The Algebraic Structure of Group Rings

Overview


"Highly recommended" by the Bulletin of the London Mathematical Society, this comprehensive, self-contained treatment of group rings was written by an authority on the subject. Suitable for graduate students, it was hailed by the Bulletin of the American Mathematical Society as "a majestic account… encyclopedic and lucid."
The three-part survey begins with an introduction that defines the trace map, considers the augmentation ideal in considerable detail, and provides all the ...
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Overview


"Highly recommended" by the Bulletin of the London Mathematical Society, this comprehensive, self-contained treatment of group rings was written by an authority on the subject. Suitable for graduate students, it was hailed by the Bulletin of the American Mathematical Society as "a majestic account… encyclopedic and lucid."
The three-part survey begins with an introduction that defines the trace map, considers the augmentation ideal in considerable detail, and provides all the necessary group ring results for later characterizations of dimension subgroups. A second section on linear identities characterizes prime and semiprime group rings, brings semisimplicity considerations into play, and offers construction techniques for obtaining primitive group rings. The final part, an exploration of finiteness properties, consists chiefly of a study of Noetherian group rings. Hundreds of exercises of varying difficulty appear throughout the text.
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Product Details

  • ISBN-13: 9780486482064
  • Publisher: Dover Publications
  • Publication date: 9/14/2011
  • Series: Dover Books on Mathematics Series
  • Edition description: Reprint
  • Pages: 752
  • Sales rank: 434,948
  • Product dimensions: 6.10 (w) x 9.10 (h) x 1.30 (d)

Table of Contents

Part 1 Introduction

1 Group Rings 3

1 Definitions 3

2 Twisted Group Rings 13

3 Tensor Products 21

Exercises 28

2 The Trace Map 31

1 Complex Idempotents 31

2 Places 39

3 The Power Map 45

4 Finite Groups 55

Exercises 63

3 The Augmentation Ideal 67

1 Augmentation Annihilators 67

2 Injective Modules 74

3 Filtrations in Characteristic p 84

4 Filtrations in Characteristic 0 92

Exercises 105

Part 2 Linear Identities

4 The Center 113

1 Finite Conjugate Groups 113

2 The Basic Reductions 120

3 Idempotents and Annihilators 131

4 The Classical Ring of Quotients 146

5 The Maximal Ring of Quotients 155

Exercises 163

5 Polynomial Identities 168

1 Standard Identities 168

2 Subsets of Finite Index 180

3 Primitive P.I. Algebras 191

4 Central Polynomials 202

Exercises 214

6 Bounded Representation Degree 223

1 Large Centralizers 223

2 Finite Groups with r.b.n 234

3 Semiprime P.I. Algebras 245

4 Matrix Embedding 250

Exercises 262

7 The Semisimplicity Problem 268

1 Nondenurnerable Fields 268

2 Finite Extensions 273

3 Abelian Extensions and Separable Algebras 282

4 Semisimple Rings 291

Exercises 299

8 The Nilpotent Radical 304

1 The Controller Subgroup 304

2 The N*-radical 315

3 A Conjecture 322

4 Radical Ideals and Intersection Theorems 333

Exercises 348

9 Primitive Rings 353

1 Extensions and the Center 353

2 Some Basic Constructions 364

3 Normal Abelian Subgroups 378

4 Simple Augmentation Ideals 387

Exercises 396

Part 3 Finteness Properties

10 Chain Conditions 405

1 Artinian Rings 405

2 Noetherian Rings 419

3 Homological Dimension 430

4 Goldie Rings 451

Exercises 461

11 Nilpotent Groups 469

1 Dimension Subgroups 469

2 The Artin-Rees Property 485

3 Hypercentral Rings 500

4 Ideal Correspondence 516

Exercises 524

12 Finite Dimensional Modules 529

1 Solvable and Finite Linear Groups 529

2 Philip Hall's Problem 537

3 Some Residually Finite Groups 545

4 Finite Endomorphism Dimension 556

Exercises 575

13 The Zero Divisor Problem 582

1 Unique Product Groups 582

2 Ordered Groups 592

3 Supersolvable Groups 605

4 Polycyclic-by-finite Groups 616

Exercises 637

14 Isomorphism Questions 644

1 Central Subfields 644

2 Finite Metabelian Groups 660

3 Infinite Abelian Groups 675

4 Fields of the First Kind 688

Exercises 697

References 703

Index 713

Appendix 721

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