Ring Theory, Volume 1 (Pure and Applied Mathematics (Academic Press), Volume 127)

569 Pages·1988·25.88 MB·English

by  Louis Halle Rowen

Ring Theory
Volume I
This is volume 127 in
PURE AND APPLIED MATHEMATICS
H. Bass and S. Eilenberg, editors
A list of titles in this series appears at the end of this volume.
Ring Theory
Volume I
Louis H. Rowen
Department of Mathematics and Computer Science
Bar IIan University
Ramat Can, Israel
ACADEMIC PRESS, INC.
Harcourt Brace Jovanovich, Publishers
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Berkeley London Sydney
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Copyright 0 1988 by Academic Press, Inc.
All rights reserved.
No part of this publication may be reproduced or
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ACADEMIC PRESS, INC.
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United Kingdom Edition published by
ACADEMIC PRESS, INC. (LONDON) LTD.
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Library of Congress Cataloging-in-Publication Data
Rowen, Louis Halle.
Ring theory.
(Pure and applied mathematics; v. 127-128)
Includes bibliographies and indexes.
1. Rings (Algebra) I. Title. 11. Series: Pure and
applied mathematics (Academic Press) ; 127-128.
QA3.P8 VOI. 127-128 510 s [512’.4] 87-14536
CQA2471
ISBN 0-12-599841-4 (v. 1)
ISBN 0-12-599842-2( v. 2)
88899091 9 8 7 6 5 4 3 2 1
Printed in the United States of America
Contents
...
Foreword Xlll
Introduction: An Overview of Ring Theory xvii
Table of Principal Notation xxiii
Chapter 0 General Fundamentals
0.0 Preliminary Foundations 1
Monoids and Groups 1
Rings and Modules 2
Algebras 5
Preorders and Posets 6
Upper and Lower Bounds 6
Lattices 7
Modular Lattices 8
Zorn’s Lemma 10
Well-ordered Sets and Transfinite Induction 11
Fields 12
First Order Logic 12
0.1 Categories of Rings and Modules 13
Monics and Epics 15
Functors 16
0.2 Finitely Generated Modules, Simple Modules, and
Noetherian and Artinian Modules 18
Finitely Generated Modules 18
V
vi Contents
Cyclic Modules 19
Simple Rings and Modules 19
Chain Conditions 21
Noetherian and Artinian Modules 22
0.3 Abstract Dependence 23
Algebraic and Transcendental Elements 25
Exercises 26
Chapter 1 Construction of Rings
1.1 Matrix Rings and Idempotents 29
Matrices and Matrix Units 29
Subrings of Matrix Rings 32
Matrices Whose Entries Are Not Necessarily in Rings 33
Idempotents and the Peirce Decomposition 35
Idempotents and Simple Modules 37
Primitive Idempotents 39
Lifting Matrix Units and Idempotents 39
1.2 Polynomial Rings 42
Monoid Rings 42
Polynomial Rings 43
Rings of Formal Power Series and of Laurent Series 48
Digression: More General Constructions 51
A Supplement: Ordered Groups 51
1.3 Free Modules and Rings 53
Free Modules 54
Independent Modules and Direct Sums 54
Modules Over Division Rings Are Free 55
Free Objects 56
Free Rings and Algebras 57
Free Commutative Ring 59
Diagrams 60
Universals 60
Invariant Base Number 61
Weakly Finite Rings 63
A Supplement: The Free Group 64
1.4 Products and Sums 66
Direct Products and Direct Sums 66
Contents vii
Exact Sequences 67
Split Exact Sequences 68
Reduced Products 69
Ultraproducts 71
Products and Coproducts 73
B Supplement: Free Products and Amalgamated Sums 76
C Supplement: Categorical Properties of Modules:
Abelian Categories 78
1.5 Endomorphism Rings and the Regular Representation 81
Endomorphism Rings 81
Endomorphisms as Matrices 84
The Dual Base 86
Adjunction of 1 87
