The Algebraic Structure of Group Rings Book

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