Endomorphism Rings of Abelian Groups

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1 Endomorphism Rings of Abelian Groups

2 Algebras and Applications Volume 2 Editors: F. Van Oystaeyen University of Antwe1p, UIA, Wilrijk, Belgium A. Verschoren University of Antwe1p, RUCA, Antwe1p, Belgium Advisory Board: M. Artin Massachusetts Institute of Technology Cambridge, MA, USA A. Bondal Moscow State University, Moscow, Russia I. Reiten Norwegian University of Science and Technology Trondheim, Norway The theory of rings, algebras and their representations has evolved into a well-defined subdiscipline of general algebra, combining its proper methodology with that of other disciplines and thus leading to a wide variety of applications ranging from algebraic geometry and number theory to theoretical physics and robotics. Due to this, many recent results in these domains were dispersed in the literature, making it very hard for researchers to keep track of recent developments. In order to remedy this, Algebras and Applications aims to publish carefully refereed monographs containing up-to-date information about progress in the field of algebras and their representations, their classical impact on geometry and algebraic topology and applications in related domains, such as physics or discrete mathematics. Particular emphasis will thus be put on the state-of-the-art topics including rings of differential operators, Lie algebras and super-algebras, groups rings and algebras, C* algebras, Hopf algebras and quantum groups, as well as their applications.

3 Endomorphism Rings of Abelian Groups by Piotr A. Krylov Tomsk State University, Tomsk, Russia Alexander V. Mikhalev Moscow State University, Moscow, Russia and Askar A. Tuganbaev Moscow Power Engineering Institute (Technological University), Moscow, Russia SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

4 A c.i.p. Catalogue record for this book is available from the Library of Congress. ISBN DOI / ISBN (ebook) Printed on acid-free paper All Rights Reserved 2003 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2003 Softcover reprint of the hardcover 1st edition 2003 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.

5 CONTENTS Preface Symbols Vll Xl Chapter I. General Results on Endomorphism Rings 1 1. Rings, Modules, and Categories 1 2. Abelian Groups Examples and Some Properties of Endomorphism Rings Torsion-Free Rings of Finite Rank Quasi-Endomorphism Rings of Torsion-Free Groups E-Modules and E-Rings Torsion-Free Groups Coinciding with Their Pseudo-Socles Irreducible Torsion-Free Groups 55 Chapter II. Groups as Modules over Their Endomorphism Rings Endo-Artinian and Endo-Noetherian Groups Endo-Flat Primary Groups Endo-Finite Torsion-Free Groups of Finite Rank Endo-Projective and Endo-Generator Torsion-Free Groups of Finite Rank Endo-Flat Torsion-Free Groups of Finite Rank 89 Chapter III. Ring Properties of Endomorphism Rings 14. The Finite Topology 15. Endomorphism Rings with the Minimum Condition 16. Hom(A, B) as a Noetherian Module over End(B) 17. Mixed Groups with Noetherian Endomorphism Rings 18. Regular Endomorphism Rings 19. Commutative and Local Endomorphism Rings Chapter IV. The Jacobson Radical of the Endomorphism Ring The Case of p-groups The Radical of the Endomorphism Ring of a Torsion-Free Group of Finite Rank 146

6 vi 22. The Radical of the Endomorphism Ring of Algebraically Compact and Completely Decomposable Torsion-Free Groups The Nilpotence of the Radicals N(End(G)) and J(End(G)) 172 Chapter V. Isomorphism and Realization Theorems The Baer-Kaplansky Theorem Continuous and Discrete Isomorphisms of Endomorphism Rings Endomorphism Rings of Groups with Large Divisible Subgroups Endomorphism Rings of Mixed Groups of Torsion-Free Rank The Corner Theorem on Split Realization Realizations for Endomorphism Rings of Torsion-Free Groups The Realization Problem for Endomorphism Rings of Mixed Groups 241 Chapter VI. Hereditary Endomorphism Rings Self-Small Groups Categories of Groups and Modules over Endomorphism Rings Faithful Groups Faithful Endo-Flat Groups Groups with Right Hereditary Endomorphism Rings Groups of Generalized Rank Torsion-Free Groups with Hereditary Endomorphism Rings Maximal Orders as Endomorphism Rings p-semisimple Groups 361 Chapter VII. Fully Transitive Groups Homogeneous Fully Transitive Groups Groups whose Quasi-Endomorphism Rings are Division Rings Fully Transitive Groups Coinciding with Their Pseudo-Socles Fully Transitive Groups with Restrictions on Element Types Torsion-Free Groups of p-ranks :::; 1 References Index

