Ring Theory V1 (eBook)

Front Cover 1
Ring Theory 4
Copyright Page 5
Contents 6
Foreword 14
Introduction: An Overview of Ring Theory 18
Table of Principal Notation 24
Chapter 0. General Fundamentals 26
0.0 Preliminary Foundations 26
0.1 Categories of Rings and Modules 38
0.2 Finitely Generated Modules, Simple Modules, and Noetherian and Artinian Modules 43
0.3 Abstract Dependence 48
Exercises 51
Chapter 1. Construction of Rings 54
1.1 Matrix Rings and Idempotents 54
1.2 Polynomial Rings 67
1.3 Free Modules and Rings 78
1.4 Products and Sums 91
1.5 Endomorphism Rings and the Regular Representation 106
1.6 Automorphisms, Derivations, and Skew Polynomial Rings 113
1.7 Tensor Products 126
1.8 Direct Limits and Inverse Limits 136
1.9 Graded Rings and Modules 143
1.10 Central Localization (also, cf. §2.12.9ff.) 154
Exercises 162
Chapter 2. Basic Structure Theory 174
2.1 Primitive Rings 175
2.2 The Chinese Remainder Theorem and Subdirect Products 187
2.3 Modules with Composition Series and Artinian Rings 190
2.4 Completely Reducible Modules and the Socle 201
2.5 The Jacobson Radical 204
2.6 Nilradicals 224
2.7 Semiprimary Rings and Their Generalizations 236
2.8 Projective Modules (An Introduction) 250
2.9 Indecomposable Modules and LE-Modules 262
2.10 Injective Modules 286
2.11 Exact Functors 295
2.12 The Prime Spectrum 305
2.13 Rings with Involution 321
Exercises 333
Chapter 3. Rings of Fractions and Embedding Theorems 372
3.1 Classical Rings of Fractions 373
3.2 Goldie’s Theorems and Orders in Artinian Quotient Rings 381
3.3 Localization of Nonsingular Rings and Their Modules 402
3.4 Noncommutative Localization 409
3.5 Left Noetherian Rings 419
Exercises 468
Chapter 4. Categorical Aspects of Module Theory 492
4.1 The Morita Theorems 492
4.2 Adjoints 502
Exercises 506
Appendix A. The Proof of Magnus’ Theorem and the Magnus-Witt Theorem (Theorem 1.3.38) 510
Exercise 515
Appendix B. Normed Algebras and Banach Algebras 515
Exercises 518
The Basic Ring-Theoretic Notions and Their Characterizations 520
Major Ring- and Module-Theoretic Results Proved in Volume I (Theorems and Counterexamples also cf. “Characterizations”)
References 532
Subject Index 556
Pure and Applied Mathematics 564