . J.L ( raea) J.L=r J.L ea. t.e., . N ow we have raea= If g
A
is an automorphism. then while considering the family of
(e~). we
idempotents
r J.L= l'tmr aJ.L .
have
rgeg=(raea)g =(rea)g = rgeg
r g = l'tm r ga'
.
i.e .•
And. finally. let r = lim rat s = lim sa' In this case. if (e a ) .(f a) A
A
are the corresponding families of idempotents. then for the sets (r a ± sa)' (r aSa) we shall consider the family of idempotents ( e a f al a
E A).
It is evident that this family is directed and. if ee a fa = 0
for all a, then. for an arbitrary we get
E A
choosing an element
(eea)fp=e(eaeY)'(fpey)= (eeyfy).eafp=O.
eea=O
(r
f3
and.
± s) eaf p = (r
hence. ea)fa
e=O.
± (s
i.e .•
sup(e a f a }=1.
fa) e a = (r a
r ~ a. f3
Therefore.
Further
on,
± sa) eafa. and. accordingly.
(rs) eafa = (r aSa) eafa' The proposition is proved.
1.6.4. It should be remarked that the latter proof of proposition 1.6.3 does not imply that RF is a topological ring. as. possibly. there exist families convergent in the above determined topology but having no limit (in the above determined sense). Therefore. there arises a question (inessential for the further material but of its own interest): if with the topology determined above? 1.6.5.
Let
F( X)
RF
is a topological ring
be a polynomial of a set of variables X with
coefficients from R F which do not commute with the variables. in the presentation of which not only the operations of addition and multiplication. but possibly the unary operations of derivation as well as automorphisms take part. 1.6.6.Lemma. The
mapping
F( x): RF
e _. e
RF
-+ RF is
quite
AUTOMORPmSMS AND DERIVATIONS
30
continuous. In particular, if F( x) turns to zero on the set F( X)
T!:: R F ,
then
product
and
-+ M 1 , VI: M1 -+ M2
are
1\
turns to zero on the closure T!::: R F . Proof. Let us first remark that the
composition of quite continuous mappings quite
continuous.
qJ2(m)=limqJ2(ma)
If
difference,
qJ l' qJ 2: M
m= lim ma , A
then
qJt(m)=limqJt(ma),
and, therefore, according to proposition 1.6.3, we have
In the same way, by the definition of a continuous transformation, we get
We should now remark that
F( x) can be presented as a sum of
products of the compositions of the projections
1& i: R Fe...
e
R F -+ R F
with automorphisms and derivations R F -+ R F' as well as with the operators of multiplications by the elements from R F . And, finally, if F( X) turns to zero on T, then the mapping F( x) is equal to zero on T e .. e T, the closure of which is certain to contain 1\
e .. e
1\
The lemma is proved. The above definition does not imply that the directed family of elements (m a) has not more than one limit. T
T.
1.6.7. Lemma. The following conditions on the C-module Mare equivalent. (a) Any directed family of elements of the module M has not more than one limit. (b) Anyone-element set is closed. (c) The zero submodule is closed. (d) M is a nonsingular module. It should be recalled that a Singular submodule of the module M is a union of all the elements having essential annihilators in the ring of operators (an ideal of a ring is called essential if it has a nonzero
CHAPTER 1
31
intersection with any other nonzero ideal). And, accordingly, a module is called nonsingular if its singular submodule equals zero. Proof. The implications of (a) -+(b) -+(c) are obvious. Let us prove that the closure of a zero sub module coincides with the singular ideal. It should be remarked the essentiality of the ideal I for the ring C is equivalent to the fact that its support is equal to the unit, e(I) = 1. Indeed, if I is essential, f = 1 - e(I), then I II fC = 0, i.e., f = 0 and, hence, e(I) = 1. Inversely, if e(I) = I and I II J = 0, then, choosing an arbitrary element f in J, we get I f ~ I II J = 0, i.e., f = f· e(I) = 0 and, hence, J = O. Let m E z( M). In this case annc(m) is an essential ideal. Let A be a set of all idempotents of this ideal. It is a directed set, in which case m . a = 0 . a for all a E A, so that m = lim ma , where ma = O. aE A
Consequently, z( M) m=
Let us show
~(O).
where
lim ma ,
aE A
to be a closed submodule. Let
Z( M)
and
let
m e;z( M),
i.e.,
f = 1 - e(annc m) :I:. O.
As for the family of idempotents [ e a' a E A), which defines the limit, the condition sup e a = 1 is valid, there is an element
f3
A
E
A, such that
e 13 f:l:. O. Since
i.e., we can find an idempotent have
U
13 E
m13 E Z(M),
then sup(anne m13) = 1,
ann e m 13' such that
mef3fuf3=mf3ef3fuf3=(mf3uf3)ef3f=O.
e 13 fu 13 E ann em.
Since
the
o= (e 13 fu 13) f = e 13 fu 13:1:. 0
idempotent
f
It
annihilates
e 13 fu 13 :I:. O. We means
that
anne m,
then
which is a contradiction.
Let us, finally, show that (d) -+(e). Let us assume that a certain directed family of idempotents ma has two limits, m(l) :I:. m(2). Let be the corresponding families of idempotents. In this case we
[ e a) ,[ fa) have (m
(1)
(2) (1) (2) - m ) e a fa = (m e a - m fa) e a fa =
= ma(e a - fa) eafa = O.
If we now recall that
proved.
sup [eafal = 1 (see 1.6.3), then the lemma is A
32
AuroMORPmSMS AND DERIVATIONS
1.6.8. If A is a ring, then we as usual denote the ring of endomorphisms of the additive group of the ring A through End( A, + ). The ring End( A, +) is the right module over the center of A, but it can be not an algebra over it. The action of the central elements is governed by the formula x.( tpc) =(x tp) c. 1.6.9.Lemma. The modules R F' End(R F ' +) are nonsingular. Proof. Let us make use of condition ( a ) of lemma 1.6.7. Let r, S be the limits of the family (r a' a E A), and let (e a ) ,(f a) be the corresponding systems of idempotents. In this case (rea)fa=raeafa=sfaea.
then
r - s
= (r
Hence, (r-s)eafa=O.
- s)sup (e a fa)
= O.
If, finally,
r,sER F ,
If
r, s E End(R F' +), then
for any x E RF the equalities (r - s)( x)e afa yield lemma is proved.
.r( x) = s(x).
The
1\
1.6.10.Lemma. If T is a subring of R F' then its closure T is also a subring. Proof. Let us denote through k( T) a set of all the limiting points of T. Let
where
/3
is a limiting number. In this case the union of all
Ta
is a closed
1\
set and, hence, it coincides with T. Therefore, according to the transfinite k(T) is a subring. Let induction, it suffices to prove that r
= 1~
ra I s
= 1 ~m
s p , r a's pET
I
{
e a} , {ep } be the corresponding
systems of idempotents. Let us consider the family (e a e pi (a, assuming that r
(a,
/3) ~ (aI' /31) <=>
± s = lim (r a ± s p) , AX B
Lemma. 1\
/3
~
/31'
AX B
Let I
Then the closure I is an ideal of
which
be an ideal of the subring 1\
T.
A x B},
In this case we get
rs= lim (rasp)Ek(T),
required proof. 1.6.11.
a ~ aI'
/3) E
is
T
~
the
RF .
CHAPTER 1
33
Proof. Like in the lemma above. it suffices to prove that k(I) is an ideal of k(T). Let r = lim ra E k( T). s = lim s f.l E k( I). then A
rs
= AX lim B
raS
f.l
I'
E k(I)
and sr
B
I'
k(I).
E
1.6.12. Definition. The module M is called complete if any family of its elements (m ~ a E A). for which there exists a directed family of idempotents (e a ). such that sup (e a) mae p = mp e p' e a ~ e p
=1
a ~ P the relations
and at
are valid. has a limit.
1.6.13.Lemma. The modules
RF and
Proof. Let us first assume that family of idempotents. such that rae a
End( RF • +) are complete.
ra E RF • and let
= r pep
(e a )
a > p and
at
, ea ~ ep
be a
Let Na = {x E Rlxe a E R, xraea E R}. Then Na is a left ideal of the ring R , in which case N a ~ I a E F ( R). The union sup (e a) N
us
=~
= 1.
Naea is also a left ideal. Indeed. if aa ea.
find
aa e a ra
an
+
element
ape p r pER.
hence. aaea+ ape p
r~ a.
P.
then
ape pEN. and let
(aaea
+
ape p) rrer=
and.
Analogously.
E Nre r .
As Nrer;J Iaea. then Besides. the annihilator of I
is an ideal of R. Indeed. if Ix=O. then
N :2 LIaea = I
equals zero. Iaeax=O. i.e .• eax=O and. hence. x=O.
Let us determine the homomorphism of left R-modules by the formula aaea;
= aaeara.
aaeara= aaearr= apepr r =
If aaea= ape p and apeprfj' i.e .•
Since the domain of the definition of ; there is an element rE RF • such I(ear- eara)=O.
This
implies
;
r~
;: N ~ R
a. P. then
is defined correctly.
contains the ideal I E F. then that aaear= aaeara' i.e .•
that
re a = raea
and.
hence.
r=lim ra. If now
rae End(R F • + ). then for any
x
E RF
there exists a limit
34
AUTOMORPlllSMS AND DERIVATIONS
of ra(x). and we can set The lemma is proved.
r( x)
= lim ra( x).
in which case
r
= lim ra.
1.6.14. Lemma. The subrings Q, C are closed in RF and. hence. they are complete modules over c. Proof. Let r= lim ra and earaIa(;;. R, eaIa s;;: R, where Ia E F. Then J = LeaIa E F, in which case reaIa = raeaIa' and, hence, rJ (;;. R, i.e., r E Q. The subring C is closed as an intersection of all the kernels of quite continuous mappings ad a: x ~ xa - ax. The lemma is proved.
1.6.15. Let us recall that the module M is called projective if it has the following property. Let 1r be the epimorphism of a module B on a module A, then any homomorphism qJ: M ~ A can be "raised" to the homomorphism",: M ~ B, such that 1r. 'II = qJ
This property is equivalent to the fact that for any epimorphism there exists a decomposition M' = ker 9 e Mil. It can be readily shown and it is well known that a direct sum of modules is projective iff all the summands are projective. Accordingly, the module M is called injective if it has a dual property. Let 1r be a monomorphism of a module A into a module B , then any homomorphism qJ: A ~ M allows an extension to the homomorphism vr. B ~ M, such that qJ. 1r = qJ 9: M'
~ M
II~----B
~T .4
This property is equivalent to the fact that for any monomor-phism 9: M ~ M' there exists a decomposition 14 = imB ED M", i.e., the module M is extracted as a direct summand of any module containing it. It can be easily shown, and this is also a well known fact, that a direct product of modules is injective iff all the cofactors are injective.
CHAPTER 1
35
It should be recalled that a ring is called (right) self-injective is injective as a (right) module over itself.
if it
1.6.16.Lemma. Any complete nonsingular module over C is injective and, vice versa, any injective module over C is complete. Proof. Let T be a complete nonsingular submodule of the module M. Let us show that T is extracted from M by a direct summand. In M let us consider a set of all submodules having zero intersection with T. This set is directed by inclusion, i.e., according to the Zorn lemma, , this set has at least one maximal element M. Let us show that M' is a closed submodule in M. Let m = lim ma' ma A
M'
E
and
let
{ eal
idempotents. If m eo M', then (M' + mC) a
suitable
tea = mea
m'
+
E M',
e
E
C.
Then
maeae E M' 11 T = (0),
module T the element , Let now m m E M + T. Let us idempotents a E c, meal v ~) = m~
11 T
the
corresponding
*- (0). Let
for i.e.,
any due
to
t = m'
aE
families
of
+ me *- 0 for we
have
nonsingularity
of the
A
t is equal to zero, which is a contradiction.
be an arbitrary element of M. Let us show that , assume that m eo M, and consider a set A of all such that ma E M + T. This set is directed: if a l v ~ = a l + ~ - ~a2 E A,
then
~, ~ E A,
be
+ (m ~)(1 -
~).
so
that
Let us prove that sup A= 1.
O. Then f eo A and, hence, mf eo M' + T and, in particular, mf eo M', i.e., (M' + mfC) 11 T *- (0). Let us find an element e E c, such that 0 *- mfe + m' E T for a suitable m' EM. In this case mfe is a nonzero element of M' + T. Since C is a regular ring, there is an element e', such that e1 = ee' is an idempotent, Let, on the contrary,
f = 1 - sup A*-
and e l e = e. The latter equality shows, in particular, that other fA
hand, mfe1 = (mfe) e' lies
= (0)
and, hence,
Thus,
mf3
then m f3 = ma f3 , t t =
= .t( fe 1) = 0,
sup A= 1. Let
f3 ::; a, then
limit
fe l
lim
aE A
in M' + T,
ma = ma
We
now
have
fe l E A.
However,
which is a contradiction.
+
where
t a,
= (ma)f3 = maf3 + t a f3. As f3 = t a f3. Since the module
tao
i.e.,
*- O. On the
fe l
(m -
the sum
ma E M,t a E T. M'
+
T,
If
is direct,
T is complete, there exists a
t) a = ma E M'
and,
hence,
36
AUTOMORPIDSMS AND DERIVATIONS
m- t
= lim E A
M',
as the submodule M' is closed. Therefore, me M' + T,
which is the required proof. Inversely, let M be an Injective C -module. Let us consider a direct sum fo!1 = M e aC, where aC == C is a free one-generated module. Let, then, (m a 1 be a family of elements from
M,
and let (e a ) be a directed family
sup e a = 1 and for f3 ~ a a m p' e a = ma e a hold. In Ml let us consider a submodule
of idempotents, such
the elements Indeed, if
that
ma e a $ ae a.
the equalities generated by
N
It should be remarked that
N 11 M =
(0).
then a ~eaca = 0, i.e., ~eaca = O. a a Let f3 be the. upper boundary of elements a included in the latter sum.
Then
0= mp~eaca=~mpeaca=~maeaca' a
a
Therefore, the natural homomorphism
a
qJ: M -+
fo!1 /
N
i.e.,
m=O.
is an embedding, and
we can write the relations qJ( ma) e a + ae a = 0 for all a. Let us apply the definition of injectivity to the following diagram:
O-+M~MIN 1 ,J..
,J.. 'If
M
M
We shall find a homomorphism Let
m=-ljI(a). Then
vr. fo!1 /
N -+ M,
mae a - me a = 'If(a)e a
=
such that qJ'If = 1 .
1f!fqJ(m a )e a
+
aea]=O.
Consequently, m = lim ma, which is the required proof. 1.6.17.Corollary. A generalized centroid of a semiprime ring is a regular self-injective ring. 1.6. 18.Note. It would be expedient to add here that any self-injective commutative ring A coincides with its generalized centroid (and is, according to theorem 1.5.3, regular). Indeed, let Q( A) be a Martindale ring of
37
CHAPTER 1
quotients. Let us consider it as a left A-module. Since A!:: ~ A), then the direct decomposition ex A) = A $ M is valid. If mEA, then, by the definition of a ring of quotients, there is an ideal I E F (A), such that I m !:: A. On the other hand, I m = (0). Hence, I m = (0) and, consequently, M = (0). Thus, A= ex A). 1.6. 19.Note. We have seen the definition of topology to depend only on the structure of E-module (here E is the set of all central idempotents), and it would, therefore, be natural to expect that the statements formulated for C modules will also be valid for E-linear subsets. And this is really the case. Let us convert E into a ring by introducing an addition by the formula e $ f = e + f - 2ef. Then the E-linear sets will be converted into modules over the ring E. And, finally, it should be remarked that the ring E coincides with its generalized centroid. 1.6.20. Lemma. ~ E) = E. Proof. Let I be an essential ideal of E. In this case sup I = 1 and, if Iq ~ E, q E ex E), let us assume that e i = iq and consider I as a directed set of idempotents. Since C is complete, we find e = lim e i. It is I
evident
that
e
is
an
idempotent
and
i(e - q) = e i - iq = 0,
i.e.,
= 0 and, hence, q = e E E. The lemma is proved. 1.6.21. Lemma. Let M be a nonsinguiar C-module. In this case any finitely-generated C-submodule in M is projective and injective, and is a direct sum of a finite number of cyclic modules. Proof. Let us first prove that this is true for a one-generated submodule. We have mC == C / ann m, where ann m = (c E C, me = O) . I( e - q)
Since M is a nonsingular module, then m· lim c a = lim me a, where c a is a converging family of elements from ann m. This means that ann m is a closed ideal in C. Any closed ideal in C is principal (and vice versa). Indeed, a set of all idempotents from a closed ideal forms a directed family, the limit of which generates the given ideal. This limit exists because C is a closed submodule of the complete module Q. Therefore, ann m = eC and, hence, me == (1 - e) C, which affords C == m C E9 (1 - e)C. As C is both an injective and projective C-module, then mC is also both an injective and projective C-module. Let us now assume that the lemma is also valid for the submodules with n - 1 generating elements. Let N be an n-generated submodule, and m be a generating element. In this case the chain 0 ~ mC ~ N ~ N / mC ~ 0 splits, since, according to the above proved, mC is an injective module. We have N == mC E9 K, where the submodule K == N / mC is generated by the
38
AlJfOMORPHISMS AND DERIVATIONS
(n - 1 )-th elements. Thus, the lemma is proved by induction.
In lemma 1.6.21 we, in particular, have shown that if m is an element of a nonsingular module, then mC == eC for a certain idempotent e E c. This idempotent will here from be denoted by e( m) and called a support of element m, which fact does not contradict the denotation 8(S) for elements
S E
R F assumed above.
1.6.22. Lemma. Let M be a complete nonsingular c-module, N be a closed submodule of M. Then factor-module M / N is complete nonsingular module. Proof. By lemma 1.6.16 the module N is injective and, hence, it is singled out from M by a direct addent: M = N $ K. One, has, finally, to remark that M / N == K is an injective nonsingular module. 1.6.23.
Let
Remark.
VI' v 2 be
VIE VI' V2 E V2 · Then vI® v2 =0 in VI® ~
Proof.
We
1.6.24.
Vi ®
iff
C -modules,
e(VI) e(V2) = =0.
v 2 C == e (v2 ) C.
c, which is the proof due
VI C ® v2 C == e (vI) e (v2 ) VI C ® v 2 C ~
VI C == e (vI) C,
have
nonsingular
to
Therefore,
the embedding
V2 ·
Remark.
Let
VI' V2
be nonsingular C-modules and let
VI' v2~-' vn E VI' Then there exist elements
~'C3'-" CnE C,
such that for
any al'-" an E v 2 the equality I v i ® a i = 0 implies ale (vI) = I
i:? 2
The proof will remark 1.6.23 affords Let now
be carried out by induction over n. At
C
i'
n= 1
a l e (VI) = a l e ( aI)e (VI) = O.
n> 1. By lemma 1.6.21, the submodule VI C is extracted
from the module I vi C by a direct summand. Let I vi C = VI C $ and vi = VI C i + ._,i
~
WzC
$ .-
2. Then
This affords vI®(aI+Iaici)=O, and, by remark 1.6.23,
-I
ai
a i C i e (VI)' which is the required proof.
aIe(vI )=
39
CHAPTER 1
1.6.25.Lemma.
Let D.
~nn
be
a
submodule
End( R F , + ), which consists of all inner for
of
the
Q derivations. Then
module D.
~nn
a closed submodule and, hence, the fiactor module End(R r ,+) / D.~nn nonsingular.
Proof. We have D.
~nn
is is
== Q / C. Since Q is a complete module and
C is a closed submodule, then, according to lemma 1.6.22, Q / C is a complete nonsingular module. We have already remarked that any module
isomorphism is quite continuous. Hence,
D inn
is also a complete module and,
consequently, it is closed in End( R F , + ). The lemma is proved. Let us call the sum of elements a + b of a nonsingular module M orthogonal if the supports of these elements are orthogonal, e(a)' e(b) = O. The subset S of the module M will be referred to as an E-subset (here E is a set of all central idempotents), if it is closed relative to multiplication by the idempotents and, together with any two elements, contains their sum, provided it is orthogonal. Accordingly, the mapping f: S --+ N is called an E-mapping if for any idempotents e l , e 2 E E and the orthogonal sum aIel + a2 e 2 the equality
.t(alel + a2e 2 ) = f(al)e l + f(a 2 )e2 holds, i.e., if
f preserves the multiplication by the central idempotents and the orthogonal sums.
1.6.26. Lemma. Let closed E-subset in
M.
Then,
M
if
be a complete nonsingular module, S be a f: S --+ R F is an E-mapping, then there is
an element s E S, such that the support of the f image coincides with that of .t(s). Proof. It should be first recalled that f either reduces or preserves the supports: e(f(a»::;; e(a). Indeed, .t(a)· (1- e(a» = .t(a - a) = 0 and, hence, e(f(a»(1 - e(a» = O.
Let us consider a set N of all pairs (s, e), such that se = s E S, e = e ( f( s», and in N let us, as usual, determine the order relation ( s, e) ::;; (s I' e l ) ~ sl e = s, e::;; e l . Since f either preserves or reduces the supports, then for any s E S the pair (s e(f(s»), e(f(s») lies in N, so that N is nonempty. If (s a' e a) I a E A} is a linearly ordered subset in N, then, according to lemma 15 and since S is closed, there exists a limit
40 s
AUTOMORPHISMS AND DERIVATIONS
= lims a E
S. In this case
e(f(s» e a
A
= e(f(s)e a) = e(f(s a)) =
e a , and,
therefore, e c;t e (f( s» ~ ea' Hence, the pair (s, e) is the upper boundary of the set considered. According to Zorn lemma, the set M has maximal elements, so let ( s, ~) be one of them. If now
f(s ) (1 - ~):f::. 0, then there is an element
Let
sp -
~ =
~).
sl E S, such that
Then
~~=O,
f(s2) = f( sl)(1 - ~):f::. O. The first of these equalities shows the sum
to be orthogonal. The sum have
since
+
s
ea::;
~;
~::; eO' i.e.,
Hence, the support on the other hand,
ea =
ea ~
+
S
is also orthogonal. Therefore, we
e(f(s2e(f(s2»+ s»= e(f(s2)+ £(s»= e(f(s2»
= =(f(s2» + ~.
i.e.,
~ e ( f( s2»
~
of the image f is the support of
+
e(f(s»=
annihilates £(s)
(1 - ~),
and, hence,
~, which is the required proof.
1.7. Extension of Automorphisms to a Ring of Quotients. Conjugation Modules Let us fix notations R, R F' Q and c for a semiprime ring R, its rings of quotients and a generalized centroid, respectively. As above, IF = IF (R) will denote a union of all essential ideals of the ring R. 1.7.1.
Lemma.
Let
g be an isomorphism of the ideal
I E
IF
into the ring R, such that I g;J II ElF. Then g is uniquely extended to an isomorphism of the ring Qand to that of the ring R F
Proof. remarked that
Let
a
be an arbitrary element from g
I2 = II( I I a) II ElF, where
R F'
It should be
I a is an ideal from
IF, such
that I aa ~ R. Indeed, if I 2 r = 0, then 0 = II(II aA)g, where A is such a
4}
CHAPTER }
set that A g= I} r. This set exists since I} is an ideal and I} ~ I g . As g is an isomorphism, then II aA = 0, which affords A = 0 and, hence, I} r
= 0,
but I}
E
F, Le.,
r
= 0,
which fact proves that I2
Let us now define the mapping a g: I2 ~ i
E
I 2 , then
i
=j
g for a certain
j
E
:? I a I,
R
E
F .
in the following way. If
and we set
ia g = ( ja) g ,
in which the right part of the equality is determined because ja ~ I. It is this mapping that determines the element a g due to property (3) (see 1.4.9). We can now easily check if a ~ a g is the sought extension of g. In this case the formula other extension,
j gag = (ja ) g g I, we
j gag - j g a g'= ( ja)g - (ja)g = 0, g =
g.
And,
finally,
shows its uniqueness, since for the would have the equality
~(a
Le.,
if
a
E
Q
g - a gl)
and
= 0,
and,
hence,
aJ E R,
then
agI}(IJ)gI}~ag(I2JI)g~(aJI)g~Ig~R, Le., g is an automorphism of ring Q. The lemma is proved. 1.7.2. It follows from the lemma that any automorphism of the ring R has the only extension on Q and on
R F' Therefore, herefrom we shall
always consider the automorphisms to be determined on introduce the following notation:
R F'
Let
us
It is evident that the group of automorphisms of the ring R is contained in A ( R). All the further considerations on automorphisms are valid not only for the automorphisms of ring R, but also for those from A ( R) and, hence, if the word "automorphism" is not followed by a special indication of the ring, the automorphism will be assumed to belong to A ( R) . 1.7.3. Lemma. The set A ( R) forms a group which contains all the inner automorphisms of the ring Q. Proof. Let a be an inverse element from Q. Let us denote through a an inner automorphism corresponding to this element a: x ~ a-I xa . According to the definition, we can find an ideal I E F ( R), such a- 1 I ~ R, aI ~ R, Ia ~ R. In this case we have that Ia-} ~ R,
42
AUfOMORPIllSMS AND DERIVATIONS
a(I2) = (a- 1 I). (Ia)!:;;; R.
aI4 a -1= (aI) I2(Ia - 1)
Besides,
hence, a(I2) ~ I4. Therefore, a e A (R). Let g, h e A (R) and we assume R ~ I g~ II' I, II' J, J I e F.
Then
J I II !:: I
A!:;;; I. Let us consider the ideal I2
12 v = 0, implies
g
then 0 = (I2 · v) = I
12 e
g
g
and,
= IAI. gg
J 1 II I
v
R;:? J
therefore,
J 1 II
!:;;;
and,
I2,
~
JI,
=A
g
where where
,
If
~ II J 1I
2g
g
I v , Le., v = 0, which
F . Then we have
Besides,
2 h- 1
The latter inclusion results from the fact that if (II J I II) II J 1Iiv h=0 and, hence, v h= 0 = v. Thus,
g h - Ie A (R).
v
= 0,
then
The lemma is
proved. 1.7.4. Lemma. Let Proof. Let
g e A (R) and
and, therefore,
case IJ1 !:: J 1 !:;;; J
g,
The ideal JAIJ
belongs to
IJ1v g
= O.
Ie F. Then R;:? J
A(I)
g~ J 1 for suitable ideals. In this
IJ1 = A g
F(I),
= A (R).
for a certain subset
since the equality
Av = 0
= O.
Therefore,
Inversely, if JF(I), then
k J.
implies
Then we have (JAIJ)g = J g IJ1(IJ)g k I. Finally,
IJv
i..e., v
A
g
g
g
-1
= 0,
e A (I) .
e A(I) and
I ~ J g;:? J 1
for suitable ideals from
43
CHAPTER }
and
3
F
IJI, IJ}I E
The lemma is proved.
(R).
1.7.5. Definition. Let g be an automorphism from A ( R). Let us denote by cP g a set of all the elements of ring R F' such that for any x
E R
the equality cP g
qJx g
= (qJ E
= xqJ
holds
R F I \;/ X E R
qJx
g
= xqJ ) •
If h is another automorphism, then one can easily prove that
(11)
and, besides, if h is an identical automorphism, then cP h = C and, hence, cP g is a module over c. The module cP g will be called a conjugation module for the automorphism g (see lemma 1.7.7 below).
1.7.6. x
E R F , qJ E
Lemma.
~
R.
Let
Then
x E RF
for
all
(qJig)x g = iqJx g ,
Let
qJ
{: Q
and the equality
qJx g = xqJ
holds for all
cP g .
Proof. Ix
cP g
and I qJ E
cP g ,
be an ideal from we
F,
such that
have
(ix)qJ = qJ( ix)g = g I(xqJ- qJxg)=O, i.e., xqJ= qJx . i E I
here from we get be a nonzero element from cP g . Then one can choose an
ideal IE F, such that IqJ r;;,. R. By lemma 1.7.4, the automorphism g lies in the group A (I), Le., one can find an ideal J E F(I), such that Jg:;J J} E F (I).
and, hence,
We get
qJ E Q,
which is the required proof.
1.7.7. Lemma. Let g be an automorphism from group A ( R) . Then cP g is a cyclic c-module cP g = qJ gC in which case the element qJg
44
AUTOMORPIDSMS AND DERIVATIONS
is invertible in the ring e (4) g) C and the action of g on coincides with conjugation by the element ~ g. Proof•• Let us show that any element ring
~E
e( 4> g)R F
4>g is invertible in the
and the action of g on e( ~) R F coincides with conjugation by
e( ~)Q
~.
