Front Cover 1
Ring Theory 4
Copyright Page 5
Table of Contents 6
Foreword to the Student Edition 16
Errata from the Original Two-Volume Edition 18
Introduction: An Overview of Ring Theory 20
Table of Principal Notation 26
Chapter 0. General Fundamentals 30
0.0 Preliminary Foundations 30
0.1 Categories of Rings and Modules 40
0.2 Finitely Generated Modules, Simple Modules, and Noetherian and Artinian Modules 43
Exercises 48
Chapter 1. Construction of Rings 50
1.1 Matrix Rings and Idempotents 50
1.2 Polynomial Rings 63
1.3 Free Modules and Rings 71
1.4 Products and Sums 82
1.5 Endomorphism Rings and the Regular Representation 91
1.6 Automorphisms, Derivations, and Skew Polynomial Rings 99
1.7 Tensor Products 111
1.8 Direct Limits and Inverse Limits 120
1.9 Graded Rings and Modules 126
1.10 Central Localization (also, cf., §2.12.9ff.) 130
Exercises 139
Chapter 2. Basic Structure Theory 148
2.1 Primitive Rings 149
2.2 The Chinese Remainder Theorem and Subdirect Products 161
2.3 Modules with Composition Series and Artinian Rings 164
2.4 Completely Reducible Modules and the Socle 175
2.5 The Jacobson Radical 178
2.6 Nilradicals 198
2.7 Semiprimary Rings and Their Generalizations 209
2.8 Projective Modules (An Introduction) 222
2.9 Indecomposable Modules and LE-Modules 233
2.10 Injective Modules 241
2.11 Exact Functors 250
2.12 The Prime Spectrum 258
Exercises 274
Chapter 3. Rings of Fractions and Embedding Theorems 300
3.1 Classical Rings of Fractions 301
3.2 Goldie's Theorems and Orders in Artinian Quotient Rings 309
3.3 Localization of Nonsingular Rings and Their Modules 327
3.4 Noncommutative Localization 334
3.5 Left Noetherian Rings 344
Exercises 369
Chapter 4. Categorical Aspects of Module Theory 386
4.1 The Morita Theorems 386
Exercises 394
Chapter 5. Homology and Cohomology 398
5.0 Preliminaries about Diagrams 398
5.1 Resolutions and Projective and Injective Dimension 404
5.2 Homology, Cohomology, and Derived Functors 423
5.3 Separable Algebras and Azumaya Algebras 444
Exercises 457
Chapter 6 Rings with Polynomial Identities and Affine Algebras 464
6.1 Rings with Polynomial Identities 465
6.2 Affine Algebras 486
6.3 Affine PI-Algebras 500
Exercises 519
Chapter 7. Central Simple Algebras 526
7.1 Structure of Central Simple Algebras 526
7.2 The Brauer Group 540
Exercises 544
Chapter 8. Rings from Representation Theory 548
8.1 General Structure Theory of Group Algebras 548
8.2 Noetherian Group Rings 562
8.3 Enveloping Algebras 572
8.4 General Ring Theoretic Methods 584
Exercises 605
Dimensions for Modules and Rings 612
Major Ring- and Module-Theoretic Results Proved in Volume I (Theorems and Counterexamples also cf. "Characterizations")
Major Theorems and Counterexamples for Volume II 622
The Basic Ring-Theoretic Notions and Their Characterizations 632
References 636
Index 644