Foundations of Commutative Rings and Their Modules (eBook)

Preface 8
Contents 10
Symbols 19
1 Basic Theory of Rings and Modules 21
1.1 Basic Concepts of Rings and Modules 21
1.1.1 Rings and Ideals 21
1.1.2 Basic Concepts of Modules 23
1.1.3 Direct Product of Rings, Direct Product and Direct Sum of Modules 26
1.2 Ring Homomorphisms and Module Homomorphisms 27
1.2.1 Ring Homomorphisms 27
1.2.2 Module Homomorphisms 30
1.3 Finitely Generated Modules and Matrix Methods 34
1.3.1 Finitely Generated Modules 34
1.3.2 Simple Modules, Maximal Submodules, and Zorn's Lemma 36
1.3.3 Jacobson Radical of a Ring 38
1.3.4 Matrix Methods 38
1.4 Prime Ideals and Nil Radical 42
1.4.1 Prime Ideals 42
1.4.2 Nil Radical and Radical of an Ideal 44
1.5 Quotient Rings and Quotient Modules 45
1.5.1 Local Rings 45
1.5.2 Quotient Rings 46
1.5.3 Quotient Modules 48
1.6 Free Modules, Torsion Modules, and Torsion-Free Modules 53
1.6.1 Free Modules 53
1.6.2 Torsion Modules, Torsion-Free Modules, and Divisible Modules 56
1.7 Polynomial Rings and Power Series Rings 57
1.7.1 Polynomial Rings over One Indeterminate 57
1.7.2 Polynomials with Coefficients in a Module 61
1.7.3 Dedekind--Mertens Formula 62
1.7.4 Polynomial Rings over Many Indeterminates and Formal Power Series Rings over One Indeterminate 65
1.8 Krull Dimension of a Ring 67
1.8.1 Basic Properties of Krull Dimension of a Ring 67
1.8.2 Krull Dimension of a Polynomial Ring 68
1.8.3 Connected Rings 70
1.9 Exact Sequences and Commutative Diagrams 71
1.9.1 Exact Sequences 72
1.9.2 Five Lemma and Snake Lemma 73
1.9.3 Completion of Diagrams 79
1.9.4 Pushout and Pullback 81
1.10 Exercises 83
2 The Category of Modules 91
2.1 The Functor Hom 91
2.1.1 Categories 91
2.1.2 Functors 94
2.1.3 Basic Properties of the Functor Hom 95
2.1.4 Natural Transforms of Functors 97
2.1.5 Torsionless Modules and Reflexive Modules 98
2.2 The Functor otimes 100
2.2.1 Bilinear Mappings and Tensor Products 100
2.2.2 Basic Properties of the Functor otimes 101
2.2.3 Change of Rings and Adjoint Isomorphism Theorem 103
2.2.4 Tensor Product and Localization 105
2.3 Projective Modules 106
2.3.1 Projective Modules 107
2.3.2 Kaplansky Theorem 110
2.4 Injective Modules 113
2.4.1 Injective Modules 113
2.4.2 Injective Envelope of a Module 117
2.5 Flat Modules 120
2.5.1 Flat Modules and Their Characterizations 120
2.5.2 Faithfully Flat Modules 126
2.5.3 Direct Limits 129
2.6 Finitely Presented Modules 134
2.6.1 Finitely Presented Modules 134
2.6.2 Isomorphism Theorems Related to Hom and otimes 139
2.7 Superfluous Submodules and Projective Covers 149
2.7.1 Jacobson Radical of a Module and Superfluous Submodules 149
2.7.2 Projective Cover of a Module 150
2.8 Noetherian Modules and Artinian Modules 153
2.8.1 Noetherian Modules and Noetherian Rings 153
2.8.2 Artinian Modules and Artinian Rings 155
2.9 Semisimple Modules and Composition Series 157
2.9.1 Semisimple Modules 157
2.9.2 Composition Series 158
2.10 Exercises 161
3 Homological Methods 167
3.1 Complexes and Homologies 167
3.1.1 Complexes and Complex Morphisms 167
3.1.2 Homology Modules 169
3.1.3 Homotopy 173
3.2 Derived Functors 175
3.2.1 Comparison Theorems 175
3.2.2 Left Derived Functors 177
3.2.3 Right Derived Functors 182
3.3 Derived Functor Ext 185
3.3.1 Properties of Ext 185
3.3.2 Dimension-Shifting Method and Isomorphism Theorems Related to Ext 189