Let us consider a mapping ,:
x~
--+ x e (
where x runs through
~),
This mapping is well defined: if x~=O, then xR~= X~Rg=O, and, hence, x e (~) = 0 (see 1.5.6). The mapping obeys all the requirements of point (4) of 1.5.9. Indeed, by lemma 1.7.3 we can find an ideal IE F, Q.
-1
such that
I g
~
and, by the definition of
R
an ideal
RF
J E F,
such
-1
that J~ ~ R. Then JI g ~ = J~I is an ideal of the ring R, lying in the domain of the ~ definition, in which case its annihilator in R coincides with that of ~. Therefore,
one
can
find
an
x~b = xe (~). Substituting into it x
= 1,
~ is right and left invertible, Le., b g construction, e (b) = e (~). The
e(~)g=
e(b)g= e(b g)= e(b)= x
g
= e (~)
g
x
g
element we get ~b -1
b E RF '
=b
= b = ~- 1, latter
-1
g
such
that
~ = e (~). Thus,
where we have, by equality affords
e(~). If now xE e(~)RF' then we have g
g
-1
= e ( ~) x = b~x = bx~ = ~
xC{).
In order to find the generator of the module tP g' in tP g let us introduce a relation of a partial order by the formula ~1 ~ ~2 ~ e ( ~1) ~2 = ~1· If ( ~ a'
we ~
put
= lim ~ a A
aE
ea=e(~a)+l-sup(e(lPa»). E
A) is a linearly ordered subset, then
In
this
case
supe a =l
and
4> g' as 4> g is a closed set (it is equal to the intersection of
all the kernels of quite continuous mappings f3 x: a --+ ax g - xa), in which case ~;::: ~a for all a. Therefore, according to the Zorn lemma, one can find a maximal element ~ E 4> g. Let 1/1 E 4> g. Then ~ + (1 - e (~» 1/1;::: ~, and, therefore,
1/1 = e (~) 1/1, i..e., 1/1 = ~(~- 11/1). It remains to remark that ~- 11/1 E C, as
45
CHAPTER 1
-1
fP
E
tP _ I' The lemma is proved. g
1.7.S. Definition. Hereafter the idempotent
tP g
= fP C
will be denoted by
i(g). This idempotent singles out the part of
g acts in an inner way, Le.,
the ring RF on which the automorphism there exists a decomposition such that
RF
=
gl
1\ E
~ Er>~,
e (tP g) = e (fP), where
~
=
~
i(g) R F ,
IutR I , tP g I ~
= O.
= (1-
i(g» RF'
In particular, we have the equality
i(g)= i(g-I).
1.7.9. Corollary. If R is a prime ring, then rp g :#. 0 iff g is an inner automorphism for Q.. In this case the space tP g is onedimensional over C. The importance of modules rp g and of the above mentioned decomposition is determined by the following theorem.
Let the elements
1.7.10.Theorem.
be such that
a l E ~tP
automorphisms from vI' ... ,
V
k' t}' ... , t k
a=
k
-I
g2
gl
+ •. +
antP
g
a}' ... ,an of
-I n
g
where
gI' g2"'"
A ( R) . Then there exists a natural E R, such that k
gl
g
L v .aIt J.;to, L v·a·t .1=0, where 'I J 'IJ~J
J=
the
k
and
ring
RF
gn
are
elements
2~ i~ n.
J=
The proof of this theorem will be deduced from lemma 1.7.12. But let us ftrst make some deftnitions. 1.7.11. denote through
Definition. L
= IL( R)
Let
Z
be a ring of integer numbers. Let us
a subring in the
generated by elements of the type 1 ® r
oP, r
tensor product
Q®
z
QOP
® 1 where the rings Q and
QOP are mutually antiisomorphic but have the same additive groups, while the elements r and r op run through the rings Rand R oP, respectively. We also assume that r -+ r op is an identical antiisomorphism. If g = A ( R) ,
AUTOMORPHISMS AND DERIVAllONS
46
r
13 =L r i ®
E RF
aT E L then let us set r'
i
13 = 2,a i rr i i
,
and
If A is a subset of L, then let
If
r
E
R F , then let us denote by
such that r·
13
r.L g a totality of all elements
E
L,
13 g= 0
1.7.12.
Lemma.
Let
A be a right ideal of the ring L; necessarily different) automor-phisms.
g, gp.' gn be a sequence of (not
Then for elements
r l , ... , r n
of the
ring
the following
RF
equality
holds:
A
Ln
.Lg
+
r.CP
i =1 ~
g~
1
= «(1 r.
.Lg.L ~
i
l(lA)
(12)
g.
r:
Proof. Let us first show that the left part is contained in the right
13
one. To prove it, it is sufficient to show that if then r.qJ· ~
13 g= 0
for all qJ E cP
-I
gig
•
=
LS k ®
fk E
gi,
We have
Proof of an inverse inclusion will be carried out by induction over the number n. If v
E
and, therefore, qJ: r l . gl
subset r l ' A modules. Indeed,
to
13
gl
RF .
[r(r 1 ·f3
= v(f3(l
.L g .L g (r l 1 Il A)
gl
4
V·
g
13 , 13
Moreover,
)] qJ= [rl · (13(1
® r»
g
and E
E All
.Lg rl 1
then
V·
13
g
= 0
A is a correct mapping from the
is a homomorphism of left R-
qJ
® r»
= rev· 13 g )=
13
gl
] qJ=
r[(r l ·
13
~
)qJ].
CHAPTER I
47
Let, then, II be an ideal from F, such that all r
-I
Then for
R:2 I i
we have the equalities
E II
g,
[(rIo f3 l)r]tp=[r1(f3(r
g ·-1 1
® l»gl]tp=
-I
- 1
= v.(f3(r gl ®l»g=(v.f3 g )r gl =
= [( r 1 .
f3
g
tp] r
I)
g-I g 1
Let us now choose an ideal g
(13)
I E F
in such a way that the inclusions
gt
R ~ J :2 I, R:2 J I :2 I, Irl !:: R, Iv!:: R be fulfilled, where -I
suitable ideals from therefore, g,
IR I!:: R
g
IR !:: (I
In this case
F. g-I
g g g R)!:: (JR) !:: J !:: R,
can find an element E V.
and,
I
. 10
gl
are
the
!:: J 1 ,
and,
same
way,
Let us restrict the domain of the definition of the mapping tp ,
Le., let us assume that f3 of the definition v forms values lies in R. Let tp(ann v) = 0. Then, due a
and
!:: J
I g
J, J 1
-I
runs through A(1 ® I2). In this case the domain an ideal of the ring R, while the domain of the us extend tp on V + ann V E F, assuming to property (3) of the ring RF (see 1.4.9), one
~ E R,
such that (ann V)~
Now equality (10) shows that for all
f3
E
= 0, A(1 ®
a~
r)
= atp
for
all
the following
equality holds:
-I
Besides, (ann V) (r~ - ~ r gl g) ~
E
tP
-I
gl g
V·
and,
hence,
Le.,
•
Then, =
=0
if
f3 g - (rr f3 gl )~ =
f3 E A(1 ® I2) V·
we
f3 g - (rr f3 g1)tp = 0,
have and,
(v-rl~)·f3g=
therefore,
48
AUTOMORPlllSMS AND DERNATIONS
Therefore, the lemma is proved for Let us now assume that .lg
A
n-I ~
+ £.J
r if/J _, = ( i= I gi g
Then, due to the case n
.1
However, A n rn g n = 1
.lg
A
+
n
= 1.
n - I .lg i .1 g df .1 g r inA) = Al
n
(14)
=1
.l n1nr. gi n ~
n-l ~ £.J r if/J _, i= I g, g
A and, by (11), we get
+ r nf/J _,
gn g
gi .lg = ( n1n r .1i n A) ,
which is the required proof. If we now set q = qI' A = L (R ), then the theorem results from the
lemma proved above. Indeed, in this case the condition of the theorem implies that a l does not belong to the left part of (9) at i ~ 2, while the conclusion implies that a l does not belong to the right part of equality (9) at i ~ 2. 1.7.13. that a l e: such that
L
i~2
Theorem. f/J _ I
g, gl
ai
If elements
a l ,•., an of the ring
then there exist elements v j' t
RF
j E R,
are such
1 $ j $ m,
This theorem is obtained from the preceding one by transition to an antiisomorphic ring R op, since rp h ( R op) = f/J identical antiisomorphism.
h-
I(
R), where
r ~ r op is an
49
CHAPTER 1
The above theorems are also valid in the case when all the automorphisms are identical, in which case all the modules rb g coincide with the generalized centroid, and we get the characterization of a linear independence.
1.7.14.
If elements
Corollary. then
alli!:~C+.-+anC
v j' t j E R, 1 S j S k,
there
a1' ... , a n belong
can
be
to
found
R F and
elements
such that
Herefrom one can easily deduce the proof of lemma 1.5.8 promised earlier. Indeed, let e (a) ~ e (b) and axb = bxa for all x E R. Then, if b Ii!: aC, there can be found elements v j' t j E R, 1 S j s: k, such that Lvjat j=O. We have
b 1 =Lvjbt j :;t:O;
axb1 = ~ ~ bxv ],at,] = bx £.... ~ v'] at ],= o. £...: axv ],bt ],= £.... ]
It means that
e(a)' e (b 1) = 0,
which contradicts
the
fact
that
e(b1)S e(b)S e(a).
Inversely,
then axb = axac = acxa = bxa e(b)= e(a)e(c)S: e(a), The lemma is proved.
1.7.15.
if
b = ac,
Definition.
independent of elements
An
element
~,-" an E RF
a1 E RF
and
is said to be rig h t
with respect to a sequence of
automorphisms gl' ... ' g n E A (R ), if
Theorem 1.7.10 can now be reformulated in the following way: if a 1 is right independent of ~ ,_" a n with respect to g1' ... , g n E A (R ), then there exists an element P E lli.. R), such that
AUTOMORPlllSMS AND DERIVATIONS
50
1.8. Extension of Derivations to a Ring of Quotients 1.8.1. Proposition. Let J.L be a derivation determined on the essential ideal I of a semiprime ring R, with its values in R. In this case J.L can be uniquely extended to the derivation of ring Q and that of ring
RF · Proof. I r r !:: R
Let
r E R F'
and denote the ideal ( I I rl
homomorphism of the left
~( ]) = (
Let us consider an ideal I-modules
through ~: J
J.
IrE IF, such that
Let us consider the
-+ R, acting by the formula
jr) JL - j JL r. It should be remarked that
Jr
~
(jr)JL is determined. .JL 2 JL JL JL ] E [(IIr)] !:: IIrI + I IIr~ I r , and , theref ore,
II r ' r
hence,
due to property (3) of the ring rl
E
RF ,
such that
RF
~
I,
and,
Besides, ].JL rE R. Hence,
(see 1.4.9), we find an element
jrI = (jr)JL - jJL r.
Let us set rJL = r l and show that
J.L is a derivation of
R F'
Let
r, SERF and let us choose the ideal J 1 = {IIrIs)2. Then for any j E J 1 we have the relations (js)r JL = (jsr) JL _ ( js) JL r, . JL r ]s
( . ) JL r - ] .JLsr, -'I ) JL J\sr = ( .]sr) JL - ].JL sr. =]S
Adding the first two equalities and subtracting the third one, we get
from which, by property (2) of the ring R F (see 1.4.9), we get
51
CHAPTER 1
i.e.,
Jl is a derivation of R F • Then, if r
r Jl., . ~
E Q
and rI'
=( r·
s::
R, we have
.,)Jl - r(.,)Jl ~ E R,
~
' ) 2 . Therelore, e where ~., E (II rJl E Q and Jl is a derivation of the ring Q. And, finally, the uniqueness of the extension follows from the equalities
where
j E (II r)2. The lemma is proved.
1.8.2. It follows from the lemma proved above that any derivation of the ring R has a unique extension on Q and R,.. From now on we shall therefore consider that the derivations are determined on Q. Let us introduce the following notation: D (R ) = (Jl
E
Der Q I 3 I E F, I Jl s:: R).
(15)
It is clear that the derivations of the ring R are contained in D ( R). All the further statements are valid not only for the derivations of the ring R, but also for those from D (R) and, hence, if the word 'derivation' is not followed by the notation of a ring, the derivation will be assumed to belong to D( R) .
1.8.3. Lemma. Let a ring R have a simple or zero characteristic. Then the set D ( R) forms a differential Lie C-algebra which contains all the inner derivations of ring Q. Proof.
Let q E Q and I Iq u qI!: R, in which case I
ad q
!: qI
be
an
ideal
from
F,
such
that
+ Iq!: R
and, hence, the inner derivation conforming to the element q lies in D ( R) . A set of all derivations of the ring Q forms a differential Lie
52
AUTOMORPHISMS AND DERIVATIONS
algebra over its center C. It, therefore, remains to check if module over C closed with respect to the Lie operations. If C E C and Ice ~ R, then
J1, v
Moreover, if
E
D(R) and
D (R)
is a
are the corresponding ideals,
I, J
then ((IJ»
2 IlV
D ( R)
Hence,
~
((IJ)
is
11
IJ + IJ(IJ)
closed
11 v v ) ~ J ~ R.
relative
to
the
operations
Lie
[J1, v] = J1v - vJ1, J1[P 1= J1 ..... J1. The lemma is proved.
1.8.4. Lemma. Let Proof. If
I E IF. Then D (I) = D (R). 11 D (R) and J ~ R, then
J1 E
211 11 (I.T I) ~ (IJ) JI
J1
and, hence,
1.8.5.
E
+ IJ(JI)
11
~
I
D (I). The inverse inclusion is obvious to the same extent.
Lemma.
The
automorphisms from
A (R) act
on
D ( R)
by the formula g
J1 = g
-1
J1g.
Proof. The fact that direct checking.
J1 g E D (R ). Let us choose an ideal
Let us show that that time J E
I 11~ R
I g ~ R.
J1 g is a derivation of ring Q is proved by I
E
IF, such
Reducing, if necessary, I one can assume that at the same Using
IF, such that J g
the equality - 1
A (I) = A (R)
~ I. Now we have
The lemma is proved.
we
can
find
an
ideal
53
CHAPTER 1
1.9.
The Canonical Sheaf of a Semiprime Ring A
In this paragraph we shall present a closure RE in ring R F as a ring of global sections of a certain sheaf over the structure space of a generalized centroid. Such a presentation is useful since it enables one to see clearly the theoretic-functional intuition employed when studying a semiprime ring. It is worth while adding that in the case of a prime ring all these constructions degenerate and their essence and meaning is to reduce the process of studying semi prime rings to that of studying prime rings (i.e., stalks of a canonical sheaf). Roughly speaking, a ring of global sections of a sheaf is a set of all continuous functions on a topological space, the only difference being that at every point these functions assume the values of their own rings. Let us give the exact definitions.
1.9.1. Let there be a topological space x, with its any open set U put in correspondence to a ring (group or, more generally, an object of a certain fixed category) 9\( U), and let for any two open sets U c V be given a homomorphism
P& :9\(V) ~ 9\(U). This system is called a presheaf of rings (groups or, more generally, objects of a fixed category), provided the following conditions are met: (1) if u is empty, then 9\( u) is a zero ring (a unit group, a trivial object); (2)
p
g is an identical mapping;
w v w (3) for any open sets fJ ~ V ~ IN we have P fJ= P fJ P v' This presheaf will be denoted by one letter, 9\. The simplest example of a presheaf is a presheaf of all functions on X with the values in a ring A. In this case 9\( u) consists of all functions on U with the values in A,
v
and for U c V the homomorphism P fJ is a restriction of the function determined on V to the subset U. In order to extend the intuition of this example to the case of any presheaf, the homomorphisms
P
& are
called restriction homomorphisms.
The elements (of a ring) 9\( u) are called sections of the presheaf
54
AlITOMORPIDSMS AND DERIVAllONS
9t over u. Sections of 9t over x are called global. Therefore, 9t( u) is a ring of sections of the presheaf Rover U, R( x) is a ring of global sections. Returning to the example of the presheaf of all functions on x, let us assume the topological space X to be a union of open sets Ua. Then any function on X is uniquely determined by its restrictions on sets ua and, besides, if on every Ua a function f a is given, and the restrictions of fa and f
p coincide on
Ua
n
U P' then there exists a function on
x, such that
every fa is its restriction on Ua. These properties may be formulated for any presheaf and they single out an extremely important class of presheaves.
1.9.2. Definition. A presheaf (of rings) 9t on a topological space is called a sheaf (of rings) if for any open set U c X and its any open cover U = U Ua the following conditions are met: (1) if
pga sl = pga Sz
(2) if
sa E 9t(Ua )
s a and
s
p on
for sl' s2
E
9t(U) and all a, then
are such that for any a,
P
sl =
Sz;
the restrictions of s E 9t( U) ,
Ua n U p coincide, then there exists an element
whose restriction on Ua is equal to s a for all a. Now we shall try to consider an arbitrary sheaf as a sheaf of functions on space x. To this end it is necessary to determine the value of s( x) for the section s E 9t( u) at any point x E u. We can determine elements s( v) = sp ~ for open neighborhoods V of point x which are contained in U and, it would hence be natural to consider their 'limit', and so we have to introduce a direct limit lim
xe V )
9t(v)
relative to the system of homomorphisms p~ : 9t( U) -) 9t(V).
This limit is called a stalk or the sheaf (or a presheaf) 9t at the point x. Therefore, a stalk element at the point x is set by any section over the open neighborhood of x, but in this case the two sections, u E 9t ( U) , v E 9t (v) are identified if their restrictions on an open neighborhood of x coincide. For any open set U 3 x a natural homomorphism
CHAPTER 1
55
can be determined, which maps a section to the stalk element determined by it. Now we can determine the value of section s at the point x as
P~s). If 9t is a sheaf and if two sections,
up u 2 E
9t( U) are such that
p u~ u1) = p u~ u2) for all points x E U, then U1 = U2 • Therefore, for sheaf 9t the elements from 9t( u) can be given by the function s: U ~ u
xE
U
9t
x' s( x) E
9t
x (or, equivalently, by the elements of a direct product
II9t x). It is evident that not every such function s determines the section. U
The following condition is necessary and sufficient:
for any point neighborhood
x E U there exists such an open W of the point x contained in U,
an element W E 9t( W), such that for all points yEW.
w s( y) = p .Y< W)
In fact, a topology can be introduced on the set
and
U
(*)
9t
x
in such a
XE X
way that condition (*) becomes equivalent to the continuity of function s. 1.9.3. Let us now go over to constructing a canonical sheaf. Let c be a generalized centroid of a semiprime ring R. Let us denote through X a set of all its prime ideals other than c. This set is called a spectrum of ring C and is often denoted by Spec C. The elements of the spectrum of ring C will be called points of the spectrum or simply points. In the theory of commutative rings there is a difference between a simple spectrum and a maximal one, but, however, as is shown by the following lemma, there is no such a difference in our case. 1.9.4. Lemma. If P E Spec C, then the factor-ring is a field, i.e., any prime ideal of ring C proves to be maximal. Proof.
Let a
homomorphism
C
~
be a pre image of a nonzero element at a natural Cp. In this case, due to regularity of ring C we can
= a and e = aa' for some a' E C. In e. However, e Cp (1 - e) = 0, and, hence, case is impossible, as a = e . a :I: 0 and,
find an idempotent e, such that ea terms of images it means either
e =0
or
Cp = C / P
e = 1.
a· a = The first
56
hence,
AUTOMORPHISMS AND DERIVATIONS
a· a' =
1. Thus,
Cp is a field, which is the required proof.
1.9.5. In order to set a topology on x, it is necessary to define the operation of closure. For the set A c x let us define the closure A as a union of all the points containing the intersection
("\
PEA
P.
The topology
obtained in this way is called spectral topology. With the view of constructing a sheaf over x, it is useful to know the structure of open sets in X (they are also called domains). 1.9.6.
Definition.
If e
is
U( e) we denote a set of all points
a central
p, such that
idempotent, e!i!O P .
then
through
1.9.7. Allowing for the fact that the product e(1 - e) = 0 belongs to any prime ideal, we see any point of the spectrum contain either e or 1 - e, but not both simultaneously. Therefore, U(e) U U (1 - e)
= x,
U(e) ("\ U (1 - e)
=0
(16)
On the other hand, the closure of U(e) contains only the point q containing the intersection ("\ p = ("\ p. The latter intersection contains eep
l-eEp
the element I - e and, hence, 1 - e E q, which affords e!i!O q, i.e., by the definition, q E U(e). Now relations (13) show U(e) to be an open and closed set simultaneously. 1.9.8. Lemma. The sets U( e), e E E form a fundamental system of open neighborhoods of x, i.e., any open set is presented as a union of sets of type u( e) . Proof. Let w be an open set of points, A= ("\ p. By E( A) let pew
us denote a set of all idempotents from A, and let us consider a family of neighborhoods U(e), eE E(A). If pE U(e), eE E(A), then e!i!Op, and, hence, p r:x. A, i.e., p does not belong to the closure of the complement of W. Since the complement is closed, the condition p E u( e) implies pEW. Therefore,
U
eE E( A)
U( e) k; W.
Let us show that the inverse inclusion is also valid. Let, on the contrary, pEW, p!i!O U U(e). In this case all the idempotents from A lie in p. Since A is an ideal, then, since the ring C is regular, it contains,
57
CHAPTER 1
together with the element a, its support e (a) = a· a'. This support lies in p and, hence, a = e(a) a E p. Therefore, p contains a and, hence, it belongs to the closure of the complement of w. As the complement is closed, p Ii!: w, which is a contradiction. The lemma proved shows, in particular, that the space x is quite disconnected. and its any open set being a union of closed domains. 1.9.9. Lemma. Any closed domain in X has the form U(e) for an idempotent e. Proof. Let U be an open and closed set. Let us consider its complement U' up to x. Let A= (, p, B = 1""1 P. The intersection 9E U
pE U
of these ideals is zero and they, hence, form a direct sum. If a proper prime ideal q of ring C contains the sum A + B, then it belongs to the closure U and the closure u' simultaneously, which is impossible. It implies A + B = c. In particular, the unit of the ring C is presented as a sum 1 = e1 + e, where
e1 E A, e E B.
Since the sum of the ideals
direct, then el' e are idempotents, and, hence,
B
= e C, A = (1 -
A+ B
e) c.
is
Since
is closed, then p E U iff p:::> A, i.e., p 31 - e, which is equivalent to the condition e Ii!: p. Thus, U = u( e). The lemma is proved. U
1.9.10. The latter lemma shows a set of central idempotents to be in a one-to-one correspondence with a set of closed domains of x. One can easily prove that this correspondence preserves the lattice operations and order relation: U(e 1e2)
= U(e l ) 1""1
U(e2); U(e1 + e2 - e 1e 2 )
U(e 1) U U(e2), u(e 1) ~ u(e2)
<=> e1:S;
e2·
= (17)
This peculiarity allows one in a number of cases to identify idempotents with closed domains and consider idempotents as the ones consisting of points, which makes many considerations extremely vivid. The correspondence under discussion also preserves the exact upper boundaries U( sup (e a) =Uu( e a ) ,
a
a
(18)
where on the right we have a closure of the union of the domains u( e a).
58
AUfOMORPHISMS AND DERIVATIONS
Let us prove this equality. Let
Ba =
p
n E
U (e a )
p. Then
point
q
belongs to the right part of equality (15) iff q::;) n B a' It should be remarked that Ba = (1 - e a ) C, since p E U(e a) ¢:} 1 - e a E p, and the factor-ring C / (1 - e) C == e a C is semiprime. Let us consider now in C the topology determined by the module structure. The ideals Ba are closed in this topology and, hence, their intersection B is also closed. The idempotents from B form a directed family whose limit e lies in B. As for any element b E B its support e ( b) = bb' lies in B, then e ( b) ~ e, and, hence, B = e C . It should be remarked that e = 1 - sup ea' Indeed, on the one hand, a 1 - sup e a ~ 1 - e a for all a and, hence, 1 - sup e a E B, i.e., 1a sup e a:5; e. On the other hand, e E B and, hence, e E (1 - ea)C, i.e., a ee a = 0 for all a. Therefore, e sup e a = 0 and, hence, e:5; 1 - sup ea' a a Thus, e = 1 - sup e a and, hence, the ideal q belongs to the right part of a equality (15) iff I - sup e a E q. The latter inclusion is equivalent to a
q
E
U( sup e a ) and thus formula (15) is proved.
a
1.9.11. Lemma. The closure of an open set of space X = Spec C is open (and. naturally. closed). The proof results immediately from lemma 1.9.8 and formula (15). 1.9.12. It should be recalled that a topological space is called extremely disconnected if a closure of any domain is open. We thus showed Spec C to be extremely disconnected. 1.9.13.It should be also recalled that the space x is compact and Hausdorff. If X = U U(e a), then any point does not contain at least one of a the idempotents ea' It means that the sum of ideals e a C is not contained in
any
proper
prime
ideal
and,
1 = ea1c1 + -. + e a n c n · Therefore,
hence, any
ideal
2,e aC = a
C.
containing
In all
particular, ea.'
"
also
CHAPTER 1
59
contains the unit. Thus,
.Pt, ~
If now
al
U( e a
i
), and the compactness is established.
are different points, then we can find elements
PI\ P2' a 2 E P2\ PI
E
= Ui
X
(it should be recalled that all prime ideals are
maximal). For the supports we also have inclusions e(a I) P2\ Pl' Let
us
e (a 2 )(1 - e (a I
i.e.,
».
consider
idempotents
e l = e (a 1)(1 - e (a 2
.PI
It
E u(e 2 ).
now
remains
»;
to
remark
r
E2 =
that
intersection of domains u(e I ) and u(e 2) is equal to u(e l e2) = u(O)
sheaf
E
e l e: P2' e 2 e: PI'
These are orthogonal idempotents and
u(e I ),
~ E
the
PI\ P2' e(a 2)
E
the
= 0.
1.9.14. Now we are completely ready to determine the canonical =
r ( R) . u be an
U is open and
Let open set. By fonnula (15), its closure has the form U( e) for some e E E. Let us set T( U)
'f!
l'( U)
1\
'f!
eRE.
(19)
Since the inclusion W S; U implies WS; U, and this inclusion, by formula (14) gives the inequality f $; e for the corresponding idempotents U( f)
from
= W,
U( e) = 1\
e RE
U, then defined is the homomorphism
onto
homomorphism
f
u
1\
which
RE,
will
be
viewed
x
as
--+ the
p w'
r
The validity of axioms (1)-(3) of a presheaf for obvious.
xf
acting
restriction
is absolutely
1.9.15. Theorem. The determined above presheaf l' is a shea/.
u = U ua
Proof. Let
a
1\
8E
1'(u) = eRE. Let also
8
Ua =
U(e a ). The equalities
= O. Since by formula (15) = 8e = 8 . sup e a = 0, which is the required a Let now
a
and
8
8
a
E
r (Ua>
fJ coincide on
a
a
8e a
written as 8
and p~j (8) = 0 for all
U,r
pg(8)=0 will be
we have
a
e
= sup
e
a' then
proof, of the first condition.
be such that for any Ua (')
and for a given
It means
a, fJ that
the restrictions 8
aea= 8a
and
60
AUTOMORPHISMS AND DERIVATIONS
Sa e f3 = S f3 e a
e), such that
( s,
a,
for all
f3. Let us consider the set
se' = s, e' ~ e and
se a = sa e'
of all pairs
M
for all
a. In this set
let us introduce the order relation ( s, e') ~ (sl' e1 ) ¢:> sl e' = s, and show that M transfonns into an inductive partially-ordered subset. If
{(s
r' e r )}
is
a
linearly-ordered
1\
s = lim
completeness of RE the limit
subset,
then,
due
e~
e'l'
to
the
with respect to the system of
sr
idempotents e r + 1 - sup e r is also determined. In this case ( s, e), where
r
e=sup e r , (lim sr) e r
r
is
= lim r
the
upper
boundary
for
{(sr' e r )}
saer= sa lim e r = sa' e, i.e., (s, e)
r
and
se a =
E M.
