Endomorphism Rings of Abelian Groups

by Krylov, Piotr A.; Mikhalev, Aleksandr Vasilevich; Tuganbaev, Askar A.

• ISBN13: 9781402014383
• ISBN10: 1402014384
• Format: Hardcover
• Copyright: 2003-08-01
• Publisher: Kluwer Academic Pub

Summary

This book is the first monograph on the theory of endomorphism rings of Abelian groups. The theory is a rapidly developing area of algebra and has its origin in the theory of operators of vector spaves. The text contains additional information on groups themselves, introducing new concepts, methods, and classes of groups. All the main fields of the theory of endomorphism rings of Abelian groups from early results to the most recent are covered. Neighbouring results on endomorphism rings of modules are also mentioned. This text has many pedagogical features: -all the necessary definitions and formulations of assertions on Abelian groups, rings, and modules are gathered in the first two sections; -each chapter begins with a brief summary of results; -there are exercises of varying difficulty in each section; -lesser known facts on rings and modules are presented with proofs; -there are comments at the end of each chapter together with a brief historical review as well as a look at the future direction of modern research; -an extensive bibliography is provided. This book will be invaluable as a background text for introductory as well as advanced graduate courses. Professional algebraists might find it useful as a first systematic presentation of results previously only to be found scattered throughout various journals.

Author Biography

Askar Tuganbaev received his Ph.D. at the Moscow State University in 1978 and has been a professor at Moscow Power Engineering Institute (Technological University) since 1978. He is the author of three other monographs on ring theory and has written numerous articles on ring theory.

Table of Contents

Preface	vii
Symbols	xi
Chapter I. General Results on Endomorphism Rings	1	(64)
1. Rings, Modules, and Categories	1	(10)
2. Abelian Groups	11	(10)
3. Examples and Some Properties of Endomorphism Rings	21	(8)
4. Torsion-Free Rings of Finite Rank	29	(3)
5. Quasi-Endomorphism Rings of Torsion-Free Groups	32	(10)
6. E-Modules and E-Rings	42	(7)
7. Torsion-Free Groups Coinciding with Their Pseudo-Socles	49	(6)
8. Irreducible Torsion-Free Groups	55	(10)
Chapter II. Groups as Modules over Their Endomorphism Rings	65	(34)
9. Endo-Artinian and Endo-Noetherian Groups	66	(2)
10. Endo-Flat Primary Groups	68	(5)
11. Endo-Finite Torsion-Free Groups of Finite Rank	73	(7)
12. Endo-Projective and Endo-Generator Torsion-Free Groups of Finite Rank	80	(9)
13. Endo-Flat Torsion-Free Groups of Finite Rank	89	(10)
Chapter III. Ring Properties of Endomorphism Rings	99	(36)
14. The Finite Topology	99	(4)
15. Endomorphism Rings with the Minimum Condition	103	(3)
16. Hom(A,B) as a Noetherian Module over End(B)	106	(6)
17. Mixed Groups with Noetherian Endomorphism Rings	112	(4)
18. Regular Endomorphism Rings	116	(5)
19. Commutative and Local Endomorphism Rings	121	(14)
Chapter IV. The Jacobson Radical of the Endomorphism Ring	135	(46)
20. The Case of p-groups	136	(10)
21. The Radical of the Endomorphism Ring of a Torsion-Free Group of Finite Rank	146	(16)
22. The Radical of the Endomorphism Ring of Algebraically Compact and Completely Decomposable Torsion-Free Groups	162	(10)
23. The Nilpotence of the Radicals N(End(G)) and J(End(G))	172	(9)
Chapter V. Isomorphism and Realization Theorems	181	(72)
24. The Baer-Kaplansky Theorem	183	(3)
25. Continuous and Discrete Isomorphisms of Endomorphism Rings	186	(12)
26. Endomorphism Rings of Groups with Large Divisible Subgroups	198	(8)
27. Endomorphism Rings of Mixed Groups of Torsion-Free Rank 1	206	(10)
28. The Corner Theorem on Split Realization	216	(12)
29. Realizations for Endomorphism Rings of Torsion-Free Groups	228	(13)
30. The Realization Problem for Endomorphism Rings of Mixed Groups	241	(12)
Chapter VI. Hereditary Endomorphism Rings	253	(118)
31. Self-Small Groups	254	(9)
32. Categories of Groups and Modules over Endomorphism Rings	263	(17)
33. Faithful Groups	280	(11)
34. Faithful Endo-Flat Groups	291	(21)
35. Groups with Right Hereditary Endomorphism Rings	312	(19)
36. Groups of Generalized Rank 1	331	(13)
37. Torsion-Free Groups with Hereditary Endomorphism Rings	344	(6)
38. Maximal Orders as Endomorphism Rings	350	(11)
39. p-Semisimple Groups	361	(10)
Chapter VII. Fully Transitive Groups	371	(42)
40. Homogeneous Fully Transitive Groups	372	(9)
41. Groups whose Quasi-Endomorphism Rings are Division Rings	381	(5)
42. Fully Transitive Groups Coinciding with Their Pseudo-Socles	386	(5)
43. Fully Transitive Groups with Restrictions on Element Types	391	(22)
44. Torsion-Free Groups of p-Ranks less than 1 400
References	413	(27)
Index	440