3.4 Derived Functor Tor 191
3.4.1 Properties of Tor 191
3.4.2 Isomorphism Theorems Related to Tor 195
3.5 Projective Dimension and Injective Dimension of a Module ƒ 197
3.5.1 Projective Dimension of a Module 197
3.5.2 Injective Dimension of a Module 199
3.5.3 Global Dimension of a Ring and Semisimple Rings 200
3.6 Flat Dimension of a Module and Weak Global Dimension of a Ring 202
3.6.1 Flat Dimension of a Module 202
3.6.2 Weak Global Dimension of a Ring 204
3.6.3 von Neumann Regular Rings 205
3.7 Coherent Rings, Semihereditary Rings, and Hereditary Rings 206
3.7.1 Coherent Rings 206
3.7.2 Semihereditary Rings and Prüfer Domains 208
3.7.3 Valuation Domains 210
3.7.4 Hereditary Rings 212
3.8 Change of Rings Theorems 213
3.8.1 Several Dimension Inequalities 214
3.8.2 Rees Theorem and Homological Dimension of a Factor Ring 216
3.8.3 Homological Dimension of a Polynomial Ring 220
3.9 Homological Methods in Coherent Rings 223
3.10 Finitistic Dimension of a Ring and Perfect Rings 229
3.10.1 Finitistic Dimension and Small Finitistic Dimension of a Ring 229
3.10.2 Semiperfect Rings 232
3.10.3 Perfect Rings 233
3.11 Exercises 239
4 Basic Theory of Noetherian Rings 244
4.1 Artinian Rings 244
4.1.1 Semilocal Rings 244
4.1.2 Basic Properties of Artinian Rings 246
4.2 Associated Prime Ideals and Primary Decompositions 248
4.2.1 Associated Prime Ideals 248
4.2.2 Primary Ideals and Primary Submodules 250
4.2.3 Primary Decomposition 252
4.3 Several Classical Theorems 255
4.3.1 Injective Modules over Noetherian Rings 255
4.3.2 Krull's Principal Ideal Theorem 257
4.3.3 Hilbert Basis Theorem 260
4.3.4 Krull--Akizuki Theorem 263
4.4 Systems of Parameters and Regular Sequences 267
4.4.1 Systems of Parameters 267
4.4.2 Regular Sequences 269
4.4.3 Auslander--Buchsbaum Theorem 273
4.5 Regular Local Rings 275
4.5.1 Definition and Properties of Regular Local Rings 275
4.5.2 Finite Free Resolutions 277
4.5.3 Characterizations of Regular Local Rings 280
4.6 Gorenstein Rings 281
4.6.1 QF Rings 281
4.6.2 n-Gorenstein Rings 285
4.7 Exercises 288
5 Extensions of Rings 291
5.1 Integral Dependence 291
5.1.1 Integral Extensions 291
5.1.2 GCD Domains and UFDs 295
5.1.3 Integrally Closed Domains 297
5.2 Dedekind Domains 300
5.2.1 Fractional Ideals 300
5.2.2 Discrete Valuation Rings 303
5.2.3 Characterizations of Dedekind Domains 305
5.3 Going Up Theorem and Going Down Theorem 307
5.3.1 Going Up Theorem 307
5.3.2 Flat Extensions 310
5.3.3 Going Down Theorem 312
5.4 Valuation Overrings and Valuative Dimension 315
5.4.1 Complete Integral Closure 315
5.4.2 Valuation Overrings 317
5.4.3 Valuative Dimension of a Ring 318
5.5 Quotient Rings R langleXrangle and R(X) of Polynomial Rings 320
5.5.1 Dimension of R langleXrangle and R(X) 320
5.5.2 Stably Coherent Rings 324
5.6 Algebras 326
5.7 Valuation Methods in Rings with Zero-Divisors 330
5.7.1 Pseudo-Localization of Rings 330
5.7.2 Valuation Methods 331
5.7.3 Prüfer Rings 336
5.8 Trivial Extensions 339
5.9 Exercises 345
6 w-Modules over Commutative Rings 351
6.1 GV-Torsion-Free Modules and w-Modules 351
6.1.1 GV-Torsion Modules and GV-Torsion-Free Modules 351
6.1.2 w-Modules 355
6.2 w-Closure of Modules and Prime w-Ideals 358
6.2.1 w-Closure of Modules 358
6.2.2 Prime w-Ideals 360
6.3 w-Exact Sequences and DW-Rings 363
6.3.1 w-Exact Sequences 363