Let (~,~) be a maximal element from M. Let us show that {J = e. If this is not the case, then {Je a :;:. e a for some e a (it should be recalled that e = sup e a ) and the pair a (~ + sa(e a - (Je a ), {J + e a - (Je a ) strictly exceeds (~, (J) and lies in M. Therefore, Thus,
~
~E T(U) and
(J= e, i.e.,
P~(S')= ~ea= sa{J= sae= sa' a
is the sought element. The theorem is proved. 1\
1.9.16. So, we have achieved the posed goal, i.e., the ring RE is presented as a ring of global section of the sheaf T. This sheaf will henceforth be called a canonical sheaf of the ring R and, if there is a necessity to underline its relation with ring R it will be denoted by T( R). This sheaf obeys another important condition: its any section can be extended up to the global One (the sheaves obeying this condition are called flabby) and, moreover, the homomorphisms of the restrictions are retractions, so that the ring of global sections naturally contains all the rings of local sections 1\
1\
eRE~RE
1.9.17. We have seen the structure of a canonical sheaf to be 1\
completely defined by the structure of the E-module of the ring RE. In an analogous way, to any complete (injective) module M over any self-injective regular ring E there can be associated a certain sheaf M:
-
v
M(U) = M (u) = Me, p u(x) = xe,
61
CHAPTER 1
where fj = U(e). In conclusion of this paragraph we shall make two remarks on stalks of a canonical sheaf.
1.9.1H. Lemma. Section s belongs to the kernel of the natural P p iff the support of the element s belongs to p 0 r ,
homomorphism
which is equivalent, factor-ring /\
/\
s e pRE. The stalk
/\
r
p can be associated with the
/\
RE/RE(lpRE. Proof. Let section s lie in kernel
p p' Then, by definition, there
is a neighborhood W = u( e ) of point p, such that the restriction of s onto its neighborhood is equal to zero, i.e., se = O. It affords e (s) e = 0 and, hence, e (s) S I - e e p, since e ~ p. Therefore, e (s) e p and /\
s=se(s)e pRE. /\
Inversely, if s e pRE, then element s is presented as a sum C; sl + .- + an s n' where a i e p. Allowing for the fact that, together with the two idempotents e 1, e2, the ideal p also contains their upper boundary el v
~
= el +
~
- e1 e2 , we get
sup e ( a) = sup ( a i a') e p, where the elements a'. exist due to the regularity of ring C. ~
Moreover,
s·sup e(a i )= s, which implies 8(S)= e(s)·sup e(a.)e p. Thus, point ~
p belongs to the neighborhood u( I - e (s ) ), the restriction of s on which
equals zero. The lemma is proved. Using the metatheorem (see 1.11) we shall show below that all the stalks of a canonical sheaf are prime rings. It can be proved with no methods of the theory of models used. We recommend the reader to try to do it, and the more efforts are made in order to achieve this purpose, the more pleasure will be derived from reading the paragraph on the metatheorem.
62
AUTOMORPlllSMS AND DERIVATIONS
1.10 Invariant Sheaves In this paragraph we shall adjust the canonical sheaf to studying groups of automorphisms. Let R be a semiprime ring, G be a group of automorphisms G ~ A ( R) . Choosing an arbitrary closed domain u of the spectrum we see that the group G cannot be a group of automorphisms for a ring of sections if U g CL U for a certain g E G and, hence, the canonical sheaf stops being a sheaf if the ring is considered together with the action of the group G. This obstacle will be eliminated by going over to another similar sheaf, which will be called G-invariant or simply invariant provided the group G is fixed. It should be remarked that the group G acts on the space X = Spec C in such a way that the corresponding transformations are homeomorphisms of the topological space C. 1.10.1. Definition. The orbit p of a point all its images under the action of the group G. -
p
E
X
is a set of
g
p=(p , gE G}.
One can easily see that any two orbits either coincide or do not intersect, i.e., the whole space X is presented as a disjunctive union of orbits. Let us denote by X / G a set of all orbits and consider the mapping n: X ~ X / G, which puts the point p into correspondence to its orbit p. 1.10.2. Definition. A space of orbits X / G is a set of orbits having the weakest topology, for which the mapping rc is continuous and open (Le., X / G is a factor-space of X in the topological sense). 1.10.3.Lemma. A set is open in X / G iff it is an image of a domain under the mapping rc . Proof. The images of domains must be open since rc is an open mapping. These images define a topology and it suffices to remark that rc is continuous in this topology; if u is a domain from X, then rc- 1 (rcC U» = u
gE G
u g is also a domain. The lemma is proved.
63
CHAPTER 1
1.10.4. Detinition. A G-invariant sheaf is a sheaf over the space of orbits X / G, the ring of sections of which over a domain W is equal to T(1r- 1(w», while the homomorphisms
Since
1r- l(W)
'/C- I
(v)
p~ are equal to P '/C-I(W)"
is an invariant set of points, and the automorphisms
from G are the homomorphisms of the space x, then the closure 1r- l(W) is an invariant domain. Consequently, the idempotent e which corresponds to this domain is fixed relative the action of the group G. We see the structure of an invariant sheaf to be determined by that of the module on the ring A
RE over the ring
E G of invariant idempotents.
1.10.5. Lemma. The ring EGis self-injective. Proof. As EGis an intersection of kernels of quite continuous mappings x -+ x - x g, g e G then this set is closed in the topology determined by the idempotents from E, and, moreover, it is closed in the topology determined by its own idempotents. By lemma 1.6.16 and lemma 1.6.20, EGis a self-injective Boolean ring. The lemma is proved. According to remark 1.9.17, one can define a sheaf over the space Spec E G, the section rings and restriction homomorphisms for which coincide with those of an invariant sheaf. In essence, they are the same sheaves, though the spaces x / G and Spec E G may be not homeomorphic. The natural mapping p-+ p G is open and continuous but it can prove to be no one-to-one correspondence. 1.10.6. exactly those reasonable to canonical one.
Notations. Since rings of section of an invariant sheaf are of a canonical sheaf over invariant domains, it would be denote an invariant sheaf with the same letter, r, as a It does not result in any ambiguity. since the stalk of an
invariant sheaf over the orbit P will be designated by of a canonical sheaf at the point p by 1.10.7. homomorphism
r
r- p'
while the stalk
p•
Lemma. A section s belongs to the kernel of a natural Pp iff the support of the element s belongs to the ideal
P G C or, which is equivalent, identified with a factor-ring A " " RE / pG RE n RE •
" . The sepG RE
stalk
can be
64
AUTOMORPJDSMS AND DERIVATIONS
Proof. Let section 8 lie in the kernel pp. Then, by definition, there can be found a neighborhood w of the orbit p, such that the restriction of 8 on n- I(W) equals zero, i.e., 8e = 0, where
e (8) e = 0 and, hence,
It affords hand, n1- e
is the neighborhood of the point
1( w)
is
a(8) S 1 - e e
a
fixed
idempotent.
U(e) =n- I(W).
pG, since on the one
p, while on the other hand,
Consequently,
e (8) e p G E
and
1\
8= 8e(8)e pG RE . 1\
Inversely, if 8 e p G RE, then the element 8 can be presented as a sum
~ 81 + .• + an 8Il, where
supports
ei
of an element
ai e p G ai
It should be remarked that the
also belong to
p G:
if
a g= a,
then
n e (a)g = e (a g) = e (a) and, besides, e (a) = aa', where the element a' is determined by the condition of regularity of the ring c. G
Then, together with its two idempotents, e l , e 2 , the set P also = el + el and, therefore, contains its exact upper boundary, e l v
ez
ez - ez,
G
e=sup eie p . Besides, 8 = 8 . sup e i' i.e., the restriction of
8 on the invariant
neighborhood U(l - e) equals zero. This neighborhood contains the point p, as e (8) S sup e i e P. Hence, the restriction of the section 8 on the neighborhood W = 1l( U( 1 - e» of the orbit p equals zero. The lemma is proved.
1.10.8. Lemma. The space x / G is extremely disconnected. Proof. Let W be a domain of the space of an orbit. In this case U = n- I( w) is an invariant domain of the spectrum. Since the elements of G act on X as homeomorphisms, the closure U will be an invariant domain, i.e., 1l( u) 11 n(x\ U) = 0 and, hence, n( U) is a closed domain of the space of orbits. It is evident that the domain contains a closure W. Moreover, since n is continuous, then n- I(W) is a closed set containing U = n- I(W) and, hence, U as well, i.e., W= n( U) is a closed domain, which is the required proof.
CHAPTER 1
1.11 The
65
Metatheorem
Our task now is to describe a possibly widest class of properties, theorems, statements, which can be transferred from prime rings onto semlpnme ones using either canonical or invariant sheaves. Let us limit ourselves by the statements, which can be presented in the elementary language, i.e., those pertaining to elements. If a statement on n elements has already been presented and we know a way of elucidating whether this statement is true for a given n -tuple, then this situation is equivalent to setting an n-ary predicate.
1.11.1. It should be recalled that an n-ary predicate on a set A is a mapping P of the Cartesian degree An onto a two-element set (T, F) ( T H 'truth', F H 'falsity') and, an n-ary operation is a mapping f from An to A. Predicates of the same name (i.e., those with one name, P, and the same arity, n) or operations of the same name can be defined on different sets. In this case we speak about the values of the same predicate, P (operation f) on different sets. A signature is a set ,Q of the names of predicates and operations put in correspondence to arities. An algebraic system of the signature ,Q is a set with the given values of the predicates and operations from ,Q of the corresponding arity. Sometimes both zero-ary operations and zero-ary. predicates are considered. A zero-ary operation on set A is a fixed element of this set, while a zero-ary predicate is either truth or falsity. The predicates and operations from ,Q defined on the algebraic system of this signature are called principal or basic. With their help one can construct new operations, the so-called termal functions, and new predicates, i.e., formula ones, termal functions and terms being defined through induction by construction. To begin with, projections of an arbitrary arity (a1"-' aJ -t ai which have standard designations xi' Yi' Z i~- pertain to termal functions. Besides, if F is an n-ary basic operation and 11'1 ,..., II' n are termal functions of the same arity m, then
F( 11'1 ,..., II' J is an m-ary termal function, which
has a standard designation, F( 1P1'-" IP J, where IPI ,..., IP n are standard designations for 11'1 ,..., II' n' A standard designation of a termal function is called a term. It is common belief that there are no other termal functions. For instance, if ,Q = ( + , - , .) is a ring signature, then terms are
66
AUTOMORPmSMS AND DERIVATIONS
the arbitrary non-associative polynomials of variables (ap _' a rJ --+ f( a p _, a rJ,
functions are the mappings
is obtained from the polynomial f
f(a l ,..., arJ
xi'
while
termal
where an element
through the substitution
xi= aiE R.
Generalized polynomials can be also viewed as terms. For this purpose it is necessary to extend the signature, putting in correspondence to every coefficient r ERa zero-ary operation a I ' whose value on R is set equal to r. The simplest formula predicates are those set by atomic formulas
where P, F are basic predicates. The formula predicate is given by a formula of the elementary language, i.e., it is 'obtained' from simplest formula predicates by employing logical connectives, &, v, -, , --+ and by quanti-fication by 3, 'r;;f of subject variables x i (which are, in our case, identical with the names of certain termal functions). 1.11.2. Let formulas
be terms of the arity
f l ,..., f n
1\
ro,
P
E D. Then the
1\
frJ
P(f l ,· .. ,
(20)
give certain predicates: _., fn
_., anJ
= T,
1\
1\
and, analogously, (fJ = f 2 ) (a p
_' anJ = T,
iff
fl(a l ,
= f 2(al ,.-, arrJ.
By induction through the construction of terms one can easily prove that these predicates can be obtained from simplest formula predicates only through quantification by 3 and the use of the connective &. For instance, if fP ~, f2 = F2 ( VIi ,..., "'2)' then the formula to the following formula fl
= F1( fPI ,_., 3
Yr- Y k + 1
3 zi _.
F J( Yl'-" Y1+ I) &
= F2( ZI"'"
= fpJ&_'&
Z1 + 1( Y k + 1 = Z1
+J=
z}& ZJ = Q!I&_& Zl = Q!l& YJ = Y k = fpk)'
1\
1\
fl = f2 is equivalent
CHAPTER 1
67
The subformulas Z i = Qli and Yi = 'Pi incorporated in the above formula, set the formula predicates by the inductive supposition. 1.11.3. Let us describe the most important for us class, that of Hom formulas. The simplest Hom formulas are the following ones: ~& ~& ... & Ap ~ A p+ 1 ;
Ai;
..., ~ v ..., ~ v ... v ..., A p'
where A i are formulas of type (17). An arbitrary Hom formula is a quantorization of the conjunction of simplest Hom formulas. The predicate set by a Hom formula is called a Horn predicate. 1.11.4. Definition. A flabby sheaf 9t over an extremely disconnected space x is called correct if 9t( u) = 9t(U) for any domain U,
g
i.e., the mappings p are isomorphisms. It should be recalled that a sheaf is called flabby if for any domain U
c v the mappings p~ (or, equivalently, p~) are epimorphisms.
According to definitions 1.9.14 and 1.10.4, canonical and invariant sheaves are correct (see also 1.9.1 and 1.10.8). 1.11.5. Let us now fix a correct sheaf 9t of algebraic systems of a signature n (the reader can view 9t as a sheaf of rings in an extended signature).
1.11.6. Definition. An n -ary predicate p( Xl ,..., X rJ will be called strictly sheaf if its values are given on all the systems (rings) of sections and stalks of the sheaf 9t, in which case 9t remains to be a sheaf in the category of algebraic systems with the additional predicate P. Analogously, an n-ary operation f will be referred to as strictly sheaf, if the (n + 1 )-ary predicate p( Xl"'.' X n+ 1) given by the formula .t( Xl ,...,
X
rJ = X n + 1
is strictly sheaf.
These conditions mean that both restrlCtlons p ~ and natural homomorphisms Pt remain homomorphisms in the category of algebraic systems obtained by adding the predicate symbol P (or the functional one, f) into the signature n, in which case Pt remains a direct limit of P ~, t
E U,
and for any covering
P Uua remains approximating.
U I:: U U a
the system of homomorphisms
68
AUTOMORPHISMS AND DERIVATIONS
1.11.7. Let us write the definition of a strictly sheaf predicate in more detail.
v
V
ro
(a) If P(Sl"'" srJ= T for si E ;1\(V), then P(slP u~-' snP u )= T for all U (b)
S;;;;
v. P(s!'_.,
s rJ =
T
for
of a point t and preimages S i (c)
If
U=U u a
P(slP a ,-" snPa)= T for all
Si
Pa=
W
such that P( S 1,..., S rJ = T.
E ~ (W),
and
iff there exists a neighborhood
E ~t
in
sl'-" SnE ~(u),
P~
a
, then
which
case
P(sl'-" srJ= T.
The importance of strictly sheaf predicates and operations is governed by the fact that they can be included into a signature, in which case all the theorems proved for sheaves (the metatheorem given below, in particular) remain valid, thus expanding significantly the possibilities of their application.
1.11.8. Zero-ary operations. Let S be a zero-ary strictly sheaf operation; s( U), St be its values on ~(u), ~ t' respectively. If = s( X), x then, by conditions (a), (b), we have s( U) = So P u' St = Pt. Inversely, if
sa
sa
S is a global section, then the zero-ary operation
s( U)
=
S P ~,
St
= S Pt
will be strictly sheaf. Therefore, setting a zero-ary strictly sheaf operation is equivalent to singling out a global section.
1.11.9. Unary predicates and operations. A unary predicate is determined by a set of the elements on which it is true. Inversely, with any subset 5 in an algebraic system there can be associated a predicate P s: P J x) = T iff XES Let us now elucidate for what sets
A
5 S;;;; RE
the predicate
PS
will
be strictly sheaf for an invariant sheaf. Condition (a) implies SE G S;;;; S. Condition (b) implies that the value of the predicate P S on a stalk is determined by the set !>peS). Finally, in the terms of idempotents condition (c) looks as follows: if sup (e a ) = e and a ex E 5, where e, e a E EG
e a XES
for all
aE
A,
then
In particular, if 5 is a closed submodule of the E G -module ~ , then the predicate P S will be strictly sheaf provided its values on the stalks
69
CHAPTER 1
are determined by the formula P s(x) = T H x E PJJs). Here only condition (c) can be doubtful, since .Ii is not a directed set in it. Let B be a set of all finite subsets A, ordered by inclusion. Let us set e p = sup (eal + (1 - e). Then xe J~ E S since x sup(e 1, e2l = xe 1 + x~ ae p - xe1 e2 . Hence, xe = l~m xe pES, which is the required proof. Going now over to the module viewpoint, we come to the conclusion that any injective E G -submodule S in ~ determines a strictly sheaf predicate P s' We know that an injective submodule is singled out by a direct addent A
A
Wand the E G -module projection n s: RE ~ S is, therefore, defined. Viewing n S as a unary operation, one can easily see that it is a strictly sheaf operation. RE = S $
Thus, any closed
E G-
submodule in
A
RE
determines a strictly sheaf A
predicate P S and a strictly sheaf projection n s: RE ~ S for an invariant sheaf. Setting G = [1 l, we get an analogous statement for a canonical sheaf.
1.11.10. Composition, support. According to the definition, strictly sheaf are the operations of addition, subtraction and multiplication, Le., the predicates given by the formulas x + y = z, x - y
= z,
xy ,= z, x
= y.
Propositions 1.11.15 and 1.11.16 to be proved below and remark 1.11.2 show that any termal operation, as well as any predicate of type 1\
1\
1\
1\
or P(f 1'-" f rJ, where P is a strictly sheaf predicate, will be strictly sheaf, Le., one can say that the composition of strictly sheaf operations and predicates will be strictly sheaf. Strictly sheaf will also be the function of support, e: s ~ e ( s). fl = f2
A
Incidentally, if the ring RE has no unit, then this function will not be an operation and, strictly speaking, it cannot be included into the signature. Nevertheless, in this case defined is the binary operation (x, y) ~ e (x) y, which will be strictly sheaf. The values of operation e on the stalks must be defined by condition (b). As the element x belongs to the kernel p p iff its support belongs to
70
AUTOMORPmSMS AND DERIVATIONS
p. then for a canonical sheaf we have of the stalk.
e ( x) = 1 for all nonzero elements
1.11.11. Definition. Let us call an n Mary predicate P she af provided its values are given on all the rings of sections and on all the stalks of sheaf 9t and the following conditions are met. (b ') If
P(§I,.. .•
s rJ =
hood w of the point t S
n P ~) = T
snP~)= a
T
Si
E Rt •
and preimages
for any domain
(c ' ) If
for
T
U = U Ua
U
si
included in
and
sl'-' s n E
E
then there exists a neighborp( sIP ~._.
9t (w), such that
W.
9t( u), in which case
for any domains va~ Ua' then
P(s!'_ .• srJ= T.
The following statement shows that it is sheaf predicates that are of primary interest for us.
1.11.12. Proposition. Let P be a sheaf predicate, sections over a domain u. Then, if a set of these points t E is dense in U, then
P(sI P t ••• sn Pt) = T
be for which
sl"" U,
Sn
P(sl'." srJ = T. In particular,
if a certain theorem is set by a zero-ary sheaf predicate and is valid on all (or nearly all) stalks, then it is also valid on the ring of global sections. The proof can be directly obtained by first applying condition (b ') to the stalks on which predicate P is true. followed by employing condition (c ').
1.11.13. Metatheorem. Any Horn predicate is a sheaf one. In particular, if the formulation of a theorem can be presented as a Horn formula, then the truthfulness of this theorem on nearly all the stalks of a correct sheaf implies its truthfulness on a system (ring) of global sections of this sheaf The importance of this theorem is strengthened by the fact that in the class of prime rings any elementary formula is equivalent to a Hom one. Indeed. when constructing Horn formulas, it is forbidden to use only disjunction, but in the class of prime rings f = 0 v g = 0 is equivalent to "if x fxg = O. Or, in more exact terms. it is necessary to reduce the quantorless part to a conjunctive normal form, and then the subformulas of the type f '::f.
0 v f2
'::f.
0 v •. v f
k '::f.
0 v gl
should be replaced with Horn formulas
=0 v
.• v g n = 0
CHAPTER 1
71
The proof of the metatheorem will follow from the lemmas and propositions presented below. 1.11.14. Lemma. Let {Ua ' a E A} be a nonempty family of closed domains of an extremely disconnected space X. Then there exists a set
{U p'
of mutually disjoint closed domains U B
Up
is dense in
Ua
U
A
13
E B},
in which case for every
such that the union
13
E
one can find
B
ex( 13) E A, such that U p t;;;, Ua( fJ) .
Proof. Let us consider a set M of all the sets satisfying the lemma conditions with a possible exception of the density in the union. This set is nonempty, since anyone-element set (ua) belongs to o
M,
where % E
A
Let
us consider M as a partially ordered set by inclusion. In this case M is inductive and, according to the Zorn lemma, there exists a maximal element M
= {Up' 13
E
B}. If now a point t
does not belong to the closure of
U
pe
belongs to the union B
U a'
which does not intersect with the domain W 11 Ua . M
Then
Wa 11 U WP =
0
U
aE A
Ua ' but
then there is a domain W 3 t,
I'
U UfJ"
Let
and the set {Wa ,
U pi
t E Ua
13
E B}
wa =
and
belongs to
and is strictly greater than M. The lemma is proved. 1.11.15.
Proposition. If p. Q are (strictly) sheaf predicates, then
p& Q is a (strictly) sheaf predicate.
Proof. By definition, predicate p& Q is true on the n -tuple sl ,_., sniff both predicates, P and Q, are true on it. Herefrom follows the validity of statements (a) and (b) for predicate p& Q when P and Q are strictly sheaf, as well as the validity of (c') for sheaf predicates P and Q. Let us check (b') for sheaf predicates. Let (P & Q) (81~_, 8 rJ =
T
for
8 i E 9t t. Then we have P(sl'-. 8 rJ = T and C(s-l"'.' 8 rJ = T at the same time. Consequently, there can be found neighborhoods u, U' of point t and
preimages
s'p_' S' nE
--, s ,n
=T
Pu) w
and
9t(U),
s"l'•. ' sIt nE
Q(s" 1 Pu' w"._' co " n P u') w'
9t(U'), such that
=T
P(s'l
. lor a11 domams
&
Wc
p~, U,
72
AUTOMORPHISMS AND DERIVATIONS
VI c u'. Since elements s'i and s" i determine the same element on the stalk, they coincide on a certain neighborhood. Consequently, there is a neighborhood
V!;;; U
(l
u'
of the
point
such
t,
that
~f
S i
s'i P ~=
P~' for all i. Therefore, for any domain W!;;; v both predicates are v v. v v true on SIP w,··, snP w, l.e., p& Q(sIPW~·' snP w)= T. s" i
Let us check the inverse statement (b) for strictly sheaf predicates. If P & Q( s1 ~.,
S
J
= T, then both predicates, P and Q are true on
and, hence, they are true on proposition is proved.
1.11.16.
31,." 3 n' Hence,
Proposition.
Let
P & Q( 31".'
'r;j xp(
x, Xl ~., xJ
T.
The
be a (strictly) sheaf
P( x, Xl ~., xJ
(n + 1 )-ary predicate. Then the n-ary predicates
3J =
s1 ,..., S n'
3 xP( x, Xl ~., xJ
and
are also (strictly) sheaf
Proof. By definition, the predicate 3 xP is true on the n-tuple sl ,•. , sniff there exists an element S from a corresponding ring of sections or a stalk, such that predicate P is true on s, sl'. S n' The validity of statement (a) for predicate 3 xP follows obviously from the same statement for predicate P. The validity of 3 xp( x, .51 '.' .5 rJ is equivalent to the existence of an element 3 E 9t t' such that P( s, .51 ~.' .5J = T. Since predicate P is strictly sheaf, there exists a neighborhood W 3 t and preimages sl ,•. S n, such that
w
w
P(spv'.' snPv)= T
3 xP( x,
sl'.
Let Sl'·' sn
E
S
rJ
us,
for all domains
v
~ W,
i.e., it implies the validity of
on 9\(v). finally,
9t(u),
check
in which case
domains va C Ua' Then for any ro
;1\( Ua>, such that
ua
pes a P Va '
By lemma 1.11.14, U f3' in which case
U
condition
(c').
Let
U
c
3 xp( x, sl p~ ,•. , snP~) a
a
U
aE A = T
Ua
for
a there can be found a section
ua
SIP va'.)
=
and any
sa
E
T.
contains a dense disjunctive union of domains
U f3!;;; Ua( {3) for suitable
a( f3)
E A.
This inclusion and
73
CHAPTER 1
the fact that the predicate P is sheaf show that P( 8 any domains
V 13 C
U 13'
Since the domains
definition of a sheaf, there is an element U
such that 8p V{3 =
Ua ({3)
8
a(
13) P v {3
U 13
8 E
9t(
a(
Ua ({3)
13) P v
~") = T
for
{3
are disjoint, then by the
Ii u13) = 9t(u u 13) =
9t(U),
and, therefore,
Condition (c') applied to predicate P shows the latter to be true on the (n + l)"tuple 8, 81~"' 8 n , i.e., :3 xP(x, 81""' 8n>= T, which is the required proof. Let us consider a predicate V xP. By definition, It IS true on the ntuple 8 1 >"., 8 n iff for any 8 from the corresponding ring of sections or a stalk, the predicate P is true on 8, 8 1 ~.' 8 rr Since the mapping P ~ , because the sheaf is flabby, are epimorphisms, then the validity of statement (a) for predicate V xP is obvious when P is strictly sheaf. Statements (c) and (c') for V xP follows from the same statement for P. Let us check condition (b'). Let us assume V xP (x, .51'" . .5n> = T on stalk 9t( t). Let us fix some global sections 8 1 ,." 8 n> such that 8 i Pt = .5 i . Then for any global section 8 the predicate P is true on .5 = 8Pt, .51 ~.' .5 n> i.e., there exists a neighborhood Vs of the point t, such that for any domain V c- V s we have P ( 8p
T
v,
T
T)
8 1 P V ""' 8 n P v =T
(21)
According to condition (c') for the predicate P, one can find the biggest open set Vs with such a property. Since, by the definition of a correct sheaf, a ring of sections over the domain coincides with that over its closure, then V s is a closed domain. Let ws be a complement of V s in the space X. The domain Ws satisfies the following property. If w is a non empty subset of ws then it contains a nonempty open
74
AUTOMORPlllSMS AND DERIYATIONS
, w,
subset
such that p(sp
x
', •. , snP
w
x
(22)
,)=F.
w
Indeed, in the opposite case the predicate P, by property (c'), will be true x &lor any d · V!:; W U Vs. on sPvx SnPv omam P .,
Let us show that the closure of a union of Ws' where S runs through all the global sections, does not contain a point t. Let, on the contrary, t E U ws· By lemma 1.11.14, the domain u Ws contains a dense disjunctive union of the domains W/3' f3 f3 and suitable G E 9t( u W/3) s( f3) P ~
fJ
B, such that W/3!:; Ws( /3) for any
E
s( f3). Since the union is disjunctive, there exists a section
= 9t(u W/3} ,
at any
such that its restriction on
f3. As the sheaf is flabby, we can assume G to be a
global section with the same properties. Moreover, the intersection va nu t
W/3 coincides with
Ws
is a neighborhood of the point
and, hence, it has a nonempty intersection W with one of the sets Wp. It
results in a contradiction, since, on the one hand, condi-tion (19), combined
w
with the equalities Gf3 ~= Gp ~ pw/ = s( f3) P ~ implies the existence of a fJ
nonempty domain
V!:: W,
a x p( S ( /J) P V
'
such that x
x
sIP v ' •. , Sn P v) = F V!:; W/3!:; Ws( /3)'
but, on the other hand, the inclusion
combined with
condition (18) implies that the same predicate is true. Thus, t does not belong to the closure u WS' and, hence, the complement U of its closure is a neighborhood of t, contained in the intersection of all Vs. Therefore, for any S and any domain V c U the predicate
P
is true on
x
x
sPvp.,snPv, i.e.,
by
definition,
'VxP(x,
SIP ~, •., SnP;) = T, which is the required proof.
Finally, let us check the remaining part of statement (b) for a strictly sheaf predicate. Let there exist a neighborhood W of a point t and elements s1'•. '
S
n E 9t( w), such that 'V xP( x, sp.' s n> = T. Then for any
S E
9t( W)
CHAPTER 1
75
the predicate P is true on s. Spot Sn. i.e .• by condition (b) this predicate is true on 8. 81~_. 811' As Pt is an epimorphism, then the predicate 'V xP is true on the stalk 9t t. The proposition is completely proved. 1.11.17. Proposition. If P is a strictly sheaf n-ary predicate and Q is a sheaf one, then P -+ Q is a sheaf predicate. Proof. By definition, the predicate P -+ Q is true on sl'-" S n' if the fact that the predicate P is true implies that the predicate Q is also true. In other words, this predicate is true in two cases: either if P is false, or if Q is true. So, let P -+ c;(sl"" 8 J = '1' for 8 i E 9t t . If P(sl"" 8 J = T, then the predicate Q on 81,_" 8 n should also be true. In this case, since Q is a sheaf predicate, one can find a neighborhood wand preimages S i E
9t(W), such that
Q(sI P ~,_" snP~) = T for all domains v!: wand,
moreover,
which is the required proof. When neighborhood
= F, then, since P is strictly sheaf, for any and any preimages sl ,_., S n we have
P(sl"" 8 J
w3
t
P(SI ,_. ,
sJ =F.