7 PREFACE Every Abelian group can be related to an associative ring with an identity element, the ring of all its endomorphisms. Recently the theory of endomorphism rings of Abelian groups has become a rapidly developing area of algebra. On the one hand, it can be considered as a part of the theory of Abelian groups; on the other hand, the theory can be considered as a branch of the theory of endomorphism rings of modules and the representation theory of rings. There are several reasons for studying endomorphism rings of Abelian groups: first, it makes it possible to acquire additional information about Abelian groups themselves, to introduce new concepts and methods, and to find new interesting classes of groups; second, it stimulates further development of the theory of modules and their endomorphism rings. The theory of endomorphism rings can also be useful for studies of the structure of additive groups of rings, E-modules, and homological properties of Abelian groups. The books of Baer [52] and Kaplansky [245] have played an important role in the early development of the theory of endomorphism rings of Abelian groups and modules. Endomorphism rings of Abelian groups are much studied in monographs of Fuchs [170], [172], and [173]. Endomorphism rings are also studied in the works of Kurosh [287], Arnold [31], and Benabdallah [63]. Various results about endomorphism rings of modules can be found in the books of Anderson and Fuller [27], Facchini [148], Auslander, Reiten, and Small/l[45], Harada [211], Lambek [289], Faith [150], [151], Kasch [248], and Tuganbaev [439], [444]. Achievements in this field are reported in reviews of Mishina [334], [335], [337], [338], [339], Mikhalev [329], Mikhalev and Mishina [332]' Markov, Mikhalev, Skornyakov, and Tuganbaev [315]. The automorphism groups of Abelian groups (that is, the groups of invertible elements of endomorphism rings) are studied in the book of Bekker and Kozhukhov [61]. The present book is entirely devoted to endomorphism rings of Abelian groups. The authors have deliberately imposed such restrictions, being sure that the subject of endomorphism rings of Abelian groups is an object on its own, and also that the theory of endomorphism rings of Abelian groups is an excellent introduction to the general theory of endomorphism rings of modules. Nevertheless, sometimes we mention neighbouring results about endomorphism rings of modules. The authors hope that their book will stimulate further development of the theory of endomorphism rings. We have tried to discuss all major parts of this area of algebra thoroughly enough to estimate its value, the variety of methods, the beauty of results, and the measure of difficulty of open problems. Contributions to the theory of endomorphism rings of Abelian groups in the early stage were made by Baer, Corner, Fuchs, Kaplansky, Kulikov, Kurosh,

8 viii Pierce, Reid, Richman, Szele, and Walker. Further development ofthe field has been achieved in works of Albrecht, Arnold, Dugas, Faticoni, Gobel, Goeters, Goldsmith, Hausen, Lady, Liebert, May, Murley, Mutzbauer, Rangaswamy, Schultz, Shelah, Vinsonhaler, Warfield, Wickless, and others. Endomorphism rings of Abelian groups often surprise us. It is hard to decide which methods do prevail here: those of group theory or those of ring theory? Various modules over associative rings are considered. Category methods and topological considerations playa great role in this theory. The main challenge of this theory is to discover connections between properties of a given Abelian group A and properties of its endomorphism ring End(A). This task is very extensive. We can impose various restrictions over the ring End(A) and try to obtain information about the group A itself. The internal structure of endomorphism rings is studied, starting with its nil-radical and Jacobson radical. One of the main problems is reconstruction of the group from its endomorphism ring. In other words, the problem is to what degree the endomorphisms determine the underlying group. The results connected with this problem, we call the isomorphism theorems. Another fundamental problem with endomorphism rings is to find criteria for an abstract ring to be the endomorphism ring of some Abelian group. The corresponding theorems are realization theorems. Any Abelian group A can be naturally considered as a module over its endomorphism ring End( A). So we obtain one more important object of study, the associated module End(A)A. Much attention is devoted to the groups with manyendomorphisms. These problems make up the main content of the book. Thus we can assume that the most part of the book is devoted to the relations between objects mentioned above: an Abelian group A, its endomorphism ring End(A), and the module End(A)A. As a result of the efforts of numerous mathematicians, we can now show these relations, not seen at first sight, in all their variety. The culmination is reached in Chapter 6, which is the highest pinnacle of the book. There intersect all the main themes of the preceding chapters. All the main fields of the theory of endomorphism rings have their part in this book. It contains early results as well as recent ones. The reader comes up to recent frontiers of research. The results included in the book were chosen up to the taste of the authors. This can be justified by proofs of some deep theorems not included in the book being based upon rather special results about Abelian groups, rings, and modules. Also excluded were some important realization theorems of the 1980's: those of Gobel, Dugas, Corner, Shelah, and other authors (with the small exception of Section 30) whose proofs essentially use technically complicated set-theoretic methods. Together with their various applications, these results constitute an independent and far advanced field based on the application of set-theoretic methods in the theory of Abelian groups and homological algebra. This field lies outside of scope of