6.3.2 DW-Rings 366
6.4 Finite Type Modules and Finitely Presented Type Modules 367
6.4.1 Finite Type Modules 367
6.4.2 Finitely Presented Type Modules 369
6.5 w-Simple Modules and w-Semisimple Modules 372
6.5.1 w-Simple Modules 372
6.5.2 w-Semisimple Modules 373
6.5.3 w-Jacobson Radical 374
6.6 Quotient Ring R{X} of a Polynomial Ring R[X] 376
6.6.1 GV-Ideals of a Polynomial Ring 376
6.6.2 Properties of R{X} 380
6.7 w-Flat Modules and w-Projective Modules 384
6.7.1 w-Flat Modules 384
6.7.2 w-Projective Modules 386
6.7.3 Finite Type w-Projective Modules 390
6.8 w-Noetherian Modules and w-Noetherian Rings 398
6.8.1 Some Characterizations of w-Noetherian Rings 398
6.8.2 Associated Prime Ideals of a GV-Torsion-Free Module 401
6.8.3 Injective Modules over w-Noetherian Rings 404
6.8.4 Krull's Principal Ideal Theorem 407
6.9 w-Artinian Modules and w-Coherent Modules 408
6.9.1 w-Artinian Modules 408
6.9.2 w-Coherent Modules and w-Coherent Rings 411
6.10 Exercises 414
7 Multiplicative Ideal Theory over Integral Domains 420
7.1 Reflexive Modules and Determinants 420
7.1.1 Reflexive Modules over Integral Domains 420
7.1.2 Determinants of Torsion-Free Modules of Finite Rank 423
7.2 Star Operations 427
7.2.1 Basic Properties of Star Operations 427
7.2.2 *-Invertible Fractional Ideals 430
7.3 w-Operations and w-Ideals of a Polynomial Ring 433
7.3.1 w-Operations 433
7.3.2 w-Ideals of Polynomial Rings 435
7.3.3 Almost Principal Ideals 438
7.4 Mori Domains and Strong Mori Domains 442
7.4.1 H-Domains and TV-Domains 442
7.4.2 Mori Domains 444
7.4.3 Strong Mori Domains 445
7.5 Prüfer v-Multiplication Domains 447
7.5.1 Characterizations of PvMDs 447
7.5.2 Several (Other) Cases of Generalized Coherence 452
7.6 Finite Type Reflexive Modules over GCD Domains 454
7.6.1 GCD Domains 454
7.6.2 Finite Type Reflexive Modules over GCD Domains 456
7.7 w-Linked Extensions 459
7.7.1 w-Linked Extensions 459
7.7.2 w-Integral Dependence and w-Integral Closure 462
7.8 UMT-Domains 465
7.9 Krull Domains 470
7.10 Transforms of Multiplicative Systems of Ideals 473
7.10.1 Fractional Ideals of an mathscrS-Transform 474
7.10.2 Global Transforms and w-Global Transforms 476
7.10.3 Mori--Nagata Theorem 478
7.11 Exercises 481
8 Structural Theory of Milnor Squares 486
8.1 Basic Properties of Pullbacks 486
8.1.1 Pullbacks of Rings 486
8.1.2 Pullbacks of Modules 488
8.2 Homological Properties of Cartesian Squares 492
8.2.1 Pullbacks of Flat Modules 492
8.2.2 Pullbacks of Projective Modules 495
8.2.3 Finiteness Conditions and Coherence in Cartesian Squares 497
8.3 Basic Properties of Milnor Squares 499
8.3.1 Localization Methods in Milnor Squares 499
8.3.2 Star Operation Methods in Milnor Squares 502
8.3.3 Prime Ideals 506
8.3.4 Weak Finiteness Conditions in Milnor Squares 508
8.4 Chain Conditions of Rings in Milnor Squares 509
8.4.1 Pullbacks of Mori Domains 510
8.4.2 Pullbacks of Noetherian Domains and SM Domains 514
8.5 Coherence of Rings in Milnor Squares 517
8.5.1 Pullbacks of v-Coherent Domains 517
8.5.2 Pullbacks of Coherent Rings 522
8.5.3 Pullbacks of Quasi-Coherent Domains and FC Domains 525
8.5.4 Pullbacks of w-Coherent Domains, w-Quasi-Coherent Domains, and WFC Domains 527
8.6 Integrality and w-Invertibility in Milnor Squares 529
8.6.1 Pullbacks of Prüfer Domains and PvMDs 529