According to properties (a) and (b) of a strictly sheaf predicate, one can find, by fixing global preimages sl ,_., S n' the biggest domain V on which P is true. This domain is closed and contains no point t, and, hence, its complement w is a neighborhood of the point t. Since V is maximal. the predicate P is false on any domain U!:: W. and, hence.
which is the required proof. Let us check up statement (c'). Let P
-+
Q be true on
SIP
U
= U ua
~a ~- for any domains va C
Let us assume that (p -+ Q)(s 1,..) =: F. Then
and let the predicate
Ua ' where
S i E 9t(U).
=T
c;(s 1'-.)
p(s 1'-.)
and
=
76
F.
AlTrOMORPlllSMS AND DERIVATIONS
As the predicate P is strictly sheaf, we have
hence, CX:8 IP ~ ,..) = T. Since Q is a sheaf predicate, then a which is a contradiction. The proposition is proved.
CX:8 1'•• )
= T,
1.11.1S. Proposition. Let P be a sheaf, and Q be a strictly sheaf predicate, and let us assume that the predicate P -+ Q is true on all systems (rings) of sections. Then this predicate is true on all the stalks as well. Proof. If the predicate P is false on the stalk 9t t' then the implication P -+ Q is true. Let the predicate P be true on the stalk 9t t. As P is sheaf, we can find a neighborhood W of the point t, where P is true. By the condition, the predicate Q must be also true on w. As Q is a strictly sheaf predicate, it is also true in the stalk over t. The proposition is proved. 1.11.19. We can now easily complete the proof of the metatheorem. It should be remarked that the negation ..., P of a strictly sheaf predicate P is equivalent to the implication P -+ x:t:. x and, hence, ..., P is a sheaf predicate. Propositions 1.11.16 and 1.11.17 prove the predicates given by the simplest Horn formulas to be sheaf. The same propositions imply that conjunctions and quantorizations do not violate sheafness. 1.11.20. Corollary. Let PI' P r ., P n,.. be a sequence of m-ary Horn predicates, such that the implications P n -+ P n+ 1 are true on certain gLobal sections 8 1", 8 m of an invariant sheaf. Then, if for any point p one of predicates Pi is true on 51'." 5 m, then one of predicates Pi is true on 81'." 8 m" Proof. By the metatheorem, each point p has a neighborhood Wp Wp, such that a predicate P iCP) is true on 8 1 P u ,.. for any domain U C
Wp. The domains Wp cover the whole compact space of orbits.
Consequently, it is possible to find a finite number of orbits such that
G = Wp u .. u Wp. The predicate
X /
1
n
A
fulfilled on RE, where
8
=
max (i(p »). k
Cp 1,.:-p n ),
psis now proved to be
77
CHAPTER 1
1.11.21. Setting G = {l } , we come to the conclusion that an analogous statement is also true for a canonical sheaf.
1.12. Stalks of Canonical and Invariant Sheaves 1.12.1. Lemma. All stalks of a canonical sheaf are prime rings. Proof. Let us fix a point p of the spectrum and assume Then
81 Tp82 =0.
the
predicate
'V x 81 x 82 = 0 is true on the stalk
r
P(sl' 82)
defined by the formula
1'. According to proposition 1.11.16,
it is a strictly sheaf predicate, i.e., there is a neighborhood u( e), e preimages
sl' s2
RS I Rs 2 Re=O,
A
e eRE,
i.e.,
such
that
A
sl (e RE) s2
=0
Q;
and,
p, and
hence,
e=Oe p. Since p is a simple ideal
8(sl)· 8(s2)··
and e ~ p, then one of the supports, e(sl) or e(s2) lies in p, i.e., by lemma 1.9.18, we have 81 = 0 or B2 = 0, which is the required proof. rings.
1.12.2. Lemma. All stalks of an invariant sheaf are semiprime Proof. Let us fix an orbit
p
and assume
predicate P(s) defined by the formula 'V x 8X8
=0
8 r- p8
= O.
Then the
is true on the stalk
r- p.
By proposition 1.11.6, this is a strictly sheaf predicate, i.e., there is a neighborhood wand a preimage s e
r
(1. - 1(w», such that
A
s( eRE) s =
0,
where the fixed idempotent e is determined by the domain n- I(W). Therefore,
sR eRs
= 0,
i.e.,
e(s)e
= 0,
which affords
8(S) S; 1 -
eel' G, since
the point p the section s belongs to the kernel Pp which is the required proof.
belongs to the neighborhood U(e) =n- l (w). By lemma 1.10.7,
Proposition. Let p be an 1.12.3. spectrum. Then the following inclusions are valid:
arbitrary
point
of the
This proposition will result from an analogous statement on an inva-
78
AUTOMORPmSMS AND DERIVAllONS
riant sheaf, since for a trivial canonical one.
group an invariant sheaf coincides with a Let
1.12.4. Proposition. following inclusions occur:
p be an arbitrary orbit.
Then
the
T,,( R) ~ T ,,(Q) ~ T,,( R,) ~ (T ,,(R» J' ' r,,(Q)!;; Q(T
,,(R».
1.12.5. Lemma. Let I be an arbitrary orbit. Then
be an essential ideal of the ring R,
PjI..
p
1\
IE) is an essential ideal of the stalk
rp. Proof. Since
1\
IR is an ideal of the ring
1\
RE, then it is sufficient to
1\
show that the annihilator of t>-;;( IE) in the stalk is equal to zero. If p
1\
t>-;;( IE ). p
s =0,
'V x(
where
P"
IE
P
then on the stalk the following predicate is true A
IE
(x) ~ xs = 0) ,
is the predicate determined by the set
1\
IE (see
1.11. 9) .
According to the metatheorem, this is a sheaf predicate and, in particular, it
is true on a neighborhood w of the orbit p. Let TC-1(W)= U(e), where e is a fixed idempotent. In this case the fact that the predicate is true
implies that for a preimage s of the element s the equality e IE s = 0 holds, which affords es = 0, i.e., the restriction of the element s on the neighborhood w equals zero and, hence, s = O. The lemma is proved. Proof of proposition 1.12.4. By lemma 1.10.7, the kernel of the homomorphism Pp consists of the elements, the supports of which lie in 1\
p G c. This condition does not depend on the fact in which of the rings 1\
the element itself lies, and, hence, for the sheaf T( R) the homomorphism Pp is a restriction of the same homomorphism for the sheaf RE , Q, R F
T (Q) which is, in its turn, a restriction of Pp for the sheaf T ( R ,). This is a proof of the validity of the first two inclusions of the upper chain.
79
CHAPTER 1
Let us now check the third inclusion of the upper chain. It would be natural to put into correspondence to an element of the stalk determined by the global section v
v:
determined by the homomorphism formula v(x)
= xv + ker
E
T( R,)'
R F , an element of the ring (T,J R» ,
PJI..
1\
IE ) ~ T-j/.R), which acts by the
Pp (here I is an ideal from
F ( R), such that
Iv ~ R).
It should be remarked that
1\
is an essential ideal of the ring
~(IE)
p
T-,JR), and it is necessary to check the correctness of the definition of the
action of v and calculate the kernel of the mapping v + ker Pp Let us check the correctness. If a and e (av)
~
e (a) e (v)
~
e
E
E
G
v.
e( a) ~
ker Pp' then
p , and, hence, (ker p-p) v
~
~
eE p G
ker P". which
proves the correctness. 1\
Let us, finally, assume that v = O. Then
IE v
s:: ker Pp'
Le., on
the stalk T p( R,) the predicate 'V x(p
is true, where
I\(x)~
IE
v = Pp (v).
neighborhood wof the orbit
xv=O),
By the metatheorem, this predicate is true on a
p.
Let -n- 1(w)= U(e) and e e: p
G.
Then we
have e IE = 0, Le., ev = 0 and, hence, the restriction of v on the domain w equals zero, Le., v E ker Pp, which is the required proof. 1\
The above mentioned embedding the lower chain as well: if v
vE
Q(T-;t
R».
E Q
v~ v
and
determines the inclusion in
vI!: R,
then
1\
v IE
1.12.6. Lemma. A generalized centroid of the stalk
s::
1\
RE,
Le.,
T ,JR) of a
canonical sheaf equals p p( C) = C / p, where C is a generalized centroid of the ring R. Proof. Let Il be a nonzero element of a generalized centroid of a
80
AUTOMORPIDSMS AND DERIVATIONS
stalk. By the definition of a Martindale ring of quotients, one can find nonzero elements of the stalk, a, b, such that a = Ab. For these two elements let us consider the predicate
According to proposition 1.11.6, this is a strictly sheaf predicate and, hence, there can be found preimages a, b of the elements a, b, respectively, such 1\
that the identity axb = bxa is fulfilled on ring RE. Lemma 1.5.8, applied to the elements e (b)a and e (a) b shows them to be proportional: e (b) a = cb for an C E C. As b is a nonzero element, then the image of e (b) equals the unit, i.e., a = cb, where c = P pi . In particular, the difference (It. - c) annihilates the two-sided ideal of the stalk, generated by the element b. For the prime ring T sl-R) it implies It. = c. The lemma is proved. 1.12.7. Lemma and notation. The group G is induced on the stalk To p of an invariant sheaf. Let us designate the induced group by
Gp. Then
Gpk A(T~.
Proof. Since the kernel
Pp
Q
~ To", Q) is equal to
P G Q, then it
is G-invariant and, hence, G is induced on the stalk T p(Q) . If and
&
I, J
are essential ideals of the ring R, such that
J k
g
E
G
I g ~ R, then
k ( f'E )g k ~ , since the automorphisms are continuous in the topology determined by the idempotents of C and a set of all idempotents E is invariant. Applying p p to the ideals of the latter chain and using lemma 1.12.5, we come to the conclusion that the induced automorphism g lies in
Ai.. To p(R». The lemma is proved. We can now consider an invariant sheaf of rings T(Q) together with the automorphisms from G, assuming them to be unary operations. 1.12.8. Proposition. The invariant sheaf T(Q) with the unary operations from G remains to be a correct sheaf, i.e., the automorphisms from G prove to be strictly sheaf operations. Proof. Let us check conditions (a), (b) and (c) on a strictly sheaf predicate.
81
CHAPTER 1
(a) If (fs I )
g
SI' s2 E
= fS2
for any fixed idempotent f
(b) Let 3
f= 32
preimages the inclusion g
detail, e (s 1 - s2) from p. Let f
e
U(f)
restricting
s.z) .
and 31,
sf- S2
= e l c 1 + .- +
=1 -
f
p, i.e.,
g (s I -
eQ for a fixed idempotent e, then
Sz E r- p.
e.
Then, by the definition, for some
ker p P' i.e., e (s
f- S2)
E
p G c. In more
e nCn for some fixed idempotents
sup( e 1'-.' e ~.
Then f
is
a
fixed
idempotent,
s.z).
f =
this 0,
neighborhood, i.e. ,
(slf)
we g
= s2 f
g
get
and,
and
p.
(s 1-
s.z) f
By
=
hence,
s.z f
are the sought neighbor-hoods and preimages. The inverse statement results from the definition of the
U( f), sl f,
e l ,.·, en
is an invariant neighborhood of the point on
g e( s I -
E
~
sf
= s2 in a ring of sections, then 3 a stalk: if (c) The checking is obvious. The proposition is proved.
f= 32
G
action on
on a stalk.
1.13 Martindale's Theorem In this paragraph we shall study the construction of a central closure of a prime ring which satisfies a non-trivial generalized identity. A generalized identity is a polynomial of noncommutative variables with the coefficients from R (or from R F ), turning to zero when substituted instead of the variable elements of the ring R. Trivial are the identities incorporated into the definition of a ring (distributivity, associativity, etc.), identities of type xc = CX, where C is an element of a generalized centroid, and all their corollaries. Here we have an opportunity to give exact definitions in terms of free products. 1.13.1. It should be recalled that a free product of algebras with a unit, A, B, over a field C is the algebra A * B, containing A and Bas subalgebras and generated by the latter, and such that for any homomorphisms p: A~ D, q: B ~ D there is a homomorphism p * q: A* B ~ D, expanding p and q. Such a definition does not yet guarantee the existence of a free
82
AUTOMORPIDSMS AND DERIVATIONS
product, but it can be, however, easily constructed as a sum of tensor products of spaces:
A
*
= L DI ®
B
n
D2
(23)
® .• ® D n' Die (A, B) , D i:t:. D i + 1 '
the multiplication on which is determined by the natural formula
1\,
(24)
,
Dl ® •. ® D n ® D n + I ® D m, A
,
,
I
where Dn®Dn+l=Dn®Dn+l when Dn:t:.Dn+l' while if
" is a product in Dn. then ®
Dn= D' l' n+
1.13.2. A generalized polynomial over R is an element of the free product R F
*
C ( x) of algebras over C, where
C ( x) is a free algebra
with free generators (x l' ~ , .. , x n~.) = x. By this definition, any generated polynomial f can be presented as a sum of monomials
(25)
and,
in particular, it depends
only on
the finite
number of variables
f= f( Xl'•• ' x~.
Let
at ,•. , a n be some elements from
R F.
Then the value of the
polynomial f( xl'•. ' X n> at Xl = a l ,.. Xn = a n is an element of ring R F obtainea when replacing X i with a i in formula (22). Strictly speaking, the value of f qJ.
is the image of f
C ( x) --+ R F
is
a
at
the
homomorphism,
homomorphism such
that
CfX.
X
1*
qJ,
where
i) = a i'
while
1: R F --+ R F is an identical homomorphism.
1.13.3. Definition. A generalized identity of the ring R is a generalized polynomial which turns to zero at all values of the variables from
CHAPTER I
83
The identity of f
R.
is called non·· trivial
if
f
-::F-
0
as an element of
RF*C(X).
1.13.4. Theorem (Martindale). If in a prime ring R the nontrivial generalized identity holds. then the central closure R C has an idempotent e, such that eRCe is a finite-dimensional sfield over C. Proof. It should be first remarked that a non-trivial multilinear generalized identity holds in ring R, i.e., the identity f, in presentation (22) of which all the monomials depend on the same set of variables, XI"'" X n , each of these variables entering each of the monomials only once. Let us consider the following operator:
- t( XI'·"
xn> -
t( XI'·" Y.·
(26)
xn>
where y is a variable not included into the set {x p.' X n)' This operator is certain to transform any identity into another one. When it transforms f into a trivial identity, f is linear over the variable xi' If a non·trivial identity results, then its degrees over xi and over y are less than those of f over xi' As a result, multiple action of operators .1 tz required non-trivial polylinear identity.
will finally give the
For an arbitrary generalized polynomial f let us denote by v f
a
linear space over C generated by all f values at xi E R. By induction over the degree of f we shall prove that if for a non-trivial multilinear f the dimension of v f is finite, then the conclusion of the theorem is valid. k
Let the power of f equal one. Then
f =
L a i xb i'
and we can
i= I
assume that a l ,•. , a k are linearly independent, while b l ,·., b k are nonzero elements, since in the opposite case one could reduce the number k in the presentation of f without changing .f proper. Corollary 1.7.14 states that r l ,.., r m; t l ,.., t m from R , such that ~ = there exist elements m
L r.a.t J.=0 . I J
J= m
L I r.J f(t J. x),
.
J=
~
i=2,3 •• , k.
at
and, hence, ~ R C b l ~
L. rJ.V J
f'
We
have
In particular,
~ R C b l is
84
AlITOMORPlllSMS AND DERIVATIONS
a finite-dimensional space. Then, there is an element O;:i:.
raa E
R 3 sbl ;:i:.
r, s E R, such that
0, and, hence, (rao)R C(sb l ) is finite-dimensional.
Let us choose nonzero elements a, b E RC in such a way that the space aRCb has a minimal possible dimension 1. As the ring R is prime, we can find an element u, such that bua;:i:. O. In this case the dimension of the space S = (aRCb) u is not greater and, hence, equal to 1, and ;} = aRCbuaRCbu ;:i:. O. Besides, for any nonzero element
we have sRCs !:: S and again, since the dimension is minimal, sRCs = S. It, in particular, implies that ;J-;:i:. 0 (in the opposite case ;} = SRCs 2 RCs = 0) and, hence, ;J- RCs 2 = S. Finally, s· S = ;J- RCs;;;2 s2 RCs 2 = S, Le., S is a finite-dimensional sfield. If e is the unit of the sfield S, then eRCe!:: S and, since the dimensionality is minimal, eRCe = s, which is the required proof. Let now the power of f be greater than one. An element f can be presented as m
I
f =
i=l
a i Xl wi
+
I
i=l
s E S
k
Pi Xl q i
+
I
i=l
vi Xl b i '
where ai' b i E R F ; Wi' Pi' qi' Vi are non-trivial generalized polynomials. As before, one can assume that the elements a l ,_., a n are linearly independent. Repeating the starting considerations for the case n = 1, we can and fl has the form find a polynomial f l , such that dim v f < 00
I
This polynomial is not equal to zero in R F presentation
Pi
and
vi
*
C ( x), since in their
have the first letter on the left other than
Xl
and
ai ;:i:. O. Then, if for any r 2 ,_., r n E R the elements wl( r 2 ,_., r J are linearly expressed through bl'_.' b k' then the dimensionality of V W1 over C is finite and we can use the inductive supposition. If this is not the case, then, by corollary 1.7.14, one can find elements sl'-.'
sm' t l ,_·, t m'
such that
Is. . b ~
~
]
·t ~. = 0 and the polynomial
85
CHAPTER 1
wi = LS i wit i i
is not an identity of R.
Now the polynomial f2
=
m
L
has the fonn
fl(xs)t i
i= 1
*
in which case v f !::LVft i , and ~ is nonzero in RF 2 1 .
Extending these considerations to the letters the first in the presentation of the monomials from element f3' which is nonzero in
RF
*
Xz'
C(
x).
x ro which are one can find an
X 3,...,
f2'
and such that dim V f <
C ( x)
and f3 has the fonn f3
00
3
= La i Xl S i + L
b i -"2 u i + .• + L d i i i i
X
(27)
r hi'
where any of the letters Xl ,..., X r is not the last for any monomial encoun· tered in presentation (24) of the element f3' and the sets of elements { a J ,{ b J ,.., ( d i) are linearly independent. Since n ~ 2 and at least one variable should be the last in the presentation of a monomial, then r < n. According to the inductive supposi. tion, one can assume that for an o is valid. Let us set
~ ,•. , an E R
S'.= S ~.(x.., •..• ~ L.
=
where the element Besides. V f
4
X n- l.an>. u'~..
f4 = La i Xl S 'i + ...
and. finally.
!:; V f ' 3
otherwise we have La i i
Xl S
i =0
find a nonzero element a. such that
f4
the inequality
sl( a 2,•. , an>
*
u ~{Xl' x..j •...• X n- l' an> ••.
+L
di
Xr
h ~ has the power
is nonzero in
RF
*
C ( x).
n - 1.
since
and. applying corollary 1.7.14. we can
i
aXl S =
O. Allowing for the fact that
86
AUTOMORPlllSMS AND DERIVATIONS
aJ *" 0,
we come to a contradiction with the primarity of the ring R. According to the inductive supposition, the theorem is proved. By way of concluding this paragraph, let us prove another three useful statements on prime rings satisfying a generalized identity. sl(a 2 ,".,
1.13.5.
Lemma.
A center of the sfield
is equal
T = eRCe
to
Ceo
Proof. Let t be a central element of T. Then for any x E R we are linearly have f( x) == txe - exet = o. If the elements e and t independent over C in ring Q, then, by corollary 1.7.14, there can be found elements Vi' riE R, such that 0= Lvieri*"Lvitr i = a. Then 0= Lv i fer i x) = axe for all x, which is the required proof.
which
is
impossible.
Therefore,
1.13.6. Lemma. For any linear over C transformation
t
E
Ce,
1: T ~ T
there exist elements ai' b i E T, such that l(x) = La i xb i . Proof. If n is the dimension of the sfield T over the center C e, then the dimension of the space of all linear transformations is n 2 . On the other hand,
n 2 of
the
linear
transformations
1 ij: x
~
a i xa j'
where
a l ,"., a n is a basis of T over the center, are linearly independent. Indeed, if XE T,
then, applying corollary 1.7.14 to ring T, we get
C ij
=
o.
The lemma is
proved.
1.13.7. Corollary. If a ring with no zero divisors obeys nontrivial generalized identity, then its center is nonzero and the central closure is a finite-dimensional sfield. Proof. It should be remarked that if a ring R has no zero divisors, then the two-sided ring of quotients C( R) has them neither. Let, on the contrary, q, v E Q, qv = O. Then for a suitable ideal IE F we have Iq ~ R, vI ~ R. It yields (Iq)· (vI) = 0 and, hence, Iq = 0 or vI = 0, i.e., q = 0 or v = O. Now, according to the Martindale theorem, R C has a primitive idempotent, but a ring with no zero divisors can have only one idempotent other than zero, i.e., the unit. Therefore, RC = 1 . RC . 1 = T is a finite-
87
CHAPTER 1
dimen-sional sfield over C.
.
on
.
Let us consider a linear over C projectIOn 1: T ~ C and assume that l(T) " R = o. Then, by lemma 1.13.6, there are elements ai' b i E T, such that l(x) =
I, a i xb i
find a nonzero element q have
1.14.
x
E
T. For the elements
R, such that
a i q, qb i
E
ai' b i
R, since
one can
T!:: Q. We
therefore, C = l(T) = 1( qTq) = l(qRCq) = (1( T)" R)C = 0, this contradiction proving the corollary.
l(qRq) !:: R
l(qRq)C ~
for all E
Quite
and,
Primitive
Rings
In this paragraph we discuss in detail the structure of rings ansmg in the Martindale theorem. As a quite primitive ring we shall call a prime ring R with a nonzero idempotent e, such that eRe is a sfield. In literature these rings are called primitive with a nonzero one-sided ideal or primitive with a nonzero socle. The idempotent e :t:. 0 of a prime ring R, such that eRe is a sfield, is called primitive. A soc/e of a quite primitive ring R is a two-sided ideal generated by all the primitive idempotems. A sode is designated by H( R) or simply by H. 1.14.1. Lemma. The soc/e H of a quite pnmltlve ring R is the least nonzero ideal of this ring. In particular, a soc/e is generated by any primitive idempotent and is a simple ring. Proof. If I is a nonzero ideal and e is an arbitrary primitive idempotent, then O:t:. eIe!:: I " eR e. As a sfield has no proper ideals, then I " eRe 3 e and, hence, I::l H. If now A is a nonzero ideal of the ring H, then H AH is an ideal of the ring R contained in A. According to the above proved, this ideal contains a socle, Le., H::: A, which is the required proof. It should be recalled that the module M over the ring R is called irreducible if it contains no proper submodules (Le., submodules other than (0) or M). 1.14.2. Lemma. Let e, f be pnmltlve idempotents of a prime ring R. Then the sfields eRe and fRf are isomorphic. The right ideals, fR and eR, as well as the left ideals, Re and Rf, are mutually isomorphic as modules over R and are irreducible.
88
AUfOMORPHISMS AND DERIVATIONS
Proof. Since the ring R is prime, there can be found an element u, such that fue i:- O. By the same reason, the set fueRf forms a nonzero right ideal of the sfield fRf, i.e., there exists an element u', such that fue· eu'f == f.
Squaring both parts of the latter equality, we see that Therefore, ~ is a nonzero idempotent lying in the sfield eRe. Thus,
(~2==
~
== eu' f fue i:- O.
eu' 3 f( fue· eu' f) fue ==
~)
eu'f· fue == e.
It is now absolutely clear that the mappings ere ~ fuereu' f ,
frf ~ eu' frfue
give the sought isomorphism of sfields. Let N be a nonzero right ideal contained in eR. Then eN = Nand, hence, since the ring R is prime, eNe == Ne i:- O. However, eNe is the right ideal of the sfield eRe and, therefore, e E eNe k Nand eR = N, so that eR (and, analogously, Re) is an irreducible module. Let us now consider a mapping q;: eR ~ fR which acts by the rule er ~ fuer, where u is an element determined from the onset. As fue i:- 0, then cP is a nonzero homomorphism of right R-modules. Therefore, the kernel of cP is a submodule of eR other than eR. Due to irreducibility, cP is an embedding. The image of cP is a nonzero submodule in fR, coinciding, due to its irreducibility, with fRo Thus, cp is an isomorphism of the modules. The lemma is proved. 1.14.3. The lemma proved above makes it possible to determine the sfields of a quite primitive ring R as a sfield T, which is isomorphic to eRe for a certain primitive idempotent e. Moreover, the lemma states that the module V = eR is also independent of the choice of a primitive idempotent. Since eR e· eR k eR, then this module can be viewed as a left vector space over the sfield T == eR e. In this case the elements of the ring R turn to transformations of the left vector space v over the sfield T. Indeed, an element vr E V is unambiguously determined for every v E V and r E R, in which case the mapping v ~ vr is a linear transformation of V over T, with which the element r is identified. This
89
CHAPTER 1
presentation is exact, since if Vr = 0 then r = 0, as R is prime. There now naturally arises a ring L of all linear transformations of the space v over the sfield T. This ring has an explicit description with (infinite) matrices over T. 1.14.4. Let I be a set. An I x I -matrix is an element of the Cartesian product T I x I (Le., the mapping ~: I x I --+ T), which will be designated by
I t apll,
where
(a, f3) -th place (Le., t ap =
matrix
I tap I
t
ap is an element of the sfield occupying the ~ a,
f3)). The line numbered lly
is the line (t avo p), f3
E
I
(Le., an element of
Analogously, the column numbered f3v is the column (taP)' a I x
E I
E
T I
of the
(av) x
I
).
(Le., an
(Pv)
element of T ). A matrix is called finite-rowed if its every row contains only a finite number of nonzero components. On a set of finite rowed I x Imatrices ring operations are naturally determined:
In the second formula the sum is determined, since it contains only a finite number of nonzero terms. It is also evident that these operations result again in finite-rowed matrices. 1.14.5. Lemma. The ring lL is isomorphic to the ring of finiteI x I -matrices over the sfield T for a set I. Proof. Let us choose a basis ( val a E I) of the space v over the sfield T. Then for any transformation cp: v --+ v the images cp(v a) can be written as finite linear combinations of basic elements rowed
It is the correspondence ~ --+ The lemma is proved.
lit a~1
that sets the sought isomorphism.
1.14.6. The embedding of a quite pnmItlve ring R into the ring L is, by itself, not enough information. The most essential is the fact that at
90
AUTOMORPHISMS AND DERIVATIONS
such an embedding R proves to be a dense subring in L. A ring s of linear transformations of the space v over the sfield T is called dense (in L), if for any finite-dimensional subspace w ~ v and any linear transformation 1 E IL, there exists an element S E S which coincides with 1 on W. In terms of matrices this is equivalent to the fact that for any finite subset
I
S
a~1
J C
I
and
any
J
x
J -matrix
S, extending 1, Le., such that
E
S
III a~1
there
exists
af3 = 1 af3 at a, f3
E
a
matrix
J.
It should also be added that on the ring L there ia natural topology, called finite topology, the density in which coincides with that given above. Namely, in finite topology the basis of zero neighborhoods are the sets wl.= (l E ILl Wl = 0) where W runs through a set of all finite-dimensional subspaces of v.
1.14.7. Among transformations from L of special importance are those of a finite rank, Le., the ones having a finite-dimensional image. In terms of matrices such transformations are set by the matrices having only a finite number of nonzero columns. 1.14.8. Theorem. A quite pnmltlve ring R is dense in L.. A set of all transformations of a finite rank lying in R coincides with the socle H of a ring R. Proof. In order to establish density, it is enough to show that for any linear independent over T = eR e elements WI ,_., W n E V = eR and any elements v I ,-., v n E V there is an element r E R, such that WI r = v I .-, wnr =
V
n·
Let us carry out induction over n. At n = 1 an element WI is nonzero and, hence, WI R is a nonzero submodule in v, i.e., due to irreducibility, there can be found an element r E R, such that WI r = VI' Let the required statement be proved for a given n. Let us consider a linearly independent set wo' wI .-, Wn and show, first, that there is an element
rO E R, such that Wo ro:f; 0,
WI r 0 = 0,..., wn ro = O.
