This book on modern module and non-commutative ring theory starts at the foundations of the subject and progresses rapidly through the basic concepts to help the reader reach current research frontiers. The first half of the book is concerned with free, projective, and injective modules, tensor algebras, simple modules and primitive rings, the Jacobson radical, and subdirect products. Later in the book, more advanced topics, such as hereditary rings, categories and functors, flat modules, and purity are introduced. These later chapters will also prove a useful reference for researchers in non-commutative ring theory.
Author(s): John Dauns
Publisher: CUP
Year: 1994
Language: English
Pages: 460
Tags: Математика;Общая алгебра;Теория колец;
Cover......Page 1
Title page......Page 3
Copyright page......Page 4
Contents......Page 5
PREFACE......Page 11
NOTE TO THE READER......Page 17
1-1 Definitions......Page 19
1-2 Direct Products and Sums......Page 23
1-3 Adjunction of 1 to $R$......Page 26
1-4 Sequences of Modules......Page 28
1-5 Exercises......Page 30
Introduction......Page 37
2-1 Definition of Free Modules......Page 38
2-2 Bases of Free Modules......Page 42
2-3 Exercises......Page 47
3-1 Properties of Injectives......Page 48
3-2 Divisibility......Page 51
3-3 Embeddings in Injectives......Page 54
3-4 Injective Hulls......Page 57
3-5 Noetherian Rings......Page 63
3-6 Examples......Page 65
3-7 Exercises......Page 69
4-1 Tensor Products of Modules......Page 71
4-2 Definitions for Algebras......Page 78
4-3 Tensor Products of Algebras......Page 81
4-4 Exercises......Page 87
Introduction......Page 89
5-1 Free and Tensor Algebras......Page 90
5-2 Exterior Algebras......Page 92
5-3 Exercises......Page 101
Introduction......Page 104
6-1 Preliminaries......Page 105
6-2 Cyclic Modules......Page 108
6-3 Simple Modules......Page 109
6-4 Examples......Page 112
6-5 Density......Page 113
6-6 More on Density and Simples......Page 117
6-7 Examples......Page 123
6-8 Exercises......Page 127
Introduction......Page 129
7-1 Characterizations......Page 130
7-2 Radicals of Related Rings......Page 143
7-3 Local Rings......Page 150
7-4 Examples......Page 153
7-5 Exercises......Page 156
Introduction......Page 158
8-1 Subdirect Products......Page 159
8-2 Dense Subdirect Products......Page 163
8-3 Exercises......Page 165
Introduction......Page 166
9-1 Prime Ideals......Page 167
9-2 Semiprime Ideals and the Prime Radical......Page 169
9-3 Nil Radicals......Page 174
9-4 Primes and Semiprimes in Derived Rings......Page 176
9-5 Exercises......Page 180
Introduction......Page 181
10-1 Projective Modules......Page 183
10-2 Projective Dimension......Page 190
10-3 Minimal Right Ideals......Page 195
10-4 Main Theorems......Page 198
10-5 Direct Proofs......Page 202
10-6 Uniqueness......Page 208
10-7 Rings with D.C.C. and Idempotents......Page 209
10-8 Exercises......Page 214
Introduction......Page 222
11-1 Completely Reducible Modules......Page 224
11-2 Radical of a Module......Page 227
11-3 Artinian and Noetherian Modules......Page 231
11-4 Direct Sums of Indecomposables......Page 239
11-5 Singular Submodule......Page 249
11-6 Exercises......Page 252
Introduction......Page 257
12-1 Algebra Modules......Page 258
12-2 Multiplication Algebra......Page 259
12-3 Tensor Products of Simple Rings......Page 264
12-4 Centralizers......Page 268
12-5 Double Centralizers......Page 277
12-6 Exercises......Page 284
13-1 Hereditary Rings......Page 287
13-2 Injectivity and Projectivity......Page 290
13-3 Finitely Generated Modules......Page 293
13-4 Examples......Page 297
13-5 Exercises......Page 299
Introduction......Page 301
14-1 Pullbacks......Page 302
14-2 Pushouts......Page 307
14-3 Pushout Application......Page 311
14-4 Exercises......Page 313
15-1 Basics of Categories......Page 316
15-2 Objects......Page 330
15-3 Pre-additive Categories......Page 333
15-4 Adjoint Functors......Page 343
15-5 Exercises......Page 351
16-1 Generators and Cogenerators......Page 353
16-2 Horn Functor......Page 356
16-3 Tensor Product Functor......Page 357
16-4 Adjoint Associativity......Page 360
16-5 Elements of Tensor Products......Page 363
16-6 Direct and Inverse Limits......Page 365
16-7 Exercises......Page 370
16-8 Exercises on direct and inverse limits......Page 373
17-1 Character Modules......Page 377
17-2 Flat Module Basics......Page 380
17-3 Exercises......Page 383
Introduction......Page 385
18-1 Systems of Equations in Modules......Page 386
18-2 Pure Projectives and Pure Exact Sequences......Page 388
18-3 Direct Limits......Page 397
18-4 Pure Injectives......Page 401
18-5 Pure Injective Hull......Page 409
18-6 Exercises......Page 414
A-1 Sets, Symbols, and Functions......Page 416
A-2 Background Review......Page 423
A-3 Exercises......Page 426
B-2 Exterior Algebras......Page 430
B-3 A Unified Approach......Page 433
LIST OF SYMBOLS AND NOTATION......Page 445
BIBLIOGRAPHY......Page 450
SUBJECT INDEX......Page 454
AUTHOR INDEX......Page 460