8.6.2 Integrally Closedness in Milnor Squares 531
8.6.3 Pullbacks of UMT-Domains 532
8.6.4 Basic Properties of D+M Constructions 534
8.6.5 Pullbacks of GCD Domains 535
8.7 Dimensions of Rings in Milnor Squares 537
8.7.1 Krull Dimensions of Rings in Milnor Squares 537
8.7.2 w-Dimensions of Rings in Milnor Squares 539
8.7.3 Valuative Dimensions in Milnor Squares 541
8.7.4 t-Dimensions of Rings in Milnor Squares 545
8.8 Exercises 549
9 Coherent Rings with Finite Weak Global Dimension 551
9.1 Fitting Invariant Ideals and Coherent Regular Rings 551
9.1.1 Fitting Invariant Ideals 551
9.1.2 w-Ideals of Coherent Regular Rings 558
9.2 Super Coherent Regular Local Rings and Generalized Umbrella Rings 559
9.2.1 Super Coherent Regular Local Rings 559
9.2.2 Generalized Umbrella Rings 568
9.3 Domains with Weak Global Dimension 2 570
9.4 Umbrella Rings and U2-rings 574
9.4.1 Structural Characterizations of Umbrella Rings 574
9.4.2 Properties of U2-rings 577
9.5 GE Rings 579
9.6 Exercises 585
10 The Grothendieck Group of a Ring 588
10.1 Basic Properties of Grothendieck Groups 588
10.2 Picard Groups of Rings 594
10.2.1 Invertible Modules 594
10.2.2 Exterior Powers 596
10.3 Grothendieck Groups of Dedekind Domains 603
10.4 Grothendieck Groups of Polynomial Rings 613
10.4.1 Grothendieck Groups in the Category of Finitely Presented Modules 613
10.4.2 Grothendieck Groups of Polynomial Rings 614
10.5 The Bass--Quillen Problem 618
10.5.1 Gluing Theorem 618
10.5.2 Bass--Quillen Conjecture and Quillen's Method 623
10.5.3 Lequain--Simis Method 628
10.6 Exercises 632
11 Relative Homological Algebra 634
11.1 Gorenstein Projective Modules and Strongly Gorenstein Projective Modules 634
11.1.1 Gorenstein Projective Modules 634
11.1.2 Strongly Gorenstein Projective Modules 637
11.1.3 n-Strongly Gorenstein Projective Modules 643
11.2 Gorenstein Injective Modules and Strongly Gorenstein Injective Modules 647
11.2.1 Gorenstein Injective Modules 647
11.2.2 Strongly Gorenstein Injective Modules 648
11.2.3 n-Strongly Gorenstein Injective Modules 649
11.3 Gorenstein Projective Dimension and Gorenstein Injective Dimension of a Module 650
11.3.1 Gorenstein Projective Dimension of a Module 650
11.3.2 Gorenstein Injective Dimension of a Module 657
11.4 Gorenstein Global Dimension of a Ring 659
11.4.1 Basic Properties of the Gorenstein Global Dimension of a Ring 659
11.4.2 Rings of Gorenstein Global Dimension 0 662
11.5 Change of Rings Theorems for the Gorenstein Projective Dimension 664
11.5.1 Gorenstein Global Dimension of a Factor Ring 664
11.5.2 Gorenstein Global Dimension of a Polynomial Ring 666
11.6 Finitely Generated Gorenstein Projective Modules 668
11.6.1 Super Finitely Presented Modules 668
11.6.2 Finitely Generated Gorenstein Projective Modules 671
11.7 Gorenstein Hereditary Rings and Gorenstein Dedekind Domains 677
11.7.1 Gorenstein Hereditary Rings 677
11.7.2 Gorenstein Dedekind Domains 682
11.7.3 Noetherian Warfield Domains 684
11.8 Pseudo Valuation Rings and 2-Discrete Valuation Rings 686
11.8.1 Pseudo Valuation Rings 687
11.8.2 2-Discrete Valuation Rings 692
11.9 Exercises 694
12 Erratum to: Foundations of Commutative Rings and Their Modules 699
Erratum to: F. Wang and H. Kim, Foundations of Commutative Rings and Their Modules, Algebra and Applications 22, DOI 10.1007/978-981-10-3337-7 699
References 701
Index 708