Let us assume this is not the case. According to the inductive supposition, there are elements r I ,_., r n> such that w·~ r ~. = e, w.~ r ] . = 0 at i:f; j. For an arbitrary
x
E
R we have the following system of equalities:
91
CHAPTER 1
wi(x -
n
n
J=
J=
L r .ew J.x)=(w.L w.r .ew.)x =0, i= 1,2~_, n. 'I J ~ 'I~J J
Therefore, wO( x -
It affords
Wo =
Lr . J
= ( Wo -
. ew . x )
J
J
n
L (wO r J. e)· j= 1
W" J
n
L
'1 J=
Wo r . ew . ) x = O•
but Wo r
contradicts linear independence. Thus, there exists an element
J
j
e
J
E eR
r O'
such that
0,... wnrn= O. Analogously, one can find elements
obey the conditions wi r i n
= 1,
*' 0,
there are elements
wi r j = 0 at S
i
*' j.
i E R, such that
e = T and this equality
Wo rO *' 0, wI ro =
rl' r2~-' rn E R, which
Like it was the case for wi r is i
= vi'
It is
now
n
evident that the element r =
L
i= 0
r is:i
is the sought one.
Let us prove the second part. As Ve = eR e is a one-dimensional subspace, then e is a transformation of a finite rank. The image of the sums of transformations is contained in the sum of images, and, therefore, the set H I of transformations of a finite rank is closed by addition. Right and left multiplications result in no increase in the dimension of the image. Consequently, H' is a nonzero ideal of the ring R and, by lemma 1.14.1, we have H':J B. Inversely, let r be a transformation of a finite rank from R. Since the image of r is finite-dimensional, its kernel K has a finite co-dimension and we can find a finite-dimensional subspace w= TWI + _. + Twn , complementing K to v = eR. In this case the elements WI r ~_, wn r will be linearly independent over
T and, due to density, for every
i, 1 ~ i ~ n
there is an element ~i E R, such that wlr~ i = e, w j r~i = 0 at well as an element VIi with the condition eVli = wi r.
j
*' i,
as
Let us consider an element I,I = r L~ i eVli' Its action on a linear transformation coincides with that of r on both K and all wi' and, hence, r
= r' E
H. The theorem is proved.
92
AUTOMORPIllSMS AND DERIVATIONS
1.15. Rings of Quotients of Quite Primitive Rings Here we shall characterize rings R F and
ex R)
for a quite primitive
ring R, preserving the denotations from the preceding paragraph, fL, H .
1.15.1. Theorem. If R is quite primitive, then
RF coincides with
the ring fL of all linear transformations of the space V = eR over the sfield T = eR e. Proof. Since the ring R has the least nonzero ideal H, then R F = Hom.. RH , R).
If
h E Hom.. R H , R),
then the restriction of h A
0
n
A
determines a linear transformation h: eR -+ eR. If h = 0, then (eR)h = 0 and, hence, (H)h = (H eR)h = H (eR)h = 0, i.e., h = 0, so that eR ~ H
h
-+
A
h
. IS
an embedding of
RF
in fL.
A
Inversely, if h EfLand a E H, then, by theorem 1.14.8, eRa is a finite-dimensional over T subspace in V and, hence, due to density, there is an element a l E R, the action of which on this subspace coincides with A
that of
h.
formula
ah
eRa,
then
Let us determine the action of h on an element a by the
= aa r
This definition is correct. Indeed, if
eR(aa 1 - a~) = 0
and,
hence,
hE Hom( RH , R) and we get an inverse mapping, proved.
al =
~
A
= h
on
Therefore, A
h
-+
h.
The theorem is
1.15.2. Let us now make a small retreat into general topology. Let us consider an arbitrary set x and a set F of its transformations. One can set any topologies on the set x. As the set F has been singled out, of special interest are the topologies for which F proves to consist of continuous transformations. It is evident that if fa' a E A is a class of such topologies (a topology is considered to be given by a set of open sets), then their inter-section (\ fa will also be a topology where all the functions from F
are continuous.
A
1.15.3. Under our conditions it would be natural to consider the weakest topology on V, which converts V into a topological linear space over the sfield T, supplied with a discrete topology and such that the set R consists of continuous transformations. It should be recalled that the linear space V (supplied with a
93
CHAPTER 1
topology) over a topological sfield T is called a topological linear space , provided the linear operations (addition and multiplication, T x V -+ v), as well as the mappings tv -+ t for all v:# 0 are continuous (here t runs through T). 1.15.4.
Theorem. is a weakest topology on v, which turns it into a topological linear space over the sfield T supplied with a discrete topology, such that R consists of continuous transformations. (b) A set of all linear continuous transformations in this topology equals c;( R) . (c) A set of all continuous transformations of a finite rank coincides with the socle of the ring R. Proof. Let t be a certain (possibly, not the weakest) topology obeying condition (a) of the theorem. As, by definition, the mappings Tv -+ T, acting by the formula tv -+ T, are continuous, then the topology induced on anyone-dimensional subspace is discrete. If now r is an arbitrary element from R, e is a primitive idempotent, such that V = eR, T = eR e, then the continuous transformation re transforms V into a one-dimensional subspace. Therefore, the preimage of zero should be both open and closed. Let us consider a topology 'r, the subbase of zero neighborhoods for which consists of the kernels of the mappings v -+ vre, r E R, while the neighborhood of an arbitrary element v is determined as sums v + w, where W is the neighborhood of zero. As the co-dimensionality of the kernel of re equals one, there is an element v E v, such that (a) There
V\ ker re =
U
o~ tE
(tv V
+ ker re).
Hence, the complement to the kernel is open, while the kernel itself is closed in 'r, i.e., the topology 'r is not greater than t. Let us show that the topology 'r obeys all the requirements of the theorem. Since kernels are subspaces, then (V, 'f) is a topological Abelian group and the multiplication T x V -+ V is continuous. Moreover, if 0:# v = er E v, then, since R is prime, there is an element r 1 E R, such that er· r 1 e :# O. In this case the intersection of kernel of r 1e with the subspace Tv equals zero, i.e., the topology induced on Tv is discrete and the mapping tv -+ t is a homomorphism.
94
AlITOMORPmSMS AND DERIVATIONS
Let us check if C( R) consists of continuous transfonnations, assuming C(R)~RF=L
qE C(R). Then q is a linear transfor-
(see 1.15.1). Let
mation and qH ~ R, since H is the least nonzero ideal of R. If neighborhood of zero, then it contains a subset of n
n
u
is a type
ker r. e, r 1 E R. Its complete preimage at the mapping q contains an
i= 1
~
intersection nker . qr.~ e. As
rie
H,
E
then
qr i e
= (qr i e) e ERe
and,
~
therefore, the preimage is again a neighborhood of zero. Inversely, let 1 be a continuous transfonnation. Then for any r E R the preimage of ker re is open. This preimage is equal to the kernel of transformations ire. Consequently, the kernel of ire contains an intersection
w = 0ker
r i
e for suitable
r i E R.
~
One can assume that this intersection is minimal relative to n, i.e., ker
r], e
n
tJ.
ker
i* j v'] r ] . e
such that
r
ie, 1 ~ j
'* 0, v
]
~ n.
. r ~,e = 0
It enables one to find elements v].
at every
i, j,i
'* j.
v,
E
Then the elements
v j ire and v j r j e belong to the sfield eRe and, therefore, there is an
element t].
E R,
such that v ] ,ire = v ] . r ] . et ] . e. Let r'
=
n
L r .et ] .e. . 1 ]
]=
Let us show that ire = r'. Indeed, the action of these elements coincide on all v j and on W. Moreover, for every v E V the images vr i e
and
vir i e
lie
in
the
same
linear
subspace
vr.e= t'.v,r,e for a certain t',E T. Now the difference ~ ~~ ~ ~
lies in the intersection of the kernels of transfonnations V
= W + LTv. . ]
r i
eRe,
i.e.,
v-Lt'v ii i
e. It implies that
and, hence, ire = r'.
]
Hence, ire lH
E
R, i.e.,
lR e I:: R. We have
= lR eR = (lR e) R ~
R,
and, hence, 1 E C( R) . Statement (c) results immediately from theorem 1.14.8. The theorem is thus completely proved.
CHAPTER 1
REFERENCES
S.Amitsur [4,6] V,A.Andrunakievich, Yu.M.Ryabukhin [8] K.I.Beidar [13,14] K.I.Beidar, A.V.Mikhalev [15,16] G.Bergman, I.Isaacs [21) S.Burris, H.Verner [25) N.Jacobson [65] V.P.Elizarov [42] J.Lambek [91] G.Levitsky [98) Lyubetsky, Gordon [100] A.I.Mal'tsev [102,103] W.Martindale [105) V.K.Kharchenko [70,71,74,80) G.Higman [59)
95
CHAPTER 2 ON ALGEBRAIC INDEPENDENCE OF AUTOMORPHISMS AND DERIVATIONS 2.0
Trivial
Algebraic
Dependences
Automorphisms of fields are well known to be algebraically independent, this fact playing an important role in the classical Galois theory. Therefore, the problem of algebraic dependence of automorphisms is of importance in a noncommutative situation as well. A simple example that follows shows that from a common point of view automorphisms can be algebraically (and even linearly) dependent. Example. Let R = S be an algebra of 2 x 2-matrices over a field of complex numbers and let diag(i, 1). Then
qJ
be a conjunction by a diagonal matrix
f/>=
(1)
for all matrices x , though the automorphisms the group Aut( R) .
q)3,
qJ2, qJ • 1
are different in
It is also evident that if g is an inner automorphism of an infinite order of any algebra R, while the element corresponding to it is algebraic, then the g powers will be linearly dependent. We shall see below that algebraic dependences among the elements corresponding to inner automorphisms (derivations) prove to be the only reason which can result in an algebraic dependence of automorphisms (derivations). Under noncom mutative conditions it would be natural to consider the algebraic dependences set by generalized polynomials. If gl'•. ' g n is a set of automorphisms or derivations, then their dependence in such a sense means that for a nontrivial generalized polynomial W( z) the equality w( x gi ) = 0 is valid for all x from the ring. If we now consider the set gl'.·' g n as a set of unary operations, then we come to the necessity to study identities with automorphisms (and differential identities as well). We can now write some identities with automorphisms which evidently
96 V. K. Kharchenko, Automorphisms and Derivations of Associative Rings © Kluwer Academic Publishers 1991
97
CHAPTER 2
hold in any ring: 1. (xyl = x g yg, (x + y)g = x g + yg; 2. (x g) gl = x( ggl) ;
3.
xg -
(fJ-I x(fJ=
0,
where g is an inner automorphism, and (fJ is an element corresponding to it. These identities and their corollaries should naturally be considered trivial. From the viewpoint of studying algebraic dependences of automorphisms, generalized identities (without automorphisms) of a ring should also be referred to as trivial identities. In this sense equality (1) presented - iii
above is trivial, since after substituting f/J xf/J for x'P we get a zero generalized polynomial (as an element of the ring of generalized polynomials R * C< The results presented in this chapter show, in particular, that a semiprime ring has no multilinear identities with automorphisms from A(R), which is analogous to the theorem on algebraic independence of automorphisms of fields. For the derivations of a semiprime ring R of the characteristic p ~ 0 we can also write identities determining the structure of the Lie iJalgebra on the set of derivations D( R) and its inner (for Q ) part:
x».
4. (x + y) Ji.= x Ji.+ Y Ji.;
5. (xy) Ji. = xJi. y + xyJi.; 6.
where
a)Jl., ~t(xl)~-(x~)Ji.,
/l. 0 are any derivations, and [/l, 0] is a commutator in the algebra
D(R);
f3
where
/l, 0 are any derivations, a,
where
/l is an inner for Q derivation and a is a corresponding element;
E
C;
AUTOMORPHISMS AND DERIVATIONS
98
. P
9. ( ... (
J.L. J.l J.l
r
X)
)
••• )
tl
= x
J.l[PJ
where Jl is any derivation, Jl[p] is the value of a p-operation in the algebra D( R) (when p=O this identity assumes the form x = x). If these identities and their corollaries , as well as the generalized identities of a ring are considered trivial, then the results obtained below for derivations imply that there are no nontrivial differential identities. This is an analog of the theorem on algebraic independence of derivations of sfields. And, finally, there arises a problem on dependences between automorphisms and derivations. The remark on the fact that automorphisms from A(R) operate on the Lie a-algebra D( R) .
«x
to.
g
-1 J.l g )
)
= x
J.l g
,
which is also to be considered trivial. It is proved that there are also no nontrivial multilinear dependences between automorphisms and derivations. No exact definition will be given on what a trivial identity is, or what a corollary of identities is (though, it is not, in fact, a problem), but let us instead show in what wayan arbitrary polynomial f( X), which includes in its presentation not only the variables, ring operations and coefficients from R F but also the derivation operations and automorphisms, can be reduced using identities 1 - 10. Then we shall show that the reduced expression turns to zero at all the values of the variables from R only when it is obtained from generalized identities of a ring using substitutions Z
ij = x
gill. j
.
As above, let us fix the following notations: R , Q , R F and C for a semlpnme ring, its two-sided and left ring of quotients and a generalized centroid, respectively.
2.1
Process of Reducing Polynomials
2.1.1. Let R be a semiprime ring. Let us designate by I P an ideal of R, which consists of all the elements of an additive order p, where p is a simple number. Let ep=e(I~ and ~= 1- sup e p . Then all idempotents
~,
e2 ,
E3,
p:
e5 ,_·, e p'...
are
mutually
orthogonal
and
the
99
CHAPTER 2
characteristic of the ring expression
ep
RF
equals
P.
.t( x) = 0 is an identity in Riff
Moreover, sup e p = I, i.e., the p
e p.t( x) = 0 is an identity of
e p R for all p ;::: O. Since all idempotents e p are invariants of all automorphisms and are constants of all derivations, then, when studying identities, one can limit oneself with the case when a ring has a characteristic p;::: 0 (the ring is assUlmed to have a zero characteristic if there is no additive torsion). So, let R be a semiprime ring of a characteristic p ;::: O. Then, as we have seen above, D( R) is a Lie a-algebra over a generalized centroid. 2.1.2.
Definition.
A set
M
of derivations from
D( R)
called strongly independent if the fact that a linear combination different derivations from
M
will be
L J.l i Ci i
of
is an inner derivation for the ring Q, affords
J.liCi=O for all i.
Let us order a strongly independent set of derivations M and in a standard way expand this order on a set of words from M, i.e., we assume that a longer word is bigger than a shorter one, while the words of the same length are compared lexicographically. 2.1.3. Definitions. A word 6 from M is called correct if it does not contain subwords of the type flt j.J'2' where J.lI > ~ and if p> 0, then it does not contain subwords of type
c\
_. ~ om'
where ~ ~ ~ equality in a row.
J.lp.
In other words,
and this chain should not contain
L1 = p
c\.- om,
signs of
2.1.4. It should be recalled (lemma 1.6.21) that for an element m of a nonsingular module we have determined e( m) as an idempotent from C, such that mC == e( m)C. If J.l E D( R), then E( J.l) = e (R Jl) is easily seen to be a support of the image of J.l. It should be also recalled (see 1.7.8) that by i(g) we have denoted the support of c]J g for automorphisms g E A (R), which singles out the part of the ring R F on which g acts in an inner way.
2.1.5. of which use operation and operations and
Definition. The polynomial .t( xl'-" xJ, in the presentation is made of not only variables, addition operation, multiplication coefficients from the ring of quotients, but also of derivation automorphisms will be called, for short, aDA-polynomial.
100
AUTOMORPIDSMS AND DERIVAnONS
2.1.6. ADA-polynomial
is called reduced
.t( x}'... , xrJ
exists a generalized polynomial F( z ijk) ,
if there
1:S; i:S; n, 1:S; k, j:S; m, such that
where Eik are central idempotents, such that Eik i (gi.1· g it) = 0 at all t> k and all i, in which case for every i the set D i = (A ijl 1:S; j:S; m) consists of mutually different correct words of a strongly independent set of derivation Mi. 2.1.7. Theorem. Any DA-polynomial can be transformed to a reduced form using identities 1 - 10. Proof. Using identities 1, 4 and 5, we can achieve a state when automorphisms and derivations are directly applied only to variables, i.e., a DA-polynomial f( Xl ,..., x rJ is transformed to a generalized polynomial, and
T J are
, TJ F (xi)' where
F'(z .. ) is ~J
the words of automorphisms and 9
derivations. Identity 10, in its equivalent form x M = xgJl. makes it possible to place all automorphisms in the beginning of the words, while identity 2 allows one to replace their product in every word with one automorphism. Therefore, .t( x ~ is transformed into a polynomial form (2)
In order to better understand the essence of the phenomenon in question, a further reduction will be first carried out for a prime ring. and then for a general case. Let R be prime. Then its generalized centroid C is a field. and 4> g 0 for an automorphism g E A (R) iff g is an inner automor-phism
'*
for Q. Let G be a group generated by all automorphisms g ik' Gin be a normal subgroup of inner for Q automorphisms. Let us choose a system (gt) of representatives of cosets of Gin. Then g ik = h ikgt(i, k), where h ik E Gin and, hence,
x
g
ik
+ + (qr 1 xqJ)
gt(i, k)
•
. ThIS enables one to reduce
f( x i) using identities 3 and 1 to form 2, where different automorphisms
differ
by
modulo
i(gilg it) = 0
at
G in'
k,* t.
i.e., since
i(g- 1 h) = 0 g ik' g it
at
g'* h.
In
particular.
are different automorphisms
CHAPTER 2
101
encountered with a variable xi. Let us make derivations. Let us fix a basis (1 fJ'
PE
B)
of the
subalgebra adQ s;: D( R) and expand it to the basis {l a l a E A ~ B) of the whole algebra D( R) . Then the derivations ( 1 al a E A\ B) = M will be strongly independent. Now, writing the derivations in the presentation of words Il
1
j
with respect to this basis and replacing the occurrences u fJ for
ud 13 - d 13 u, where add 13 = 1 13'
one can reduce
the
DA-polynomial
.t( xl'-.) to form 2, where Il i j are words of the strongly independent set M.
Let us fix a complete order on table of multiplication in D( R) [1 a' ly] P
'"
1(15) ='" £,..,Aayl 15'
M.
Using identities 5, 6, 8 and the
1(15) Aay E C;
15
(6)
1 a =£..Pa 1 6 ,
(6)
Pa
E C.
6
we can transform the words /1 j into a linear combination of correct words. Therefore, the initial polynomial is reduced to a reduced form (in which case all sets Mi will even coincide). Let us now consider a semi prime ring. Let us first transform the DA-polynomial (2) to a form where Il i j are correct words of a strongly independent set of derivations. For every monomial m from a polynomial presentation (2) of f( xl'-.) let us denote by l(m) a sum of the lengths of words of derivations occurring in the presentation of m. Let D k( f) be a set of all derivations occurring in the presentation of the monomial m, such that l(m) = k. Let derivations
D1 = Dif)C D l( f) ,
where
be a submodule in End(RF'+)' generated by the
1 = 1( f) is the maximum among the numbers
l(m) relative all monomials m from the presentation of f. According to lemmas 1,6,9, 1.6.21 and 1.6.22, the finitely-generated submodule A = D1/ D1 () Din of the module End( R F' +) / Din is injective and, hence, the exact sequence
o---7
D1 () Din ----7 D 1 ----7 A ----7
0
102
AUTOMORPIDSMS AND DERIVATIONS
splits, i.e., D1 = (D 1 11 Did ED A where A == A !:: D1. Then, A' is a finitely-generated submodule and, hence, it splits into a direct sum of cyclic I ,
modules A'
= LED
I
O.C. ~
Therefore, using identities 6 and 7 one can reduce polynomial form F', where derivations from independent.
DiF')
Let us order a strongly independent set (0i)
= (oJ
f( x)
are
to
a
strongly
and extend this order
in a standard way onto a set of words from ( OJ. Then, using identities 5 and 8 one can reduce F' to a form F", where all the words occurring in the presentation of monomials m, such that l(m) = 1, are correct words from (Oi) and l(F''):5: l(F '). Let us consider a submodule
~
in
End( R F' +)
generated by
derivations Di F") U D1- I( F " ). Let A in = ~ 11 Din. Then, as has been shown above, A in is singled out from Al by a direct addent and is, hence, Ain + D iF ')C
a finitely-generated submodule. Therefore,
is also a finitely-
generated submodule and, hence, it is singled out from Al by a direct addent ~
=(Aint
ED DC F') C) ED B, where B is a finitely-generated submodule, Le.,
by lemma 1.6.9, B = LED J.l j
c.
Let us order a set (0i) U ( J.l j) in such a
0i > J.l j ' and, using identities 6 and 9, let us transform the expression F " to a form F"', where all the words encountered in the way that
presentation of the monomials m , such that l(m) from the strongly independent set ( 0 i)
U (
~
1- I
are correct words
J.l j) .
Continuing this process, we obtain an expression P(X) of form (2), which is equivalent to the initial one by modulo of identities 1, 2, 4 - 10, and such that the words occurring in its presentation are correct words from the strongly independent set of derivations D . Let us now transform the automorphisms. Let g ~·1' g ~'2~-' g.~n be all the automorphisms occurring in the presentation F( X) with a letter xi' Let, then, cP _ 1 = qJ kt C, e kt = e( qJ kt) , in which case one can assume (see gik g it
1. 7 .8) that qJ ktqJtk = e kt system of equations: e kj x
gik _
= etk . git
- qJ kt xi
From the definition of
qJtk' k
<
t
< n.
cP g
we get a
103
CHAPTER 2
Let
Eik=
1- e k , k+
t·· °e kn'
where,
by
the
definition,
aOb= a+ b- ab=sup(a, b). Then
(3)
where ).,~) are central idempotents. In this case the idempotents Eik are characterized by the following property: E ik' q, _ 1 = 0 at all t > k. gik g it
Let us now substitute into F(X) the right-hand part of 3 at instead of x
gi1 •
Then the automorphism
gil
k =1
will occur in the presentation
only in the terms with the coefficients EiI' such that t> 1. In the expression obtained all the occurrences
i (gj}g it) = 0 at
X~12 will be replaced
with the right-hand part of 3 at k=2, etc .. As a result, we get a reduced DA-polynomial. The theorem is proved.
Linear
2.2.
Differential
Identities
with
Automorphisms
-~
2.2.1.
Definition. Automorphisms
g, h are called mutually
outer
if i(gh-l)=O (see 1.7.8). 2.2.2.Theorem.
Let AI"" A n be pairwise different correct words
from a strongly independent set of derivations M, such that e( J1.) = 1 for all J1. EM. Let. then. gl'-" g m be pairwise mutually outer automorphisms. In this case , if the following identity holds on R:
then the following equalities are valid in the tensor product R, ® c R,: s(i, j)
L
a(.k~ ®
b(.k? =
for all i, j, 1 ~ i!:; m,l
~ j~
k = I
~J
~J
0
n(i).
(5)
104
AUTOMORPIDSMS AND DERIVATIONS
The proof will be divided into several steps.
2.2.3. One can assume the supports of all coefficients to coincide. In a Boolean ring of central idempotents E let us consider a subring T generated by the unit and the supports of the coefficients. This subring is finite (since any Boolean ring is locally-finite). Let el'-" e r be all minimal idempotents of the ring T . Since eqe t Seq, et , then these idempotents are pairwise orthogonal and, as 1 E T, their sum equals the unit Therefore, it is enough to show in the theorem the validity of the following equalities:
Let us multiply all coefficients of identity (4) by
e t . We have
e(a~]et) = e(a~;). e t S et: As a result, the supports of all nonzero coefficients of the obtained identity are equal to each other (and to e t ), and it is now sufficient to prove the theorem for the identity obtained.
2.2.4. One can assume that none of equalities (5) takes place, since in the opposite case one can eliminate a definite number of members without violating identity. Namely, if LaC k) 0
b(k) = 0
sum LaC k) xgl!. b( k), since in this case La( k)
then one can cross out the
yb( k) =
0 at all
y
E RF .
The same considerations allow one to assume that for any fixed
i, j
the elements {a~1} are linearly independent over C , i.e., none of them
belongs to the module generated by others. Let us now carry out induction over a senior word fil which occurs in the presentation of identity (4). 2.2.5. Let fll be an empty word. Then derivations take no part in the presentation of the identity and it acquires the form: m
f( x) =
L
= 1 k= 1 As, by the condition i (g i be written as i
k
g -; 1) = 0
f( x) = '" L. a i x g.L b i = 0, i = 1
at i:t:- t, then this identity will
(6)
105
CHAPTER 2
where the element at is right-independent of ~,_. an with respect to the sequence of automorphisms gl' g2~·' gn (see 1.7.15). By theorem 1.7.10, one
can ai
.
find
Pgl= 0 2.2.6.
variable x,
an
PE
element
lJ. R),
df
such
that a = at·
Definition. If .t( x) is a DA-polynomial with a singled out
P =L r i
g./i
®
S
then we set .t( x) .
i E L(R),
b)·
a
fJ
= (a .
ag
fJ
)
x
ga
b
+
s . (a .
a g81
fJ
where dots denote a sum of the terms dxgt,. b, is a natural number, S
'* 0,
at i ~ 2 Let us recall the following definition.
f3 = S i f(r i x).
2.2.7. If P E lJ. R), g is an automorphism , and A = correct word, then the following identity holds: (ax
gl
P
such that
51
) X
g8z.. 8m
<\.. 5m
+ .• ,
is a (7)
where A < ~ .• 5 m, and s
= ~ = .. = 5 5 ,*
55 + 1
and,
hence,
1= O(mod p). This equality results from the Leibnitz formula
where summation is carried out over all correct subwords AI' obtained from
<\.. 5 n
by crossing out some letters, and where A2 is a correct word extending A1 to A. For instance,
If A is empty, then formula (7) assumes the form (8)
get
2.2.8. Applying the obtained in 2.2.5 element ax
g
1
q = t( x) . P = O.
P
to relation (6), we
106
AUTOMORPIDSMS AND DERIVATIONS
Hence,
e(a)' e(b l )
=O. However, since the element a is expressed through
aI' then e(a) $; e(a 1) = e( b 1) (see 2.2.3). Thus, contradiction which serves a basis for induction. Let us carry out an induction step.
e(a)
= e( a) . e( b l ) = 0,
a
2.2.9. In addition to 2.2.4 one can assume that the senior word Al can occur in the presentation of identity (4) but once. Indeed, an element (2) ( n:I all ,... , all
»'
relative a sequence of automorphisms
aW
because tP
_I
g1 gJ
eIement k
-:I;.
/3 e
at
j.
-:I;.
u.. R),
in
= e(a)e (b (1) ll ) -:I;. 0, an
and
According to theorem 1.7.10, one can find an
suc h th at
which
substituting gi
i
1. According to (7),
AI' e(a)
=0
(2) n(1) all ,_., all
is not expressed linearly over C through
f·
(1) /3g1 -:1;., 0 = all'
a
/3 has the only term
case
e(a)
$;
0 1 'f =,
(k)/3g1
ail
e(a~l/) =
ax
g, A
e( bi?) (1 )
i.e., by lemma 1.6.23, a ® b ll
x
element
g-I
instead
I
of
x
(1)
I I b il
one
; .... .... 1 or
~
'
containing
and,
hence,
-:I;.
Therefore,
O.
can
assume
= 1, nO) = m(1 ) = 1. 2.2.10.
J.l171~-, J.lnfl
Let
occur in
u..
J.ll'~~-,J.ln
the
be
all
presentation of
the f
derivations, ,
/3 e R), then, according to (7), the sum of terms will have the form s .
(a~l) p (\x.1 b~I) +
"", (k)
a l "P)x +.U. k ,]
.1
(k)
7l =
where f·
such ~._ OIr/"
that If
/3 containing ;J.
L (a~~)r' p Jlr)x.1 b~~~ +
r, k
b l , ] .,
(9)
107
CHAPTER 2
A = II J.,
where
J.L r 71
= II r
and we have, for simplicity, omitted the second
index 1 at all coefficients a, b (it should be recalled that gl
= 1).
If a~l). 13 = 0, then f( x) . f3 = 0 is an identity to which one can the inductive SUpposluon. It means that in the tensor product
apply
RF ® RF
valid is the equality obtained from (9) by substituting all
;;J.
occurrences of
with the sign ®. According to lemma 1.6.24, one can
find elements '2'." C v E C which are 13 -independent and such that the first left coefficient a in (9) is linearly expressed through the remaining left coefficients using '2'." clJ" Here account has been taken of the fact that
e(sa~l). f3Bl)~ e(a~l)= e(b~l»
e(a)=
and,
hence,
a= a· «a)=
(1)
a e(b 1 ). Using linearity of the operators 13 and 13 denotations, we get a relation of the following type
Jl.