9 this book and certainly deserves special consideration. Such a consideration is partly given in the book of Eklof [144]. While preparing this manuscript, the authors have met one serious difficulty. The topic of the book is, as stated above, the middle ground between the theory of Abelian groups and ring theory. Not surprisingly, proofs of many theorems require various results about Abelian groups, rings, and modules. In addition certain elements of topological algebra and category theory are used. An attempt to make the book self-contained would have most certainly failed. In any case it would have led to an inappropriate increase in size. The authors have chosen another route. For the convenience of readers all the necessary definitions and formulations of assertions about Abelian groups, rings, and modules are gathered in the first two sections. At the beginning of every chapter (and sometimes also at the beginning of a section) we also give (possibly repeating something already said) the necessary definitions and results. As a rule, they are standard and quite well known and are contained, for example in one of the books listed below. The monograph of Fuchs Infinite Abelian groups (Fuchs [172] and [173]) is a celebrated manual of the theory of Abelian groups; in addition the book of Arnold [31] might be useful. For results about rings and modules we recommend the following books: Jacobson, Structure of Rings (Jacobson [240]), Faith Algebra: Rings, Modules, and Categories, I, Faith, Algebra II, Ring Theory (Faith [150] and [151]), Lambek Lectures on Rings and Modules (Lambek [289]), and Kasch Modules and Rings (Kasch [248]). As a rule, less familiar results taken from books and journal papers on rings and modules are presented with proofs. Inside proofs we usually avoid references to journal papers. This book is written for a reader somewhat familiar with the foundations of Abelian groups, rings, and modules. It can be used as a background text for introductory and advanced graduate courses. Professional algebraists might find it useful as a first systematic presentation of results scattered through various journal papers. The authors hope that the book will also be of interest to a wide audience of mathematicians. The introduction to each chapter contains a brief summary of the results. All sections contain exercises of varying difficulty. Some of these are results from journal papers, proofs of which are omitted in the text for some reason. At the end of each chapter comments are given together with a brief historical review. Also we recount some directions of modern research, additional results, and extensions of the chapter's results to modules. Some extended sections contain comments as well. We also single out a number of unsolved problems. Some of these are well known problems, whilis others are formulated for the first time. The authors have put their best efforts into compiling a comprehensive bibliography, although the resulting list is far from being complete. ix

10 x We accept the Zermelo-Fraenkel axiomatic set ZFC of set theory (including the choice axiom and the Zorn lemma). The terms 'class' and 'set' are used in the ordinary set-theoretic sense.

11 SYMBOLS Al E9.. E9 An ASJ! or ~A SJ! Ab A[n] the (finite) direct sum of the modules All.. I An the direct sum of m copies of the module A the category of all Abelian groups the subset {a E A I na = O} of an Abelian group A a:n an element a of an Abelian group A is divisible by an integer n Ap the p-component of an Abelian group A.4p the field of p-adic numbers Aut(G) the automorphism group of the group G BiendR(M) the bi-endomorphism ring of an R-module M End(A) the endomorphism ring of an Abelian group A QEnd(A) the quasi-endomorphism ring of a torsion-free Abelian group A EndR(M) the endomorphism ring of an R-module M (R) the class of all E(R)-groups H(a) the height matrix of an element a h(a) the height of an element a XA(a) or x(a) the characteristic of an element a of A h:(a) or hp(a) the p-height of an element a h;(a) the generalized p-height of an element a Jp J(R) the Jacobson radical of a ring R (M) mod-r N N(R) the group of p-adic integers the subgroup generated by a subset M of a group the category of all right R-modules the set of all positive integers the nil-radical of a ring R

12 xii SYMBOLS o(a) p I1(A) II Ai iei Q Qp Q; R+ r(a) the order of an element a a prime integer the set of all prime p with pa # A direct product of the modules Ai the field or the group of rational numbers the ring of all rational numbers with denominators coprime to p n Rl X... X Rn or II Ri i=l SA(B) Sing(M) Soc A T(A) ta(a) or t(a) t(a) T(A) Walk Z Z (n) Zn Zp or Fp Z(R) the ring of p-adic integers the additive group of a ring R the rank of a torsion-free Abelian group A the ring of all n X n matrices over the ring R the p-rank of an Abelian group A the product of the rings R 1,., Rn the trace of an Abelian group A in an Abelian group B the singular submodule the pseudo-socle of an Abelian group A the direct sum of the modules Ai (i E I) the torsion part (the torsion subgroup) of an Abelian group A the type of an element a of A the type of a homogeneous torsion-free Abelian group A the set of the types of all nonzero elements of A the Walker category the ring or the group of integers the cyclic group of order n the residue ring modulo n the residue field modulo p the center ofthe ring R

Exercises in Basic Ring Theory

Exercises in Basic Ring Theory Kluwer Texts in the Mathematical Sciences VOLUME 20 A Graduate-Level Book Series The titles published in this series are listed at the end of this volume. Exercises in Basic