1.6 Automorphisms, Derivations, and Skew Polynomial Rings 88
Automorphisms 88
Derivations, Commutators, and Lie Algebras 89
Skew Polynomial Rings and Ore Extensions 92
Principal Left Ideal Domains (PLID’s) 93
Skew Polynomial Rings (Without Derivation) Over Fields 96
Differential Polynomial Rings Over Fields 97
The Weyl Algebra 98
Skew Power Series and Skew Laurent Series 100
Skew Group Rings 101
1.7 Tensor Products 101
Tensor Products of Bimodules and of Algebras 103
Properties of the Tensor Operation 105
Tensors and Centralizing Extensions 106
Tensor Products Over Fields 108
Tensor Products and Bimodules 110
1.8 Direct Limits and Inverse Limits 111
Direct Limits 112
D Supplement: Projective Limits 115
The Completion 116
1.9 Graded Rings and Modules 118
Tensor Rings 121
B Supplement: Constructing Free Products 124
viii Contents
1.10 Central Localization (also, cf. 92.12.9ff.) 129
Structure Passing From R to S-’R 132
Examples of Central Localization 134
Central Localization of Modules 134
Exercises 137
Chapter 2 Basic Structure Theory
2.1 Primitive Rings 150
Jacobson’s Density Theorem 151
Prime Rings and Minimal Left Ideals 153
Finite-Ranked Transformations and the Socle 154
Examples of Primitive Rings 158
E Supplement: A Right Primitive Ring Which Is Not Primitive 159
2.2 The Chinese Remainder Theorem and Subdirect Products 162
Subdirect Products 163
Semiprime Rings 164
2.3 Modules with Composition Series and Artinian Rings 165
The Jordan-Holder and Schreier Theorems 165
Artinian Rings 167
Split Semisimple Artinian Algebras 171
Central Simple Algebras and Splitting 173
Zariski Topology (for Finite Dimensional Algebras) 174
2.4 Completely Reducible Modules and the Socle 176
2.5 The Jacobson Radical 179
Quasi-Invertibility 180
Examples 182
Idempotents and the Jacobson Radical 183
Weak Nullstellensatz and the Jacobson Radical 184
The Structure Theoretical Approach to Rings 186
“Nakayama’s Lemma” 188
The Radical of a Module 189
F Supplement: Finer Results Concerning the
Jacobson Radical 189
Wedderburn’s Principal Theorem 192
F Supplement: Amitsur’s Theorem and Graded Rings 194
F Supplement: The Jacobson Radical of a Rng 196
Galois Theory of Rings 198
Contents ix
2.6 Nilradicals 199
Reduced Rings 20 1
Nilpotent Ideals and Nilradicals 202
The Nilradical of Noetherian Rings 205
Bounded Index 206
Derivations and Nilradicals 207
Nil Subsets 207
G Supplement: Koethe’s Conjecture 209
2.7 Semiprimary Rings and Their Generalizations 21 1
Chain Conditions on Principal Left Ideals 212
Passing from R to eRe and Back 215
Semiperfect Rings 217
Structure of Idempotents of Rings 219
H Supplement: Perfect Rings 22 1
G Supplement: Left x-Regular Rings 224
2.8 Projective Modules (An Introduction) 225
Projective Modules 225
Projective Versus Free 227
Hereditary Rings 228
Dual Basis Lemma 229
Flat Modules 230
Schanuel’s Lemma and Finitely Presented Modules 232
H Supplement: Projective Covers 233
Categorical Description of Projective Covers 236
2.9 Indecomposable Modules and LE-Modules 237
The Krull-Schmidt Decomposition 238
Uniqueness of the Krull-Schmidt Decomposition 240
Applications of the Krull-Schmidt Theorem to
Semiperfect Rings 242
Decompositions of Modules Over Noetherian Rings 244
I Supplement: Levy’s Counterexample 245
C Supplement: Representation Theory 249
The Brauer-Thrall Conjectures 25 1
Quivers and f.r.t. 259
The Multiplicative Basis Theorem and the
Classification Problem 260
The Categorical Language 26 1