1
and introducing new
(10)
element
In this case the elements dr' h are independent of the choice of the 13, since '2'-" C v are independent of it. Equality (10) shows the mapping ;:
a~l) . 13 ~ a~l) . 13 Bl + L
dr' 13 Jl. r
+
h· 13 ,
r
where
13
runs through the set l1.. R) to be well-defined. Let us choose an
ideal IE F
in such a way that Ia~l) ~ R,
Id r I:: R,
Ih
~
R, IJl.r
~
R. If
now 13 runs through T = l1.. R)(l ® r), then the domain of the definition of the corresponding mapping is an ideal in R , while the domain of values lies of R. Moreover,
;(v(a~l) . 13»
=
;(a~l) . 13(1 ®
v» =
+Ld r ' f3Jl.r(1 ® v)+ h· /3(1 ® v)= r
a~l) . 13 \1 ® v;(a~l).
13).
v) +
108
AUTOMORPHISMS AND DERIVATIONS
Therefore, property (3) of the ring
(see 1.4.9) determines an
RF
element t, such that (11 )
Let
x E R, then
+(ail ). {3)x ol+L(d r ' {3J.i.r)x +L(d r · {3)xJ.i.r+(h· {3)x. r
r
Let us multiply relation (11) from the right by x and subtract it from (12), getting (13) M= ail)c + LdrC.
Let us now consider a module
By lemmas
r
1.6.9 and 1.6.21, it can be presented as a direct sum of cyclic modules _ (I) M - al C $
In
this
case
if
{3
a 2 c $ ... $ E
TI =
(\
i~
2
anC.
a10 (\ ~
T
(it
should
and dr=arar+··,arE C, then and equality (13) assumes the form T=l.{R)'(10;h
be
recalled
d r ·{3=
that
a r ai l ){3,
(ail). {3)([x, t]- xOl-LarxJ.i.r)=o.
(14)
r
As through
r
= 0 and the element
~, •. , an'
then,
by
lemma
ail) is not expressed linearly over C 1.7.9, at
ail) . TI is a nonzero two-sided ideal of R. Let equality (14) yields
g= gi= 1
eo
we
see
that
be its support. Then
109
CHAPTER 2
Since
(-'t' J.L r I
E1A = 0
is a strongly independent set of derivations, then
which contradicts the supposition that ce( -'t) = 1. Theorem 2.2.2 is proved. This theorem immediately yields a corollary, which the reader can easily prove using corollaries 1.7.14. 2.2.11. Corollary. On a semiprime ring R the following generalized identity
linear
n
L
a.xb.=O ~
i= 1
~
with the coefficients from
Rp
ho Ids,
La i ® b i = 0
iff
in
the
tensor
product RF ®c R F ·
2.3.
Multilinear
_
automorphisms R ,
Identities
with
Automorphisms
Theorem. If a multilinear reduced differential identity with
2.3.1.
ring
Differential
gjkAjJ
f( x1'-" x rJ = F( xi
then on
valid, where e i j
RF
the
_
£ ik) -
0
generalized identity
= e( l1})e (~).-
e( J.L t ), d i j
holds F(
Z
on
a
semiprime
i j k£ ike i }
= 0 is
= J.Lr · J.Le
Proof. It is sufficient to show that for any nonzero central idempotent e there is a nonzero central idempotent e l ~ e, such that elF
(z ijk£ ik e i } =
0
a=
F(a . 'k£ 'k
e ' .)
'1=
~J
~
~J
is an identity on
RF
Indeed, in this
0, we can find an idempotent
ae1 = 0, which is impossible (since
e 1 = e1e (a)
case, if
e 1 ~ ce(a) , such that
= 0).
Moreover, it would be enough to consider the case when f( xl'-" xrJ depends on a single variable. Indeed, giving concrete values to all the variables but the first one" we see that the identity F(
zl ] 'k f:.k -1
letters
Exten d'109 t hese conSl' d ' e ~J ' " x gjkAjJ ' £ ~'k) = 0, i> era-tIOns to t he ~ - 2.
~ ,_.,
we get the required result. Therefore, herefrom the index i
110
AUTOMORPmSMS AND DERIVATIONS
can be omitted everywhere. Let us fix a nonzero central idempotent e. Our nearest aim is to find a nonzero idempotent u ~ e and divide uf( x) into a sum of several addends
= L uf 1..( x),
uf( x)
such that
uf 1..( x) is
an identity of an ideal
and all the automorphisms participating in the presentation of f I..(x) act in the same manner on the set of all idempotents which are less than or equal to u. IE
F
can set
If all the automorphisms act in the same way on {u I.. ~ e}, then one u = e. If this is not the case, then one can choose an idempotent
u1 ~ e,
on which not all automorphisms act in the same way.
f( xu l ) =
L f I.. (x) ut...
where
u~ A, are different (central) idempotents. On a
finite Boolean ring generated by elements
in which case
~
U~A let us choose a minimal
A we have either
idempotent ~. Then for any
Then
gA u1 u2 = 0 or
gA u1 u 2 = ~,
can be chosen in such a way that neither the first nor the
A. Therefore, we have the decomposition
second equality hold for all
where all the expressions in square brackets turn to zero on I ElF, and contain the least number of terms (here I is an ideal, such that IU 1 ~ R ). Thus, we have got the required decomposition by induction. In different addends uf I..(x) no similar automorphisms can occur, and therefore,
uF(
Z
k .) =
]
LI.. uF(
Z
k .), where
]
f I..( x) =
uF 1..( x
g II k
Jf ~
is
a
reduced polynomial, and it is sufficient to establish the validity of the theorem for the addends uf( x). Let h be one of the automorphisms occurring in the automorphisms from this (h
case
-I
g k)
-I-I
h
uf I..(x)
uf I..(x h
is
-1
)
a
uf I..(x). Then all
act on {e I.. ~ u} in a trivial way. In reduced
D A -polynomial,
as
-I
g j = g k g j'
Reducing, if necessary, the idempotent u, one can assume that for any k we have either uf k = u or ef k = 0, and for any j we have either ue j
= u or ue j = O. This fact, in particular, implies that the automorphisms
111
CHAPTER 2
for which
gk
ue k
u, as well as the words L1 j' for which
'#
ue j '# u,
h- 1
"
take no part whatsoever in the presentation of f (x) == uF A. (x ). Therefore, going over to a ring uR, one can assume that all ek and e j are equal to the unit; all the automorphisms occurring in f"( x) act on E in an identical way, and
f"( x) turns to zero on IE F. Let us now apply theorem 2.2.2 to the identity f' (x) of the ring I. We have every possibility to do so, since ek = 1 and, hence, the
automorphisms in the presentation of
f"
Moreover,
e( J.l.) = 1 for all the derivations in
e(~) _. e(
Ilt) = 1 and, hence,
are pairwise mutually outer.
the presentation of :t'. Let m
f "( x)
r(i)
= '~ ""
i=1
then,
according
R, R, ®
to
'"" ~
s(i, j)
j=1
theorem
'"" ~
k=1
..1
( k)
a i jX
gi
j
b( k) i j'
2.2.2, in the tensor product
I, ® I, =
the following equalities are valid:
for all i, j, 1 S is ro,l S jS n(i). Hence,
at
any
Z
'J. E RF •
we
have
m ~
r(i) ~
~ ~ i=1 j=1
and
k
consequently,
" F (zij)=
'"" a( k) z b - 0 ~ijijij-"
s(i, j) (k)
L k=l
(k)
a . . z .. b .. =0, ~J
~J
~J
which is the required proof. 2.3.2. Corollary. If a multilinear differential identity with automorphisms is fulfilled on a ring R, then it is also fulfilled on the ring of quotients R F . Indeed, all identities of type 1. - 10. are valid for any x, y E R F ' i.e., by theorem 2.1.7, it is sufficient to establish a corollary for reduced identities. As in theorem 2.3.1, z ijk can assume any values from R F' the corollary is proved.
112
AUTOMORPlllSMS AND DERIVATIONS
2.3.3.
Corollary.
Let /1 = " .• On be a correct word of strongly
sets of derivations. In this case e(/1) = e(,,) .• e(orJ. Proof. It should be recalled that the support of /1 is, by definition,
equal to the support of the image e( JIl). Let f = 1 - e(/1). Then fx1 = 0 is a reduced identity, i.e., by theorem 2.3.1 , we have
fze(") .• e(orJ
=0
and, hence, e(,,) •. e(orJ::; e(/1). On the other hand,
and, therefore, e(/1)::; e (") .• e (0 rJ. The corollary is proved. Example. In theorem 2.3.1 the condition of multilinearity of an identity cannot be omitted. Let R be a perfect field of a characteristic p> O. Let us consider an automorphism g: x ~ x P and let w( x, y) = xP_y. Then
W(x,xg)=O, while W(x,y)*O on R. For the case of the zero characteristic the condition of multilinearity in theorem 2.3.1 is, naturally, inessential, since then any identity is equivalent to a system of complete linearizations of its homogenous components. Nevertheless, in the case of an arbitrary characteristic the presence of a (nonmultilinear) nontrivial identity with automorphisms also gives enough
information on the structure of the ring R. For instance, using operators /1 ~ arising in the process of proving the Martindale theorem, one can obtain a nontrivial poly linear identity for a prime ring. The theorem proved above shows that in this case the initial ring obeys a nontrivial generalized identity, i.e., by the Martindale theorem, its central closure is quite primitive, while its sfield is finite·dimensional over the center. In the next two paragraphs we shall see that for differential identities (without automorphisms) the condition of multilinearity in the theorem on independence excessive.
2.4. Differential Identities of Prime Rings In this paragraph R is assumed to be a prime ring. 2.4.1. a prime
Theorem. Let
ring R,
where
f( Xl ,•. , x rJ = 0 be a differential identity of d
j
f(xp= F(xi)
is a reduced D-polynomial.
113
CHAPTER 2
Then
F(
Z ij)
= 0 is a generalized identity on
RF .
Proof. If the characteristic of a field C equals zero, then any identity is equivalent to a system of complete linearizations of its homogenous components. Therefore, we assume P > O. Since operators /:}. ~
(see formula (23) in 1.13) do not violate
reducibility (if from the onset we replace (x + y)~ with ~ + 0), then the complete linearization tp( Yl '"., Yd of the polynomial t( Xl ,..., x rJ will be a reduced identity and, hence, in the ring R the generalized identity ~
where tp( Yl'"') = '1'( Yl 1), holds. Let us assume that this identity is trivial, Le., it follows from identities of type cx = xc, where c E C. Identities of this type do not change the order of sequence of the variables. It implies that if in the '1'(
Z
ij)
= 0,
expression '1'( Z i} we collect all the terms with the order of sequence of the variables fixed, then their sum will be also a trivial identity. Identifying the variables in this sum in an appropriate way, we get a senior homogenous .. ), Le., by induction, F( z ~] .. ) = 0 is a trivial identity in R part of F( Z ~] and, hence, in
RF
as well.
Let us, finally, assume that tp( Z i} = 0 is a nontrivial identity. According to the Martindale theorem, R C is then a primitive ring with a nonzero socle H, the sfield of which, T, is finite"dimensional over the center C.
2.4.2.Lemma. Let a prime ring R satisfy the nontrivial generalized identity. In this case any derivation acting in a trivial way on a generalized centroid C will be inner for Q. Proof. Let e be a primitive idempotent from R C. Then the sfield T is isomorphic to eRCe and is finite-dimensional over the center ec == C. By lemmas 1.13.5 and 1.13.6, the algebra of multiplications of the sfield T is equal to a complete ring of endomorphisms End c T. In particular, a . on C has the form l(x) = '" projection 1: T ~ £... a i xb i' where ai' b i E T. Let now J1 be a derivation acting trivially on C, x, Y E R. We
have
l(exe) = cle, l(eye) = ~e,
1( eye) Jl.=
S. e
cl '
~ E
C.
Hence,
Jl.. This affords
l(exe)l(eye)
Jl.
=
l(eye)l(exe)
Jl.
=
cl
~ee
i.e., we have obtained a polylinear differential identity.
Jl.
,
Jl. l(exe) =
Jl.
cle ,
114
AU1OMORPIDSMS AND DERIVATIONS
Let
+
l(exe)
Jl.
Then, if multilinear reduced identity A(x)
l(ex Jl. e).
l(exe)[A(y)
~ Jl. Jl. Jl. df = "",[(a ie) xeb i + aiex(eb i) ] + l(ex e) =
J1 is an outer derivation, then we get a
+ l(eYJl. e)] = l(eye)[A(x) + l(exJl. e)]
and, hence, by theorem 2.3.1, in the ring R the following identity is valid: l(exe)[A(y)
+ l(eze)]
= l(eye)[A(x) + l(ez:ze)].
Substituting into it x = 0, we get 0 = 1(eye)1(ez 2e), which is impossible, since 1 is an epimorphism onto the center. The lemma is proved. Let us continue proving the theorem. It follows from lemma 2.4.2 that if two derivations coincide on C , then they are equal modulo Din' In particular, the derivations induced on C by strongly independent derivations are linearly independent over C. The inverse statement is also valid: if the induced derivations are linearly independent, then the initial ones are strongly independent (it should be recalled that now C is a sfield). If the field C is finite, then c ~ e P is its automorphism and (el) Jl. = pcp-1cJl.= 0, i.e., all derivations are trivial on C. It means that all derivations of the ring R are inner and, hence, the reduced identity does not include in its presentation derivations and f == F, and it now remains to show that F=O is an identity on R F . We know (theorem 1.1S.1) that RF
=L
is a complete ring of linear transformations of a space over the
sfield, in which case RC is dense in L. Since C is finite, one can find a nonzero ideal I of R, such that IC!:;;; R and, hence, F is an identity on IC. However, IC contains the socle of the ring RC and, therefore, IC is also dense in L. Since ring operations are continuous in a finite topology, the identity F is transferred from IC to IL, which is the required proof. Therefore, the only case left to be considered is that if an infinite generalized centroid C. Step 1. Let us first show that our theorem is valid when R = C is an infinite field of a characteristic p > O. Since the field C is infinite, then infinite is the fields C P which consists, as we have seen above, of the constants of all derivations, and, therefore, one can assume that f( x1'-" x n> is a homogenous (over each of the variables) identities (which does not of course imply that F is a homogenous polynomial over each of the variables). It should be further remarked that it would be sufficient for us to consider the identities depending on one variable. Indeed, if f depends on
115
CHAPTER 2
several variables, then one can single out one of them and consider the rest variables to be coefficients. In this case the coefficients prove to be differential polynomials which depend on a less number of variables and, therefore, allowing for the fact that a polynomial over an infinite field is uniquely determined by its values, one can carry out an evident induction. Let us begin with the simplest case. m l) P' Let t(x)= ri(x i) , where riE C,{c5 i } are deriva-tions
L
i=l
which are linearly independent over C. Then, if t( x) = 0 is an identity of C, then for any C E C we have an equation that has a nonzero solution Z
i= ri in C:
Lm
i= 1
l)
Z
.(c i)
P'
~
= O.
If c runs through the whole field C, then we get an infinite system of equations, each having a zero solution in C, which fact implies that the rank of every finite subsystem is strictly less than m, and since all the n
coefficients lie in the field C P , then every finite subsystem has its solution n
in
CP
.
Choosing now a subsystem in such a way that the space of its n
solution over c P had a maximum possible dimensionality, we come to the conclusion that the solutions of this subsystem will be those of the whole infinite system. Therefore, there can be found elements which are nonzero and such that
"'o.c. = 0, ~ ~
It affords £..
derivations
<\'_"
cfn ~_, c~n E
c pn ,
which contradicts linear independence of the
Olll'
L
Let now t( x) = rix ~ i) pn, where a i are pairwise correct words from a linearly independent set of derivations. Let us carry out induction over a senior word
a o'
There are no empty words among
a i'
since the
substitution x = I shows the corresponding coefficient to equal zero. If the senior word is as long as 1, then the prove is given above. Let us consider the expression G( x, y)
= F( x, y) -
xpn F( y) - F(x) yp'.
116
AuroMORPIDSMS AND DERIVATIONS
All the tenns containing the word .6. 0 in this expression are cancelled. Let .6. 0 = ljs.6.', where .6.' does not begin with lj and
s < p. Then the
s- 1
word .6. = lj .6.' is nonempty and the terms containing this word in their presentation will have the fonn n pn 0 0 pn sp 16(xO y.1+ x.1 y O) + L.rk(x k y.1+ x.1 y k) =
=[s~ro(xO)~ +L.r~xOk)pn](y.1/n + ... where summation is carried out by those indices k '# 0, for which the word .6. k has the fonn ljk.6. (it should be recalled that all the words are correct and .6. k < .6. 0 ), Therefore, according to the inductive supposition and the remark made at the beginning of this step, we have
which is impossible due to the case considered above. An arbitrary homogenous differential expression can be reduced to the form
where
(i)
nl
+._ + n m( i )- _n_ - const, .6. 1 < .6. 2 < _. < .6. m are diffe-rent correct
words from a certain finite set of derivations. Let us expand the indices n~) in the powers of p: (i) ,0< - r. k< p. ].
Let us then transfonn our identity to the fonn L.ri i
m}1)
III x Llk]
0,
r(l)
1:. 1)
k [(x Llk) p] I, k [( X Llk) Ji] 2, k .•. =
O.
(15)
k=1
Of importance for us here is the fact that the expressions in square brackets are additive homomorphisms, while the powers over them are strictly less than p.
117
CHAPTER 2
Let us carry out induction over a number q = max ( q i = If q = 1, then, due to homogeneity, the expression considered above. Let us consider the e)(pression G( x, y) = F( x
assumes the form
+ y) - F(x)- F( y).
If this expression is reduced to form (15), assuming that Yi depend
on x, then we see that the number q for the expression GX< y) = G(x, y) is reduced. The same is valid for the variable y. Moreover, the terms G( x, y) for which q i = q - I will emerge only from the terms F( x), for which qi = q, in which case, according to Newton binomial, the corresponding coefficients will be further multiplied by some of the numbers (i) r j, k'
Let us now put to a lexicographical order all the m-tuples of the (a) num bers «n mi) ,_., n 1(i» ' f or wh'IC h qi:= q, Yi:F- 0 , and let (n (a) m .-, n1 ) be
n<;)
a maximal m-tuple, with and
r =
being the first on the right nonzero number
)t~)s being a nonzero number with the least possible number t.
Then in Gx< y) the coefficient at
will be equal to
rra<x
.d
S)
pt"
+
~
1:5;]<s
Yi( j, 1)(
.d
x
j)
pl"
+
£..
O:5;1
Yi(s, 1)(x
.d
~
pl
,
0:5; 1
where
tl
i(j,l) is such an index i, that v.=(x j) ~
pl
T(x).
A homogenous component of the power p t of this coefficient has the form
118
AUTOMORPfllSMS AND DERIVA nONS
Therefore, since the field of constants is infinite and allowing for the inductive supposition and the case considered above, we come to the conclusion that Ya = O. This is a contradiction showing that all coefficients Yi equal zero.
Step 2. Let us now assume that the coefficients of t( X) lie in a wider ring L containing C as a subring of the center of L , while the derivations are determined only on C. In this case we, naturally, assume that the variables are commuting with L, i.e., FE L®c C[
Z
i}'
Let us show that if t( x I '-" xJ turn to zero on C, then a zero polynomial. Let us present 1k
are elements of the ring L
Then, by the condition, elements therefore,
F
as a sum
'"
F = ~lk ® F
F(k)(
Z
ij)
Z . .) ~J
(zij)'
is
where
which are linearly independent over C.
"L.Jl kF(k) (c t:. i j) = 0
for
Ci E
are linearly independent, we see that
1k
(k)
F(
are zero polynomials. Hence,
F
C, F
(k)
i.e., since t:.
j
( xi ) =
0,
the and,
is also zero.
Let us go on considering a general case.
Step 3. Let us prove that t( xl'-' xJ turn to zero on H. Let us choose a variable components homogenous over Xl
Since the field constants ideal
Xl
and expand
f( xI '-'
I E F, such that
J into a sum of
C P is infinite, one can choose different central
where d is a general degree of
~,_., ~d'
X
i
I ~ j C R 0 ~ i, j ~ d. Then
f.
for
Let us choose an any
rEI
we
have
As these equalities are valid for all j
detll~~11 = .I1.(~ j ] >
~
-
~)"I= 0,
and the Wandermont's determinant,
then we have the following equalities:
CHAPTER 2
119
foe r) =0, f l ( r) =0, ... ,
frJ..r)
=O.
Extending these considerations onto the rest of the variables "2'." xn we come to the conclusion that all multihomogenous components fk(x P '.' xrJ of the polynomial t( If f k(x 1".'
type
k X rJ
ef xi
= (k I
p-
over
Xl ,_., X rJ
k rJ
is
Xl ,..., X n'
turn to zero on
I.
a set of the
degrees
of the
component
respectively, then, substituting an element of
instead of xi' we get
fk(ef x)
= II(ef/ i
.
i
fk(X I , ... , xrJ
i.e., all multihomogenous components and, hence, t( x) turns to zero on the elements of type ef r i' where e i Then, instead of the sums
n
L
of the polynomial
L x ( .1')
will
~
e . E C, r(.J') ]
~
ri
E I.
xi let us substitute into the D-polynomial
f
x~J). In this case the multihomogenous components
j= 1
t(
E C,
E I.
hence, t( Xl"'" xrJ
t(
L l-/»
also
turn
Therefore,
=0
will be identities on
f(
to
zero
at
L xC .j) e ~ = 0 ~
]
I.
l-,j) = ~
f( X)
k)(x~j»
It implies that r(,j) e~, ~
where
]
is an identity on
I
and,
is an identity on IC P .
Let us prove, finally, that Ie P :2 H. To this end, it suffices to prove that B r;; RC p. Indeed, in this case IC P is a two-sided ideal in RC P , while the socle is a simple ring and the least ideal. Let e be a primitive idempotent, 0
*' II
be an ideal in R, such that
eII , II e r;; R. Then A= e:zic P e is a subring of a finite-dimensional (over the center) sfield T = eRCe, in which case AC is an ideal in T and, hence, AC = T. The ring A has no zero divisors and obeys a generalized identity (even a polynomial one, as a subring of a finite-dimensional sfield). Hence, according to corollary 1.13.7, there is a nonzero element E An Ceo We P RC contains all primitive have e = (ee) P (e- 1) PEA. Therefore, idempotents. If e is a primitive idempotent, x E RC, then f = e + + ext(1 - e) is also a primitive idempotent, as qr. t --+ t f is a homomorphism from T
ee
120
AUTOMORPIDSMS AND DERIVATIONS
on
Consequently,
fRCf.
(1 - e)RCe = eRCe.
e)
eRC(1 -
and, by analogy,
RCP;;J
Analogously,
eRC(1 - e).
RC P is a subring, then it also contains the product
RC P :2 (1 - e)RCe. As
Hence,
RCe S;;;; RC P . And, finally,
eRC = eRCe
+
e) S;;;; RC P ,
eRC(l -
H = RCe· eRC S;;;; RC P.
Step 4. Let us now finish proving the theorem. A standard proof is 0f
re d uce d to th e case
. bl· one vana e. 1·f 0 = F( XlLl. j
Ll. j ),
, •. , X n
h . . ten, gIvmg
concrete values to all the variables but the first one, we find an identity Ll. j
F( zi j ' x 2 ~. ); the next step may be fixing the values of all the variables
but the second one, etc. .
Ll.
So, let f( x) = F(x j) = 0 be an identity on H. Let us arrange all correct words from the derivations in the order of diminishing .Ill> .112 > •. > .11 n and put in correspondence to each of them its own variable .11 j H Z j" The monomials 111, ~ of zi ,•. , Z n will be called
equivalent provided their degrees over each of the variables are equal. SO, F(
z;>
falls
into a sum of addends, in each of them the equivalent
monomials are assembled in multihomogenous components) F(
Z
;> =
F
(1)
a
pairwise
fashion
(Le.,
into
a
sum
(z;> + ... + F (k) (z j).
of
(16)
Each class of equivalence of monomials is determined by a set of degrees over the variables. For definiteness, let us assume that the monomials from
F
(1)
(z j) are corresponded to by the largest in the lexicographical
\ L et us show that sense set ( 11 ~., lnJ.
F (1 )( Z j ) .IS an 1·d· entIty on
Let m be an arbitrary natural number and let
H.
hl'•. ' h m be
elements from H. Let us consider m commutative variables
~I ,•. , ~ at
any Then
the following identity (see step 3) is fulfilled on C:
If we apply step 2 to this identity, then we
polynomial G( ~lj ,•. , ~
m;>
will see that the
equals zero as an element of the tensor product
121
CHAPTER 2
zero is the sum degree
G(1)
(~1 j
,_.,
~ m)
of all monomials of G which have a total
over the variables
11
~11'-.' ~ ml ;
a total degree
over the
~
variables ~12"-' ~m2'-.; a total degree 1n over the variables ~ln"- ~1lIl1" Now we have to remark that (1) " " G(1) (~1.' ] ... '~ m ] .)=F (~h·~·I,···,~h.~. . ~ ~ . ~ ~ rI\. ;L
(17)
~
Indeed, L1
(L h 3i) J = i
Lhi~i
L1 J
i
+ ... ,
(18)
where dots denote sums of the terms in which
~i
is included with a
word less than A j. Therefore, terms of the degree 11 over
~1
~I
~1
,.-, ~ m
arise only from monomials of the polynomial .t( X) of the degree X
~
I,
. SInce
occurrences
A 1 the most senior word, while x ~1
11
11
can over
is the largest of the
in the monomials from .t( X).
Of these monomials, only those with the ~I
~I
both the degree 11 over ~I ,._, ~ m and the degree ~ over the substitution x
= Lh i~).
By analogous considerations for words A3 , A4 ,etc., one can show that the terms of the necessary degree are obtained only from the monomials /1)(
x~ J).
Formula (18) shows now the validity of equality (17). Since the left-hand part of equality (17) turns to zero at all the
values of variables, then elements
H.
If
F(1) (
z ) = 0 at all
h I .-. h m were chosen arbitrarily, then
we
now
apply
the
same
Z j E hI C
/1) =
considerations
+ _. +
hm
c.
As the
0 is an identity on to
the
difference
122
AUTOMORPHISMS AND DERIYA TIONS
z .) = 0 is an identity on
F(2) (
Continuing this process, we see that all
F
]
(k)
H.
F( z} are
(z j)' and, hence,
identities on H. Since ring operations are continuous in a finite topology, and H is dense in fL, then F( z ) = 0 is an identity on fL = (RC ) F :2 R F' The theorem is proved.
Differential
2.5
2.5.1.
Identities of Semiprime Rings
Theorem. Let
f( xl'.' xrJ = 0
a semiprime ring of a characteristic p a reduced polynomial. Then F( ring R F , where e j = e(.1 ).
Z
ij
be a differential identity of _
e)
L'1
j
0, where f( xI ,•. , x rJ = F( xi) is = 0 is a generalized identity of the
~
Let us first make some remarks.
2.5.2. Any central idempotent is a constant of every derivation. ? 2 e:e
If if = e, then (if) P- = 2 ee P- = e P-, wherefrom P-_ Pd hence, ee p - ee , an, =0, .I.e., e P--2 - ee P--O - .
we
get
2.5.3. Any ideal of a generalized centroid C is differential. Indeed, since C is regular, any ideal is generated by its idempotents which, as has been shown above, prove to be constants.
J1
2.5.4. Let
be
an
arbitrary
derivation is induced on on stalks belonging to D( Indeed,
r
r
derivation from
p( Q), the
induced derivation
(IE) P-
!;;;;
/\ PRE)
the
RE
!;;;;
fi
p(R».
if
a
E
ker p p'
then
e(a)
E
derivation and,
since
/\ RE and, hence,
fi
correctIy. /\
P p( IE)
2.5.5. Now the derivation
P-
!;;;;
r
PJx)ii= =
M oreover,
deri vations
the
are
therefore,
and,
p
aP-=(ae(a»P-= aP-e(a)E ker Pp' i.e., the formula
determines
D(R). This
I' f
IP- !;;;; R ,
continuous,
p' i.e., by 1.12.5,
pJx~
fi
we
th en have
E D( r~.
J1 E D( R) can be viewed as a unary
123
CHAPTER 2
operation. This operation is strictly sheaf Conditions (a) and (b) of strict sheafness (see 1.11.7) are obvious, condition (b) is checked directly: if jill = y, then p} e( xll_ y) and f
=1-
e( xll_ y)
p E U(f), in which case (x Il_ y)f
p, i.e.,
eO
=0
or
(Xf)1l = yf, i.e., the equality xll= y is valid on T(U(f», which is the required proof.
