Endomorphism Rings of Abelian Groups Algebras and Applications Volume 2 Editors: F. Van Oystaeyen University ofAntwe1p, UIA, Wilrijk, Belgium A. Verschoren University ofAntwe1p, 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. 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. A c.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-90-481-6349-6 ISBN 978-94-017-0345-1 (eBook) DOI 10.1007/978-94-017-0345-1 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. CONTENTS Preface Vll Symbols Xl Chapter I. General Results on Endomorphism Rings 1 1. Rings, Modules, and Categories 1 2. Abelian Groups 11 3. Examples and Some Properties of Endomorphism Rings 21 4. Torsion-Free Rings of Finite Rank 29 5. Quasi-Endomorphism Rings of Torsion-Free Groups 32 6. E-Modules and E-Rings 42 7. Torsion-Free Groups Coinciding with Their Pseudo-Socles 49 8. Irreducible Torsion-Free Groups 55 Chapter II. Groups as Modules over Their Endomorphism Rings 65 9. Endo-Artinian and Endo-Noetherian Groups 66 10. Endo-Flat Primary Groups 68 11. Endo-Finite Torsion-Free Groups of Finite Rank 73 12. Endo-Projective and Endo-Generator Torsion-Free Groups of Finite Rank 80 13. Endo-Flat Torsion-Free Groups of Finite Rank 89 Chapter III. Ring Properties of Endomorphism Rings 99 14. The Finite Topology 99 15. Endomorphism Rings with the Minimum Condition 103 16. Hom(A, B) as a Noetherian Module over End(B) 106 17. Mixed Groups with Noetherian Endomorphism Rings 112 18. Regular Endomorphism Rings 116 19. Commutative and Local Endomorphism Rings 121 Chapter IV. The Jacobson Radical of the Endomorphism Ring 135 20. The Case of p-groups 136 21. The Radical of the Endomorphism Ring of a Torsion-Free Group of Finite Rank 146 vi 22. The Radical of the Endomorphism Ring of Algebraically Compact and Completely Decomposable Torsion-Free Groups 162 23. The Nilpotence of the Radicals N(End(G)) and J(End(G)) 172 Chapter V. Isomorphism and Realization Theorems 181 24. The Baer-Kaplansky Theorem 183 25. Continuous and Discrete Isomorphisms of Endomorphism Rings 186 26. Endomorphism Rings of Groups with Large Divisible Subgroups 198 27. Endomorphism Rings of Mixed Groups of Torsion-Free Rank 1 206 28. The Corner Theorem on Split Realization 216 29. Realizations for Endomorphism Rings of Torsion-Free Groups 228 30. The Realization Problem for Endomorphism Rings of Mixed Groups 241 Chapter VI. Hereditary Endomorphism Rings 253 31. Self-Small Groups 254 32. Categories of Groups and Modules over Endomorphism Rings 263 33. Faithful Groups 280 34. Faithful Endo-Flat Groups 291 35. Groups with Right Hereditary Endomorphism Rings 312 36. Groups of Generalized Rank 1 331 37. Torsion-Free Groups with Hereditary Endomorphism Rings 344 38. Maximal Orders as Endomorphism Rings 350 39. p-Semisimple Groups 361 Chapter VII. Fully Transitive Groups -371 40. Homogeneous Fully Transitive Groups 372 41. Groups whose Quasi-Endomorphism Rings are Division Rings 381 42. Fully Transitive Groups Coinciding with Their Pseudo-Socles 386 43. Fully Transitive Groups with Restrictions on Element Types 391 44. Torsion-Free Groups of p-Ranks :::; 1 400 References 413 Index 440 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 endomor­ phism 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 develop­ ment 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 stu­ died 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 Mishi­ na [332]' Markov, Mikhalev, Skornyakov, and Tuganbaev [315]. The automor­ phism 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. Nev­ ertheless, 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, 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 endo­ morphism rings is studied, starting with its nil-radical and Jacobson radical. One of the main problems is reconstruction of the group from its endomor­ phism 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 endomor­ phism 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.