2.5.6.
J.; ,...• J.l n
Let
from
D(R). P
Then
fit,...
be a strongly independent set of derivations
be a point of the spectrum, such that
p
n E
(l
i=
U(e ( J.l i»'
1
r p-
is a strongly independent set of derivations of the stalk
fin
L;i xiii -
Let, on the contrary.
~ X + x~ = 0 for all
x
E
r
p'
i:2: 1
and suitable
;i'
~ from the ring
C( r p)' in which case
find an element a of the stalk, such that a;i E
;1:t:.
O. We can
r p' i = 0,1,._, n;
a;1 :t:. O.
Let a;i = iii' Now we get
~ £.J
-j:
a~i
xa- ) ili -
(
i:2: 1
-j:
a~O
xa- +
-!! axa':l() =
0.
It means that on the stalk the following n + 2 -ary predicate is true:
Since the operations of derivations are strictly sheaf. then this predicate is also strictly sheaf and, hence. for a certain e eO p and suitable preimages a, b i e .(
1\
E RE
L
we have the identity
b i xJi.i a +
i:2: 1
in which case e· e( b 1)
eO p,
L b i xa Ji. i -
b O xa + axbO ) = 0
as hl:t:. O. If we apply theorem 3.3.1, we get
e· qZlae(J.;)=O, i.e., e· e(b1)e(a)' e(J.ll)=O. However. not a single idempotent in this product belongs to p and. hence. it cannot turn to zero, which is a contradiction.
124
AUTOMORPHISMS AND DERIVATIONS
Proof of theorem 2.5.1. Let us consider a Boolean subring in E, generated by all supports of elements and derivations of
f( Xl ,•. , X rJ.
e l ,.. , em be a set of all minimal idempotents of this ring. Then
pairwise orthogonal idempotents and
1 k\X)
=
.t( x)
(e k)
Ie kf(X) = I l k \ X),
stands for the expression obtained from f
Let are
where
by way of substituting
all coefficients t for e kt and all derivations 0 for oe k' It is also evident that
F(
Z .• ~J
the identities
e ] .)
j
= I/ k)(z k)(x 1'.
X
.. e .).
~J
]
n> = 0 ,
Therefore, if we prove the theorem for
then it will be proved for the general case
e k R, one can
as well. Going over to considering a ring e(t)
= e(o) = 1 As
f( xl'"
X
any
n>
for all t, 0 which enter the presentation of .t( Xl'·" xn>. idempotent from E is a constant, then the identity
is also valid on the ring
a i = I b i j e j , b i j E R,eijE E.
idempotents
e i j , e(f(a I ,··, an>),
idempotents
~ ::; e(f(a 1'." arJ)
ei
j ~
=~
assume that
or e i
j ~
In the latter sums j
= 0,
Indeed, let f( a 1'." a rJ
RE.
*' 0,
where
In a Boolean subring, generated by the let
us
Then
choose
a
any
e ij
for
minimal we
have
nonzero either
and, hence,
runs through the set
e i j ~ = ~. By lemma 1.6.6, f( of the ring RE in Q.
Xl ,.., X
of those indices for which
Ai
rJ is an identity of the closure
r
According to proposition 1.12.3,
peR F) ~
(r p( R»
applying the homomorphism Pp, we see that on the stalk
r
F p
/\
RE
and, hence, valid is the
identity I( Xl , •. , X rJ = 0 obtained from f by substituting all coefficients with their images and by substituting the derivations J1 with the induced ones, ji . As e( J1 i)
= 1,
then, by remark 2.5.6,
I(
Xl""
X
n>
is a reduced polynomial,
i.e., by theorem 2.4.1, F( z ij) = 0 is an identity on the stalk implies that the domain of values of through
F(
Z
ij)'
r peR p).
It
where the variables run
R F' is contained in the intersection of all kernels
p p' which equals
125
CHAPTER 2
zero. The theorem is proved.
2.5.7. Corollary. Any differential identity fulfilled on a semiprime ring R will also hold on a ring R F . Proof. Let e 2 , ~~-, ep~_be pairwise orthogonal idempo-tents,
eo,
such that
sup e p = 1 and the ring p~O
2.1.1). The identity t( X)
=0
e p RF
has a characteristic
holds on R F iff every identity
(see
p
e p t( X)
=0
is valid on e p R F. Taking into account the fact that (e pR) F = e p R F' it would be sufficient to establish the validity of the corollary for the case when the ring R has a characteristic p ~ O. By theorems 2.1. 7 and 2.5.1, we have to remark that the trivial identities 4. - 9. are fulfilled on RF in an obvious way. So, the theorem is proved. The theorem on the algebraic independence of derivations proves to be sometimes useful in a somewhat different formulation. 2.S.S. Corollary. Let AI"'.' A m be different correct words of a strongly independent set of derivations of a semiprime ring. Then, if the identity of the following type is valid on R
where
f l ,..., f m are multilinear generalized polynomials, then the identities
e(AI)f l = 0,..., e(A zn}f m= 0 hold on Rp-
Proof. Let us choose a variable xl. As the identity F( xl"'.) = 0 is equivalent to two identities: F(O,...) = 0 and F( Xl"'.) - F(O~_) = 0, then, with no generality violated, one can assume that Xl is encountered in all polynomials fl'_.' f m. Let us give concrete values to all the variables but Xl.
. to F( Xl111 ) Let us reduce the initial I·dentIty
= 0,
where {A il 1:5 i:5 n }
are all the subwords of the words AI"'. A m. Then, by theorem 2.5.1, the identity
F( zl)
=0
is valid on R. Let A 1 be a senior word among
AI"'. A n. Applying the Leibnitz formula to the expression (a l Xl a 2 ) 11 Xl 1
see when reducing, that the expression of the polynomial fl. Therefore, we have
111
,
we
arises only from the monomials F(zll'O~-,O)= fl(Zll) =0, i.e.,
126
AU1OMORPlllSMS AND DERIVATIONS
.t( zIl) = 0 is an identity on R. The rest of the proof is completed through induction (and corollary 2.5.7).
2.5.9. Remark. The above corollary does not always hold for nonmultilinear identities. For instance, in the field of the characteristic p > 0 the identity (x l') J.l =0 holds for any derivation J1., while x P ,* O. Nonetheless, considerations from the proof of corollary 2.5.8 show that in an arbitrary case partial linearizations of the identities f , = 0 over every variable will be fulfilled. ~
2.6 Essential Identities In this paragraph we shall consider polylinear differential identities with automorphisms of arbitrary (not necessarily semiprime) rings with a unit. Let .t( xl'0" x rJ be an arbitrary poly linear differential polynomial with automorphisms and coefficients from a given ring R: .t(X)= 1C E
L
Sn.
ISjSm,
(19)
IS is k, IS rS l.
2.6.1. Definition. A generalized monomial f 1C of the polynomial f( X) corresponding to a permutation n is a sum of all the monomials occurring in .t( X) in whose presentation the order of the letters x i is fixed and equal to x 1C (1)'o., x1C(n): f 1C
=
~
,~
~,
]. k
(0)
a ijr,
gj,l.1 i ,1 (1) gj,2.1 i ,2 1C x 1C (l) a ijr, 1C X ijr, 1C
It is evident that f( X) falls into a sum of its generalized monomials. For semiprime rings the process of reducing a DA-polynomial takes place in every generalized monomial individually, since not a single of identities 1. 10. changes the order of variables. If in the process of reducing all the generalized monomials turn to zero, then, naturally, f will be a zero DA-polynomial. On the other hand, one often can, using a given (generally 0
CHAPTER 2
127
speaking, non-reduced) identity, easily discover the nonzero values of its generalized monomials under concrete values of the variables. It immediately gives information on .t( X) being a nonzero DA-polynomial, in which case there is no need to reduce it. By the same reason it proves to be useful to study the identities for which there is enough information on the values of generalized monomials. 2.6.2. Definition. Let .t( X) be an arbitrary multilinear differential polynomial with automorphisms and coefficients from a given ring R. Let us call a stempel of f a two-sided ideal I F of the ring R generated by all the values of all generalized monomials. Analogously, a stempel of a set
r = (f ;.(X) )
will be an ideal
Ir= LI f · ;.
2.6.3. Definition. The identity .t( x) = 0 of a ring R is called essential if I f = R. Analogously, a system of identities is called essential if its stempel coincides with R. 2.6.4. Theorem. If a ring R with a unit satisfies an essential system of polylinear identities with automorphisms. then R has a polynomial identity
where the coefficients A. n can be chosen equal to ± 1. Let us first prove this theorem for a prime R, and then, using a canonical sheaf, transfer it to semiprime rings (making meanwhile sure that the theorem is also valid for differential identities with automorphisms of a semiprime ring) and, finally, employing the Amitsur's method, we can obtain it in a general form. 2.6.5. Proposition. If a semlprzme ring with a unit, R. obeys an essential system of DA-identities. then a certain standard identity holds in R:
The case of a prime ring. Let us transform all identities to a
AUTOMORPIDSMS AND DERIVATIONS
128
reduced form
F
( l)
gl.1- J
(x i
k
).
In this case the stempel of the system does not
change. By theorem 2.3.1, the system of identities
F
( l)
(z'k') ~ ]
=0
is valid
in the ring RF' If the left part of any of these identities is zero as a generalized polynomial, then its generalized monomials are also zero. On the other hand, the generalized monomials of polynomial
F
sums of generalized monomials of the polynomials
F
0)
( l)
g
.1-
(x lk J) are the
(z ijk)' in which
· . glk.1- J have been rnade . B y the same reason a 11 t e h su bStltUtlons Z ijk = xi generalized monomials of the corresponding DA-polynomial will assume nonzero values on R. Therefore, one of the monomials F( l)(z ijk) is nontrivial and we can make use of the Martindale theorem. According to this theorem
and
theorem
1.15.1 , we see that
is an essential system of identities of the ring R F = L of all linear transformations of a space V over a sfield ~ which is finitelydimensional over c. Let p be the dimensionality of V over ~. Let us denote by P a set of all linear transformations of the space V , the rank of which is strictly less than p.
r' = (F ( l) ( z . 'k») ~]
the
Let us show that if p is infinite, then in the factor-ring L = L / P condition of minimality for one-sided ideals does not hold. Let
( ey,r E A) be a basis of the space V and A= ~ ::::> ~ ::::> •• ::::> An::::>.. be an infinitely descending chain of subsets in A, such that the powers of A i' A i+ 1 are equal to
p,
and let
Then II
+P/ P
::::>
12 + P
/ P ::::> ••• ::::> In + P / P ::::> •••
is an infinitely strictly descending chain of the right ideals of the ring L. Indeed, if In + P = In + 1 + P, then for a transformation v, such that
129
CHAPTER 2
we should get a presentation v
= a + p,
where
a
E In +
l' pEP. Let VI
be a subspace generated by (e yl rEI n\ In + I)' Then
dim Vi p S dim V p < f3, which is a However, dim Vi = f3 , while contradiction. It should be further remarked that r is a simple ring with a unit. For this purpose it is sufficient to show that P is a maximal ideal. If 1 ~ P, then the dimensionality of V1 equals f3. Let W = ker 1 and ~ is a complement of W to V , i.e., V = W e WI' Then 1 maps WI on V 1 in a one-to-one manner and the dimensionality of WI can find a one-to-one linear transformation 11: 1 1 1: V -+ V1 ~: V1
is also
one-to-one
equals
f3.
V -+ WI C V,
Therefore, one in which case
, i.e., there is an inverse mapping
-+ V. Defining this mapping further on an extension to V1 one can
assume ~ E L. Thus, if 1 ~ P, then 11 112 = 1 for suitable 11, 12 E L, i.e., P is a maximal ideal. Since L is a simple ring with a unit, it coincides with its Martindale ring of quotients, and a system of identities r", obtained by substituting all coefficients with their images under a natural homomorphism L -+ L, is fulfilled on it. The stempel of the system r" is equal to a homomorphic image of the stempel I r and, hence, there are nontrivial identities among
r". By the Martindale theorem, L is again a primitive ring with a nonzero socle and, moreover, as LF = L, then L is a ring of all linear transformations of a vector space v over a sfield A. Since L is a simple ring, then the dimension of V over A must be finite (when the dimension is infinite, the transformations of a strictly less than the dimension rank form an ideal) and, in particular, the condition of minimality for one-sided ideals is fulfilled in L. As has been shown above, it is impossible for an infinite f3. Therefore, we come to the conclusion that R F = L is a ring of all
linear transformations of the finite-dimension space iT over the sfield L\, in which case the dimension of L\ over C is also finite. Hence, L is finitedimensional over C and, in particular, in the ring L and, hence, in R as well, a certain polynomial identity is fulfilled. For instance, if m is the dimension of Lover C, then a standard identity of the power m + 1 holds: Sm+ I( x) ==
L( _1)'" x"'(1)· .. · . x",(m+ 1) =0 '"
130
AU1OMORPIllSMS AND DERIVAnONS
The case of a semi prime ring. Let p be an arbitrary point of the spectrum. It suffices to show that on the stalk over this point valid is a
r
system of identities p' the stempel of which contains the unit. Indeed, in this case on the stalk, according to the above proved, a certain predicate is true: V xl ... xnSJ..x l , ... , Xd
=0,
where n
= n( p)
Since it is a strictly sheaf predicate (1.11.16), then there can be found a neighborhood up of a point p , such that this predicate is true on the ring of sections over up. Since the spectrum forms a compact space (1.9.13) and it has an open covering X = U up, then it is possible to find a finite subcovering X
=U k
up
Jc
as well. Let r
= max
n( p k).
k
Then on the
ring of global sections the standard identity S z{ x p.' x;J holds, which is the required proof (see also corollary 1.11.20).
r p'
In order to construct the system of identities
let us transform
all identities from r to a reduced form F( A)(X ~iJc tJ. j E .k). In this case the stempel of the system is not subject to any changes. By theorem 2.3.1, in ~
the ring RF valid is the system of identities where
i
ej
= e(.1 } .
A)(X :iJc tJ. j Eik)
Generalized
rp =
monomials
Z
ikj = xigiJctJ.
j
(.It
in particular, implies that the stempel of the system
r
i
A)(Z ikjE ik e } =
of
the
i.e.,
rp
tJ.
j
F( A)(Z ikjEik e j)'
= y
rp
O} ,
polynomial
should be rec.all ed that e j
determined as the least idempotent, such that e·] y
r,
{
are sums of generalized monomials of
. h the su b ·· StitutlOns wit
of the system
~
tJ.
j
can be
for all y). It,
contains the stempel
is also an essential system. In this case the
p obtained from r p by substituting all coefficients with their system images under a natural homomorphism is fulfilled on every stalk. Now the
stempel of the system obtained is the image of the stempel of
r
r p'
i.e., it
p is the sought essential system. The contains the unit. Therefore, proposition is proved. Now we can complete the proof of theorem 2.6.4, using the Amitsur method.
131
CHAPTER 2
Let us consider a direct product
IT
R =
AE
RA '
where A is a set
A
of countable sequences of the elements belonging to
R, R A = R
If
q> E R,
then we set q> ~ A) = (q>(A)) g, where g is an automorphism of R. In the ring R valid is a system of identities r' obtained from r by substituting coefficients a with ring
IT
AE
a A ' where A
R lies in the ideal
I
a A= a. In this case the unit of the
r' i.e., r' is an essential system of identities of
R.
Let 1L be a Baer radical of the ring R. By theorem 1.1.4 (b), 1L is the least semiprime ideal. However, 1L g is also a semiprime ideal, and, hence, 1L g
!;;;;;
g-1
1L. Applying the automorphism .
g- 1
to this inclusion, we get
g-1
,I.e., 1L = 1L again, due to the minimality. Therefore, the Baer radical is invariant with respect to all automorphisms of the ring R , i.e., the action of all automorphisms is induced on the factor-ring R / 1L. In this case on the factor-ring valid is an essential system of identities r" obtained by substituting the coefficients with their images under the natural homomorphism, and by substituting the automorphisms with induced ones. By proposition 2.6.5, the factor-ring obeys a standard identity Sm' Let us now choose elements q>1"" q> m in such a way that 1L::1 1L
q> iA) = ACi). We have
elements
(see.
S
me q>p_, q> rrJ
E
1L. Since
1.1.8), then for a certain k
Choosing now an arbitrary A = (x p_'
X
m~-)' we
1L consists of nilpotent
we have get
i.e., a certain degree of a standard identity holds in the ring theorem is proved.
R.
The
2.6.6. Remark. The only reason for eliminating derivations in the formulation of theorem 2.6.4 was that a Baer radical can prove to be no differential ideal, which makes direct application of the Amitsur method impossible. It would be of interest to overcome this obstacle in some way. The situation becomes clear when R has no additive torsion. 2.6.7. Lemma. If a ring R has Baer radical is a differential ideal.
no
additive
torsion.
then
a
132
AUTOMORPIllSMS AND DERIVATIONS
Proof. By theorem 1.1.4 (a), it is sufficient to show that the sum of all nilpotent ideals N I is a differential ideal and the factor-ring
R/ NI
has no additive torsion. The second statement is obvious: if nx e N I' then n( xl) generated by n X is nilpotent,
the ideal (x)m
= 0,
x
n 1i'{x ) m= 0
and, hence,
e Nr
Let Jl be a derivation, a e N r Then the ideal (a) generated by the element
a
is
nilpotent,
(a)m =
O.
In
particular,
for
any
xl ,..., x meR U (1 ), the following equality holds
Let us differentiate this equality m times:
where i.e.,
n A. are some integer coefficients. Among all these addends only one, nJ.l( xl a)._ J.l(x m a)
contains
no
cofactor
of
the
type
x
a,
i.e.
nJ.l(xla) ... J.l(xma)e(a). Since J.l(xa)=f.l{x)a+ xJ.l(a)5 xJ(a)(mod(a», then
This means that the ideal generated by nJ.l(a) is nilpotent by modulo (a), i.e., nmc.Jl(a» d ~(a) m =0. Therefore, J.l(a) e N I , which is the required proof. Now, according to remark 2.6.6, we come to the next theorem. 2.6.8. Theorem. If a ring R with a unit obeys an essential system of differential identities with automorphisms and there is no additive torsion in R, then a polynomial identity holds in R.
133
CHAPTER 2
2.7 Some Applications. Galois extensions of PI-rings, algebraic automorphisms and derivations, associative envelopings of Lie d-algebras of derivations Here we shall consider some immediate corollaries from the theorem on algebraic independence of automorphisms and derivations. It should be recalled that a ring R is called a PI-ring if there exists a polynomial f( x) with integer coefficients, one of which at the monomial of the highest degree is equal to one, which identically turns to zero on R
2.7.1. Theorem. Let G be a finite group of automorphisms of a ring R which has no additive I GI -torsion. Then, if a subring of fixed elements is aPI-ring, then R is also aPI-ring. Proof. Using operators .1 ~ (see formula (22) in 1.13), one can easily construct a poly linear polynomial which turns to zero on a ring of fixed elements, one of whose coefficients equals the unit:
Let us assume, for definiteness, that the coefficient ~ corresponding to an identical permutation, equals the unit. Let us first assume that the ring R has an element r, the trace of which,
t(r) = Lrg =1
(i.e., R has a unit as well). Since for any x E R an element t( x) belongs to a ring of fixed elements R G , then in the ring R the following identity with automorphisms holds (20) the generalized monomial of which, corresponding to a unit permutation, has the form
Substituting into it Xl = "2 =
N.
= Xm = r
we see that (20) is an essential
134
AUTOMORPmSMS AND DERIVAnONS
identity, Le., by theorem 2.6.4, R is aPI-ring. In order to complete the proof of the theorem, it suffices to embed the ring R, which has no additive I GI-torsion, into a ring ~, which
t(~) = 1), the ring of invariants of which
contains an element ~ (as
obeys the polynomial identity. It can is done in a standard way. Let us consider a set of formal r+ m rE R, m, k expressions of type integers, , where are
nk
k ~
0, n
=I GI.
Let
assume
us
r+ m
--;r=
that
k
r1+
n
k
111
iff
1
n k r 1, n 1m = n k ~ and let us define the operations of addition and multiplication in a standard way. One can easily see that we get a ring, in which case the mapping
r r: ~
0 will be an embedding
R C
use should be made of the absence of additive n-torsion). Further on, the action of the group G is extended onto formula rg+ m nk '
for fixed
r+m)g rg+m (~ = nk '
Le.,
nk r = n k r
g
For a fixed or
r = r
g
.
element we
have
(here
Rl ~
by the r+ m
--;;-k=
This, in particular, implies that
y, Z E RI their commutator, [y, z] = yz - zy, lies in the ring of
invariants of R and, hence, the following identity holds in the ring
R~ :
f( [ Yl ' Xl] , ... , [ Y m' z mD = 0, which is the required proof. It should be recalled that in the first part of the proof we made use neither of the restriction on the additive structure of the ring R, nor of theorem 2.6.4 in its full volume. Therefore, we can formulate a more general though less subtle statement.
2.7.2. Theorem. Let G be a finite group of automorphisms of a ring R with a unit. If an ideal t( R) of the ring RG obeys a polynomial identity of the degree m and I
E
Rt( R) m R, then R is a PI-ring.
2.7.3. It should be recalled that an automorphism s of the algebra R over a field F is called algebraic if there is a polynomial qX t) E F[ t], such that qXs) = in the ring End F R (i.e., s is an algebraic element considered as a linear transformation of the space R over the field F). The result presented below was obtained by V.E.Barnaumov under somewhat more severe restrictions.
°
CHAPTER 2
135
2.7.4. Corollary. Let s be an algebraic automorphism of the algebra R over a field F, such that the unit is not a multiple root of its minimal polynomial. Then, if the subalgebra of invariants of s obeys a polynomial identity, then R is aPI-algebra.
Proof. Let qXt) = Ao + A1 t + •. + Antn be the minimal polynomial for s. Then
for all a, x
E
R. It implies that (aoa, •. , ana) E V(sO,
J~., s~ (see 1.3.4).
*
Therefore, by lemma 1.3.4, the ideal RqX 1) is nilpotent. If qX 1) 0, then this ideal coincides with R and there is nothing to prove. Therefore, herefrom one can assume that qX 1) == 0 and qX t) = (t - 1) lp( t) , in which case lp(1) * O. As qXs) = 0, then lp(s)· s = ",(s), Le., al{l(s) is a fixed element for any a E R. Let If = R $ F be an algebra obtained from R by an external addition of a unit. Let us extend the action of s on R#, assuming that (a + a)s = as + a,a E R,a E F. Let f( xl'.' xrJ = 0 be a polylinear identity fulfilled in the subalgebra of fixed elements. In this case the following poly linear identity is valid in the subalgebra of fixed elements of the algebra R#:
This means that the following identity with automorphisms holds in
R#:
- g( 'l'(S) 'l'( S) 'l'(S) 'l'(S» - 0 h= ~ , Y1 , ... ,xm , Ym -.
Generalized an
monomials
x~N Y~:/-. x~~~ Y~~~' l{I( s)
of
the
polynomial
h
In particular, the stempel
have the form I h(
If) contains
2m
l{I{s)
elements anI •. 1 = anlp(I) E F, and, hence, IE I h , i.e., by theorem 2.6.4, the ring R # and, therefore, the ring R, obey a polynomial identity, which is the required proof. 2.7.5.
Definition.
The
derivation
Jl
is
called
0
ute r ,
if
136
AUTOMORPmSMS AND DERIVATIONS
pc ('\ Din = 0, i.e., the product
pC can not be a nonzero inner derivation This definition is equivalent to the fact that the one-element set ( p) is strongly independent.
for
C E
c.
2.7.6. Lemma. Any derivation p falls into a direct sum of inner, p in' and outer, Pout' derivations
Proof. Let us consider an exact sequence of the homomor-phisms of C-modules
o~
Din
n
pC ~ pC ~ pC / Din
n
pC -----7 O.
By lemma 1.6.21, the last but one module in this chain is both projective and injective, i.e., the chain splits and. hence. pC
We
have
= Pin C $
Pout C.
PoutC('\DinE(PC('\Did('\PoutC=O.
Pin C == e( P id c. Pout C == e ( Pout)C, and. hence. The lemma is proved.
Besides,
e( Pit!· . e( Pout) = O.
2.7.7. Definition. An endomorphism cp of the abelian group < Q. + > will be called algebraic if there is a polynomial n
t(t)= L riti.r1,_ .• rnEQ, such that Lricpi=O and the coefficients i=l r 1... ·• r n generate an ideal which has a zero annihilator in Q
(Le .• sup( e(
rl)~-' e( rrJ)
= 1).
2.7.8. Theorem. Any algebraic derivation
R with no additive torsion is inner for Q.
P
of a semiprime ring
Proof. By condition. L riP i = 0 for suitable r i
E
i
Q. i.e.. a diffe-
rential identity L r i x J.l. = 0 holds in the ring R. Let Pin = ad~. Then we have the following reduced identity:
Therefore. by theorem 2.3.1. the following identity is valid
137
CHAPTER 2
Substituting into this identity x = Z j = 0, j:t:. i, we get r i e ( Jlout) = O. Therefore, the ideal generated by the elements r 1,..·, r n annihilates the idempotent e( Jlout), Le., e( Jlout) = 0 = Jlout: It implies that Jl = Jl in is an inner derivation, which is the required proof. It should be remarked that the theorem allows enhancements. One can, for instance, require that Jl be algebraic only on certain subrings (or termal sets). 2.7.9. Corollary. Let R be a prime ring of a zero characteristic and let us assume that the restriction of derivation Jl on a nonzero left ideal, Rb, proves to be algebraic. Then Jl is an inner derivation for Q. Proof. By analogy with the above proof, we get a reduced identity
wherefrom we have corollary is proved.
r i zb e ( Jlout) =
0 for all i
and, hence,
JLout = O. The
2.7.10. D( R). By
all inner derivations from L. /8 (L) will designate an associative Csubalgebra in Q, generated by the elements corresponding to the derivations from Lin. In the considerations to follow we shall assume R to have a characteristic p ~ O. 2.7.11. Corollary. If a derivation Jl belongs to
Proof. By condition, Jl= q,(~,..., ~d, where ~,_., ~nE L; q, is a polynomial with the coefficients from lc. Let us transform the expression ~(01'.-' on) f(O; .-~ o~) x to a reduced form x . Then f will be a polynomial with the coefficients from 1 B ( L) r B (L) written to the right from the words (by rb we denote here the operator of right multiplication by b). Let us consider a c-module M generated by a (strongly independent)
138
AUTOMORPfllSMS AND DERIVATIONS
= M + pC.
set of derivations 0; ,_., O~, and let
M'
submodule of inner derivations from
M'.
M: n
Let, then,
In this case
M'.
~n
11 M =
be a
0, since
0; ,_., O~ is a strongly independent set.
A=
By lemmas 1.6.21, the finitely-generated submodule the module
o -+ M'. -+ M' -+ A -+ 0 splits. In particular, and, hence, an injective module.
M'.
~n
~n
Therefore, we see that the Injective submodule M'
by a direct summand
that
p' == ,u'(mod
o=LO:A i E
M: n of
is projective and, therefore, the exact sequence
End Q / D in
M
M' /
+
M' = (M
M:J,
+ M:J ED
p'C.
is a finitely-generated M
+
M:
n is split from
Besides, one can assume
,u = ,u' + 0 + ad;,
i.e.,
where
L, Ai E C, ; E Q.
0; ,_., O~, p'
The set
is,
by
construction,
xad; + xJJ.' + LA
Therefore, the identity
.x8~ -
strongly
x f (8) =0
~
independent. is a reduced
one. As in the presentation of this identity the derivation p' occurs but once, then, by theorem 2.3.1, the following identity is valid in R: x ad;
+ ye( ,u') + LA .e(o'.)z . + F( z. " x) = o. J: ~
= z i = z ij = 0 0 = ad;. Then,
Substituting x ,u'
= O.
Thus,
,u -
(21), we obtain an equality
Thus, }_ r~=Ll we
obtain
an
b1
r
di
equality
~
(21)
~]
into it, we get e(,u') = 0, i.e., substituting
xad;=xL1b r d i
Zi
1
,
= Z ij = 0 where
. Allowing for the fact that 1 ~ = Lei 1 b 1
'
where
ci
into identity
bi,diEJJJ(L).
lQrQ= lQ®C r Q, E
C.
Hence,
; =
LCib i
E B( L). The corollary is proved.
2.7.12. Corollary. Let L be a differential Lie c-subalgebra in D( R), where R is a semiprime ring of a characteristic p ~ 0, and C is its generalized centroid. Then. if L contains no nonzero inner derivations for Q. then cP( L) is isomorphic to the universal enveloping for L. A
Proof. Let T be a tensor algebra of a space L = L+ lc over a field F of central constants. The identical mapping L -+ L determines the
139
CHAPTER 2
epimorphisms of the associative F-algebras '1': T --+ (L). Let A be an associative enveloping for L. Then we have a C-linear homomorphism of (restricted) Lie algebras ~ A: L --+ A( - ), such that the action of coincides with that of I. to
generates
C, II
L,
A
~iL),
and the set
~iL),
L on C
together with the space
as an associative algebra. Let us extend the mapping ~ A
setting ~ i 1e) = I . e. It is sufficient now to prove that the kernel of
~ A is contained in the kernel of '1'. If f E T, then one can c:onsider
as a formal associative polynomial of derivations with non-commutative coefficients from 1e. In this f
case the condition f E ker~ implies that the linear differential identity x f =0 holds in the ring R. By theorems 2.1.7 and 2.3.1, this identity results from identities of type 4. - 8. (It should be recalled that there exist no nontrivial linear generalized identities, which can, for instance, be seen from theorem 2.2.2). Therefore, the task now is to show that the elements T corresponding to identities 4. - 10., belong to the kernel of '1'. The elements corresponding to identities 1. - 7., 9. lie in the kernel of 'I' because ~A is a C-linear homomorphism of restricted Lie algebras. Corresponding to identity 4. are the elements of the type 1e Jl- Jl1 e = 1 p. e
which are transformed, under the action of (IA'
e)(Jl~~-(Jl~~e-IA'
~A
to the elements of the type
e ll = e(Il;)_ ell=O. Identities of type 8 are
absent as, by condition, L contains no nonzero derivations which are inner for Q. The corollary is proved. 2.7.13. Corollary. Let G be a finite group of automorphisms of a semiprime ring R and let Jl be a derivation of R acting in a trivial way on a fixed ring RG. Then, if e( t( R) ) = I, then Jl is an inner derivation forQ.Heret(x)=
Lx g
.
gE G
Proof. Let p be an arbitrary simple number or zero, e p be a corresponding idempotent (see 2.1.1). Let us transform the expression ept( x)
ring
=
L
e pxg to a reduced form
gE G
~. a i j x gi bijE i ' Then in the ~]
e p R the following reduced identity is valid: e p t( x )
Il -
=
"" ( Il gj ~. a i j x b i j ~,
]
+
a ij x
gi
Il) b i j Ei
+
AU1OMORPlllSMS AND DERIVATIONS
140
+ '" f-'.
:2,J
a .. [xgiadJ:]E.b .. ~J ':. ~ ~J
+
L
.. :2,J
a "xgllloutb"E.=O. ~J
~J ~
By theorem 2.3.1, we get the following identities in e p
Substituting into these identities the elements summing over al i, we get:
o=
~. a ij x g1 bije(Jl.out)Ei = ~,
J
As e( t( R) ) = 1, then sup{ e p}
= O.
e p e( Jl.out)
x g1
R:
instead of y
and
e pt(x)e (Jl.out).
=0
and, hence,
Jl.out = 0 because
The corollary is proved.
It should be remarked that the condition e( t( R) ) = 1 is equivalent to the fact that the annihilator of the ideal Rt( R) R in the ring R equals zero. This condition is, for instance, fulfilled if R has no additive I GI -torsion, in which case it is the annihilator of the set t( R) that equals zero (see lemma 1.3.4). REFERENCES
S.Amitsur [5,7]; V.E.Barbaumov [12]; K.I.Beidar [14]; K.I.Beidar, A,V,Mikhalev [15]; V.K.Kharchenko [71,73,74,75,78,81,84]; E.Posner [138];. L.Rowen [142]; A.M.Slin'ko [145].
CHAPTER 3 THE GALOIS THEORY OF PRIME RINGS
The contribution of the Galois theory for a class of rings is commonly understood as a proof of the principal correspondence theorem of definite types of finite (or reduced-finite) groups of automorphisms and those of subrings from a given class. Let R be a ring, S a subring and G a group of automorphisms (i.e., a subgroup of the group of all automorphisms) of the ring R. An element a E R is called G-invariant if a g = a for every g E G. A set of all G-invariants is denoted by RG or by lI(G). A set of all automorphisms for which the elements of S serve as invariants is denoted by A(S). It is evident that lI(G) is a subring of the ring R, and A(S) is a group of automorphisms. As was the case in the classical theory of fields, the group A(S) is called a Galois group of the ring Rover S. A subring S is called a Galois subring of R, and R is, respectively, a Galois extension of the ring S, if S = II (G) for a group of automorphisms G. The correspondences G --+ II (G) and S --+ A(S) invert the inclusion relations,
i.e.,
if
Gl
S;;;; ~,
then
II (G 1) :2 II ( G2)
and
if
SI S;;;; ~,
then
A(S 1):2 A( S2)' We also have A(lI(G»:2 G , lI( A(S»:2 S ,
which immediately yield lI( A(lI(G»)
= II (G)
, A(lI( A(S»)
= Ei,.S).
Therefore, the mappings under discussion set a one-to-one correspondence between Galois groups and Galois subrings. And, hence, the groups A(S) and the subrings lI(G) are of primary interest in the Galois theory. In order to prove the correspondence theorem in a class of rings ~ ,
141 V. K. Kharchenko, Automorphisms and Derivations of Associative Rings © Kluwer Academic Publishers 1991
AuroMORPIDSMS AND DERIVAnONS
142
it is necessary (and sufficient) to answer the following questions: (1) Under what conditions does a subring of fixed elements for a group G of automorphisms of a ring R E ~ belong to ~? (2) When will a group G which obeys condition (1), be a Galois group? (3) Under what conditions is an intermediate ring S E~, !I(G) s: S s: R a Galois subring? An analogous approach is also possible for studying derivations. In this case the role of Galois objects is played by differential (restricted) Lie zalgebras and the subrings of constants of such algebras. In this case the same problems as for groups, i.e., (1) - (3) arise. In the three chapters to follow we are going to develop the Galois theory for automorphisms and derivations in classes of prime and semiprime rings. As above, we shall consider a somewhat more general situation, assuming that the automorphisms lie in A (R), while derivations in D( R) (see 1.7). 3.1. Basic Notions Let R be a prime ring, G be a group of automorphisms. It should be recalled that by rp g we denote a set of all elements qJ E R F such that xqJ
= qJx g
for all
x E RF
(see 1.7.5, 1.7.6). For these sets valid are the
relations cP gCP h!;;;; cP gh (see formula (8) in 1.7) which, in particular, afford that cP g is a linear space over a generalized centroid of the ring R. Moreover, corollary 1.7.9 states that CPg will be nonzero iff g is an inner automorphism for Q. 3.1.1. Definition. The algebra of a group of automorphisms G is a C-algebra /8(G)
= L
CPg.
gE G
Therefore, the algebra of the group has a basis of elements which correspond to the automorphisms which are inner for Q, i.e., it is an inner part of the group in a ring form. If G is a finite group, then its algebra /8 (G) will be finitedimensional over
c. Of a finite order will also be a factor-group
where Gin is a normal subgroup of all inner for
Q
G / Gin'
automorphisms.
3.1.2. Definition. A group G is called reduced-finite if its algebra /8 (G) is finite-dimensional, while the factor-group G / Gin is finite. In this case the number dim c /8 (G) . I G / G.~nI is called a reduced order of the group G.
CHAPTER 3
143
3.1.3. Definition. A reduced-finite group G is called an M-group (a Maschke group) if its algebra is semiprime (or, which is equivalent, semisimple). 3.1.4. Definition. A group G is called an N-group (a Noether group) if any inner automorphism of the ring Q corresponding to an invertible element from B (G) belongs to G. 3.1.5. Lemma. Any Galois group is a Noether group. Proof. Let G = A(S). If S E S, then for all g E G,qJ
E
tP g
we
have sqJ = qJs g= qJs, i.e., rp g and, hence, B(G) as well, are element-wise commutative with the subring S. In particular, if b is an invertible element from
B(G)
then the inner automorphism
£(x) = b- lxb
acts trivially on
£ E G, which is the required proof. The value of Maschke groups will be discussed later ( see 3.6), and now we go over to the central notion. S, i.e.,
3.1.6. Definition. A Maschke group G is called regular if it is also a Noether group. 3.1. 7. Definition. A regular group G is called quite regular if its algebra is simple. Let us pay attention to the fact that all these notions (but that of a reduced order) refer only to the inner part of the group G. Hence, any finite group of automorphisms which contains no non-trivial automorphisms inner for Q, will be quite regular. Such a group will be naturally called a (finite) group of outer automorphisms. The basic notions referring to subrings will be introduced later. Here we are going to dwell only on the finiteness of the ideal over a subring. The point is that in the classical Galois theory of importance is the fact that the field has a finite dimension over a Galois subfield. Under general conditions this is not the case, while a certain relation of finiteness (developed by A.I.Shirshov when studying rings with polynomial identities) is valid and even plays an important part in a number of cases. 3.1.8. Definition. A subring A of the ring R is called rightShirshov finite over a subring DE R, if there are elements r p _" rn E R, such that A~ r 1 D + .• + rnD. 3.1.9. Definition. A two-sided ideal I of the ring R is called right-locally finite over a subring D ~ R if any finitely-generated right ideal of the ring R lying in I is a right-Shirshov finite subring over D.
144
AUTOMORPHISMS AND DERIVATIONS
3.2
Some Properties of Finite Groups of Outer Automorphisms
This paragraph is of an auxiliary character and all its statements will be further significantly generalized. However, the information accumulated in the preceding chapters makes it possible to obtain these particular cases, which could be of great help for theoretical endeavours, with no special effort at all.
3.2.1. Theorem. Let G be a finite group of outer (for Q) automorphisms of a prime ring R. Then the centralizer of a ring of invariants R G in the ring of quotients R F is equal to the generalized centroid C. Proof. Let an element a commute with all fixed elements of the ring R. Then the following identity with automorphisms is valid a under all
x
L,xg- L,x g a==O gE G gE G E
R, such that
xg
E
R
(1)
for any
g E G.
As
G s:;; A(R) and G
is finite, then there exists a two-sided ideal I::t. 0, such that I g s:;; R for all gE G, i.e., (1) is an identity on I. By theorem 2.2.2, we have a ® 1 - 1 ® a == 0, i.e., a E C, which is the required proof.
3.2.2. Theorem. Any finite group of outer automorphisms of a prime ring is a Galois group. Proof. Let h be an arbitrary automorphism , h E A (R), which acts trivially on R G, i.e., h E All ( G). Then, as was the case in the preceding theorem, we can find an ideal I on which the following identity with automorphisms is fulfilled: L, x gh gE G
L, x g == 0 gE G
This identity cannot be a reduced one, since theorem 2.2.2 would immediately yield the equality I ® 1 == O. The irreducibility of this identity
CHAPTER 3
implies
145
that
one
of
the
automorphisms
(g i h)(g j h)
- 1
,
- 1 (gih)gk' git:gje G is
and, hence, b theorem,
- 1 inner. Let gihgk=b. Then be AI(G) commutes with all fixed elements, i.e., by the preceding
E = 1. Thus,
h
=
gk gi Ie G, which is the required proof.
3.2.3. Theorem. A fixed ring of a finite group of outer automorphisms of a prime ring is prime. Proof. Let aR G b = O. Then we have an identity on an ideal It: 0:
By theorem 2.2.2, we have a ® .b = 0 and, hence, either a which is the required proof.
= 0 or
b
= 0,
3.2.4. Theorem. Let G be a finite group of outer automorphisms of a prime ring R. Then there exists a nonzero ideal of the ring R which is right-locally finite over R G Proof. By theorem 1.7.10, there are elements a
r
= Lt . ] .v ] . t: 0; Lt . ] .v = 0 ] ]
for all
ge G, gt:1.
t j' v j' such that
Setting
'l(x)=
Lx g ,
gE G
we get
ax
= ~t j'r(v j
x).
]
If now x runs through the ideal I, such that
I g r:;;;, R for all
ge G, then we see that
aI C " t . R G. It remains to notice that the ideal - 'L. ]
Ral is locally-finite over
R G, since
is proved.
L r i aI r:;;;, L . . r.t
. ;
~.
]
~
]
. R G.
The theorem
Theorem. Let G be a finite group of outer automorphisms and let R;;;2 5 ;;;2 R G. Then any automorphism qr. 5 ~ R. which is trivial on R G, is extended to an automorphisms from G Both this theorem and somewhat more complex next ones will be deduced below from general results (see 3.9 and 3.11). 3.2.5.
3.2.6. Definition. A subring
5
is called rationally
complete in a
146
AUI'OMORPlllSMS AND DERIVATIONS
prime ring R if the condition Ax!:: S for x E R, and a nonzero ideal A of S imply XES. For a prime subring S this condition is seen to be equivalent to SF n R = S. 3.2.7.
Theorem. Let G be a finite
group
R:! S:! RG. Then S will be a Galois subring of R
of
automorphisms
iff it is rationally
complete. The above theorems allow the reader to easily formulate the correspondence theorem for outer groups, and show the situation to be convenient for this case. Let us now try to elucidate , in general terms, another extreme situation , that of inner automorphisms, from the point of view of the notions proved earlier. Since a fixed ring of a reduced-finite group of inner automorphisms coincides with a centralizer of the finite-dimensional algebra B (G), then we come to the necessity to study (from a general standpoint) centralizers of finite-dimensional algebras in prime rings. It is still more important since practically any finite-dimensional algebra with a unit is generated by its invertible elements and can, therefore, serve as an algebra of a reduced-finite group (with the only exception for algebras over the twoelement field GF(2) , the factor-algebras by the radical of which have two or
more direct addends isomorphic to GF( 2) ).
3.3
Centralizers
of Finite-Dimensional
Algebras
3.3.1. Definitions. A finite-dimensional algebra with a unit, B, over a field C is called centralizable if for any prime ring R with a generalized R), the centralizer of B in R is nonzero. centralizer C, such that B!:: It should be recalled that a finite-dimensional algebra with a unit is called Frobenius if it has a non-degenerate associative bilinear form. We will call the algebra B partially Frobenius (p.F.) if it has a partially nondegenerate associative bilinear form, i.e., if there exists a nonzero right, p, and a nonzero left, A, ideals, and a bilinear form p x A ~ C, such that (rb, 1)= ( r, b1) for all rep, 1 E A, bE B, and if the equality (r, A) = 0 is valid only at r = 0, while the equality (p, A) = 0 is valid only at 1 = O. These notions can be also formulated in terms of conjugated modules. It should be recalled that for the left B-module, A, we denote by A* the
ex
conjugated right B-module,
A*=
given by the formula (qJ b) (1)
= qJ( b1).
Hom
C(A,
C),
with the module structure
Accordingly, for the right B-module,
147
CHAPTER 3
p, by
p* we shall denote the conjugated module
fI' = Hom c( p, C), with
the module structure given by the formula (bl/f)r = l/f(rb). The latter formula can be more conveniently presented as r( bl/f) = ( rb) l/f, by locating the argument to the left from the function. On conjugated modules, there naturally arises a non-degenerate associative bilinear form A*X A -+ c: (qJ, 1) = qJ( 1) and, respectively, p x fI' -+ c: (r, l/f) = rl/f. In the finite-dimensional case, the inverse is also valid. If for a couple of modules, p and A, right and left, respectively, there is a non-degenerate bilinear associative form p x A -+ c, then A == p* and p == A*. The definition of a Frobenius algebra can be now given by the formula B*== B, where, on the one hand, B is a left regular module, and on the other hand, it is a right module. Being partially Frobenius means the existence of a nonzero one-sided ideal p, such that its conjugated module is isomorphic to IUlother one-sided ideal. It would now be natural to give one more definition, that of algebras which are "Frobenius by parts", i.e., such that the sum of all right ideals conjugated with the left ones coincides with the whole algebra B. Such algebras have been studied elsewhere and are called quasi-Frobenius (QF). Both Frobenius and quasi-Frobenius algebras originated in the theory of presentation of groups and finite-dimensional algebras. The two following theorems characterizing these algebras are well known, and we refer the reader to see the proof elsewhere ([36]. theorems 61.2 and 61.3). 3.3.2. Theorem. The following statements on a finite-dimensional algebra with a unit, B, are equivalent: (1) the algebra B is Frobenius; (2) there is a linear form E E B* the kernel of which contains no nonzero one-sided ideals of the algebra B; (3) for all left ideals A and right ideals p of the algebra B the following equalities are valid: dim r( A) + dim A = dim B , dim 1(p) + dim p = dim B, where r( A) = [ bE B I A b = 0), 1(p) = [b E B I bp = 0 are the right and left. respectively, annihilators in B (sometimes they are denoted by ann ~(A) and ann ~( p), respectively). 3.3.3. Theorem. The following statements on a finite-dimensional algebra with a unit, B, are equivalent: (1) the algebra B is quasi-Frobenius;
148
AU1OMORPlllSMS AND DERIVAnONS
(2) the left B-module B is injective; (3) the right B-module B is injective; (4) for every left ideal A and right ideal p of the algebra B the following equalities are valid: l(r(A»=A,
r(l(p»=p
3.3.4. Examples of Frobenius algebras. (a) A group algebra of a finite group is Frobenius.
If B
= C [G]
and
b
= 2,c g g
is an arbitrary element of a group
algebra, then we set e( b) = C e ' where e is a unit element of the group G. Then e is a linear form, the kernel of which contains no nonzero one-sided ideals. (b) The universal p-enveloping of a finite-dimensional Lie p-algebra is Frobenius. If J.; ,..., J.l n is the basis of a Lie p-algebra L, then the basis of its
p-enveloping are the words of the type
J.l( ._, J.lr;)n, where 0 ~ mi < p. Let
us denote by e( b) the coefficient at J.li - 1__ . J.l f: - I in the decomposition of the element b
E
U (L)
over this basis. Then e is the sought linear form.
(c) A semi-simple algebra is Frobenius. It suffices to remark that all irreducible left (right) modules of a
simple algebra are mutually isomorphic and any module is a direct sum of irreducible modules. Hence, from dimensional considerations, (B F!) == B B (provided B is simple). For the semi-simple algebra B we have a decomposition into a sum of simple modules, B = ~ EEl _. EEl Bll" Therefore,
If
= B~ EEl ._ EEl
B~ == BI EEl _. EEl Bn == B.
3.3.5. Lemma. In a quasi-Frobenius algebra the annihilator of the intersection of one-sided ideals is equal to the sum of their annihilators: l(Pl
n P2)
= l(Pl)
+
1(P2) ,
r( Al n ..'S) = r(A I ) + r(..'S) .
Proof. Let us, for instance, check the first equality. According to statement (4) of theorem 3.3.3, it is sufficient to prove that the right annihilators of both parts of the equality coincide, i.e., that PI n P2 =
149
CHAPTER 3
r(l( PI)
PI
II
+
1(P2)'
However,
P2' which is the required proof. Thus, the lemma is proved.
3.3.6. In linear algebras well known is the fact that the conjugated modules P, A have mutually dual bases, i.e., the bases ( a i) : = 1 ' (b i) : = lover C , such that i= .1, i i= j,
where (p, A) is the corresponding form. Moreover, for any basis (a i) of the module P there is a dual basis
p*. The most important property of the operation of module conjugation is that it is an exact contravariant functor, i.e., for any exact sequence of the homomorphisms of modules (b i) of the conjugated module
O~ E~ F~ D--70
we have an exact sequence
and, besides, the conjugation is involutory, i.e., (A *)*= A for a finitedimensional module A. We shall leave these facts as exercises for the reader (if he is not familiar with them) and go over to the basic topic. 3.3.7. Theorem. An algebra Frobenius. Let us first prove two lemmas. 3.3.8.
is centralizable iff it is partially
B
Lemma. If the dimension of the algebra B is greater than
one, then the free product lized centroid equals C.
B
*C
C[ xl
Proof. The basis of the algebra
C;;
B ( X)
B (
is prime and its genera-
x) consists of the words of the
type f3 il x f3 i2 x •. x f3 in' n ~ 1, where f3 ij are elements of a fixed basis of B. Let us order the basis of B in a standard way and extend this order to the basis B ( X) (Le., we assume that a longer word is bigger than a shorter one, while the words of the same length are compared
150
AU1OMORPlllSMS AND DERIVAnONS
lexicographically). Then if u, v are elements from B ( x), then the senior word of the product uxv is ii xv, where ii, v are the senior words of u and v, respectively. Therefore, u B ( x) v"# 0 and ring. Let z be an element of a generalized centroid,
B ( x)
is a prime
z e c. We can find
a nonzero element fEB ( X) , such that h = fz E B ( X). Let us assume that the element z e C is chosen in such a way that the senior word of h be the smallest word possible. If h = 0, then B ( x) fB ( x) z = 0 and, hence, z = 0 E c, which is impossible. Let u be an arbitrary element from B ( x) . Then fuhz=fzuh=huh=hufz. Therefore,
fuh
= huf.
Let us show that the latter equality implies the equality of the senior words f and h. If the degrees of the words f and h over x coincide, then these words are of the same length and fxfi =fxh=hxf = hxf, which affords f = h. Let, for instance, deg x f < deg x h. Then, since the dimension of B is greater than one, one can find an element b 1 , such that fxb 1 is not the initial subword for fxb 1xh-
= fxb1xh = = huf=
h. If we assume
hxb1 xh, which means that
u = xb1x, then
fxb 1 is the initial
subword of the word h, which is a contradiction. Therefore, f = h, i.e., one can choose A E C in such a way that the senior word of the element h - Af be smaller than h. Then f( Z - A) = h - Af, in which case z - A. e C. This contradicts the choice of the elements z and h and the lemma is thus proved.
,
3.3.9. Lemma. Let B be a subalgebra of an algebra B over a field C. Then the centralizer of B in the algebra B' ( x) is generated by the centralizer of B in B' and by the elements of type £oJ ~ a ~.ub., where ~
( a) ,( b) are dual bases of the conjugated left B-submodule A. in B' and the right B-submodule p in B', respectively. Proof. Let (a i) ,( b i) be dual bases of conjugated B-submodules in B' ba k
If
be B,
= L A. ki ai' i
bib
then
= L A. ~k b k'
ba i
E
A, bib E P
and,
hence,
Since the corresponding linear form is
k
associative, valid are the equalities A. ~k = (b i b,a k) = ( b i' ba R) hence,
= A ki'
and,
CHAPTER 3
151
i.e., the elements of the algebra centralizer of the subalgebra B.
B' ( X), used in the lemma, lie in the
Let us consider the inverse case. Let us fix a basis {f3 i} B. Then any element f of the free product is presented as
algebra
~~ (j) bo+~a.
. . ]
~
(j)
u.b. ] ~
of the
,
~
where u j are mutually different subwords of the basis words, which begin and end with x. If the element f lies in the centralizer of B, then for any bE B we have
This implies that if all the elements
oo(,j), a(,fJ, b(,j) b, b(.J') ~
~
~
~
are written
over the basis (f3 J, then in the obtained linear combination of basis words of the type f3 i1 x f3 i2 x_. x f3 im all the terms are cancelled. Since all u j are mutually different, then only the terms arising in the same square bracket can be cancelled. It means that the sums in the square brackets are zero. Therefore, lU belongs to the centralizer of B in B " and all the sums
La< ,j) u ,b( ,j) , J ~
~
commute with B, so the index j
can be henceforth
~
omitted.
n
Let
f =
L
a i ub i
' assuming the minimal possible number n in
i= 1
this presentation of the element f. In this case the elements {a J and {b J will be linearly independent over C, Let
P=
L b i c, A = La i C.
Let us define a non-degenerate bilinear
form on p x A, assuming that (b i' a j) = 0ij is a Kronecker symbol. The task now is to show that p is the right, and and that the form is associative,
A is the left B-submodule,
152
AUlOMORPlllSMS AND DERIVAnONS
of the
Let be B. Let us extend [ak)~=1 to the basis [a k) kE I algebra
B
,
n
and write the products
bai=
L
aika k , k= 1
1~
i~
basis. In the same way let us extend [b i) ~ = 1 to the basis [b i) m
write the products b k b
= L
a'ki b i ' i= 1
n
bf- fb=
i=lk=1
n
L
i,k=1
n
(a ik -
m
- L
~
k=li=n+1
n
aikakub i -
+
a'ki)akub i
a~i a k ub i
i
E I
and
1 ~ k ~ n. Then
m
L L
n in this
m
L L
a'kiakubi=
L
aikakub i -
k=li=1 n m
L
i=lk=n+1
(2)
=0
This means that if we decompose the left and write coefficients at u over the basis [ft i)' then, after cancelling the similar terms, all the terms should be cancelled, i.e., substituting in a formal way the letter u by the sign ® in equality (2), we get a correct equality in the tensor product B' ® B'. However, the elements a k ® b i' k, i e I form a basis of this tensor product, i.e., equality (2) shows that Il ik
=0
at
k
>
n;
Il ki
=0
at
i> n.
a~k = a ki
at 1 ~ k, i
~
nand
The latter two conditions are
equivalent to the fact that p is the right, and Il is the left B-submodule in , B , while the former is equivalent to associativity of the form n
(b k b,
o
a . ) = ( ~ a~ . b . , a . ) = a'k . = a. k = ~o
i =1
o~
~
~o
0 ~o
~o 0
n
= (b k ' ~ a i k o k= 1 0
a k) = (b k ,ba i ) . 0
0
The lemma is proved. Proof of theorem 3.3.7. Let B be a centralizable algebra. In the free product B ( x) let us consider an ideal R generated by the basis words which contain a variable x (i.e., R are polynomials without free terms with the coefficients from B of the noncommutable variable x). In
CHAPTER 3
153
this case R is a prime ring as an ideal of the prime B < x). Besides, ex R) = Q(B ( x») and, therefore, the generalized centroid is equal to C,
and
ex R)::1
B. If B has no nonzero conjugated one-sided ideals, then, by lemma
3.3.9, the centralizer of B in B ( X) is equal to the center of B which does not intersect with R, i.e., Z R(B ) = 0, which is a contradiction. Inversely, let B be a partially Frobenius algebra, and p, A, be its conjugated right and left ideals, respectively. In these ideals let us choose
p=
conjugated bases
m
L
b i C, A, =
i= 1
m
I,
a i C. Then, by lemma 3.3.9, the
i= 1
m
element
La. xb.
i= 1
~
~
commutes with the elements of B in B ( x).
B s;;; ex R) for a prime ring R with a generalized centroid c, then for any r E R the mapping x ~ r is extended to the homomorphism B ( X ) ~ Q, which is identical on B, i.e., the element If now
m
f( r) =
L
a i rb i lies in
ZQ(B).
i=1
Let
us
choose
an
a i IS;;; R, IbiS;;; R. Then
assume
f(I2) = 0.
elements t
j' v j E
f( r)
° of
Z R( B)
E
at
the r
ring
E ;.
In this case, by corollary 1.7.14,
r,
such
R,
Let one
us,
that
therefore,
can
choose
such that
ao = ~t j a l v j :t:. 0; ]
Hence,
.r :t:.
ideal
~t j
a kv j =
]
ao rbl = Lt j f( v j r) = 0,
i.e.,
°
at
k:t:.
ao RbI = 0,
1.
which contradicts the
primarity of R. The theorem is proved.
3.3.10. Theorem. Let B be a non-centralizable algebra, S be an arbitrary algebra over C. Then there is a prime ring R with a generalized centroid C, such that B
S;;;
ex R)
and Z R(B ) == S.
B' = B 0 c s', where S' is obtained from S by an external joining of a unit. Let us show that B' cannot contain conjugated nonzero finite-dimensional B-submodules. Since C is a field, then S' is a free C-module and, hence, B' is the free left and free right B-module. If now p, A, are a couple of nonzero Proof. Let us consider an algebra
154
AUTOMORPmSMS AND DERIVATIONS
,
conjugated B-submodules in B , then we can construct a homomorphism of
fIX. p) :t O. For the epimorphism -+ 0 we have a conjugated monomorphism 0 -+ q>( p)* -+ p* == A, can be considered to be a submodule of the module A. In the
right B-modules,
fIX. p) i.e., fIX. p)* p -+
fP: B' -+ B, such that
same way we can construct a homomorphism of left B-modules, VI: B' -+ B, lfI{ fIX. p)* ) :t O.
such that
In this case the module
[VI< q>(p )*) ]* can be
identif