Wang / Kim Foundations of Commutative Rings and Their Modules

1. Auflage 2017
ISBN: 978-981-10-3337-7
Verlag: Springer Nature Singapore
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 22, 714 Seiten

Reihe: Algebra and Applications

This book provides an introduction to the basics and recent developments of commutative algebra. A glance at the contents of the first five chapters shows that the topics covered are ones that usually are included in any commutative algebra text. However, the contents of this book differ significantly from most commutative algebra texts: namely, its treatment of the Dedekind-Mertens formula, the (small) finitistic dimension of a ring, Gorenstein rings, valuation overrings and the valuative dimension, and Nagata rings. Going further, Chapter 6 presents w-modules over commutative rings as they can be most commonly used by torsion theory and multiplicative ideal theory. Chapter 7 deals with multiplicative ideal theory over integral domains. Chapter 8 collects various results of the pullbacks, especially Milnor squares and D+M constructions, which are probably the most important example-generating machines. In Chapter 9, coherent rings with finite weak global dimensions are probed, and the local ring of weak global dimension two is elaborated on by combining homological tricks and methods of star operation theory. Chapter 10 is devoted to the Grothendieck group of a commutative ring. In particular, the Bass-Quillen problem is discussed. Finally, Chapter 11 aims to introduce relative homological algebra, especially where the related concepts of integral domains which appear in classical ideal theory are defined and investigated by using the class of Gorenstein projective modules. Each section of the book is followed by a selection of exercises of varying degrees of difficulty. This book will appeal to a wide readership from graduate students to academic researchers who are interested in studying commutative algebra.

Autoren/Hrsg.

Wang, Fanggui

Kim, Hwankoo

Inhaltsverzeichnis

1;Preface;8
2;Contents;10
3;Symbols;19
4;1 Basic Theory of Rings and Modules;21
4.1;1.1 Basic Concepts of Rings and Modules;21
4.1.1;1.1.1 Rings and Ideals;21
4.1.2;1.1.2 Basic Concepts of Modules;23
4.1.3;1.1.3 Direct Product of Rings, Direct Product and Direct Sum of Modules;26
4.2;1.2 Ring Homomorphisms and Module Homomorphisms;27
4.2.1;1.2.1 Ring Homomorphisms;27
4.2.2;1.2.2 Module Homomorphisms;30
4.3;1.3 Finitely Generated Modules and Matrix Methods;34
4.3.1;1.3.1 Finitely Generated Modules;34
4.3.2;1.3.2 Simple Modules, Maximal Submodules, and Zorn's Lemma;36
4.3.3;1.3.3 Jacobson Radical of a Ring;38
4.3.4;1.3.4 Matrix Methods;38
4.4;1.4 Prime Ideals and Nil Radical;42
4.4.1;1.4.1 Prime Ideals;42
4.4.2;1.4.2 Nil Radical and Radical of an Ideal;44
4.5;1.5 Quotient Rings and Quotient Modules;45
4.5.1;1.5.1 Local Rings;45
4.5.2;1.5.2 Quotient Rings;46
4.5.3;1.5.3 Quotient Modules;48
4.6;1.6 Free Modules, Torsion Modules, and Torsion-Free Modules;53
4.6.1;1.6.1 Free Modules;53
4.6.2;1.6.2 Torsion Modules, Torsion-Free Modules, and Divisible Modules;56
4.7;1.7 Polynomial Rings and Power Series Rings;57
4.7.1;1.7.1 Polynomial Rings over One Indeterminate;57
4.7.2;1.7.2 Polynomials with Coefficients in a Module;61
4.7.3;1.7.3 Dedekind--Mertens Formula;62
4.7.4;1.7.4 Polynomial Rings over Many Indeterminates and Formal Power Series Rings over One Indeterminate;65
4.8;1.8 Krull Dimension of a Ring;67
4.8.1;1.8.1 Basic Properties of Krull Dimension of a Ring;67
4.8.2;1.8.2 Krull Dimension of a Polynomial Ring;68
4.8.3;1.8.3 Connected Rings;70
4.9;1.9 Exact Sequences and Commutative Diagrams;71
4.9.1;1.9.1 Exact Sequences;72
4.9.2;1.9.2 Five Lemma and Snake Lemma;73
4.9.3;1.9.3 Completion of Diagrams;79
4.9.4;1.9.4 Pushout and Pullback;81
4.10;1.10 Exercises;83
5;2 The Category of Modules;91
5.1;2.1 The Functor Hom;91
5.1.1;2.1.1 Categories;91
5.1.2;2.1.2 Functors;94
5.1.3;2.1.3 Basic Properties of the Functor Hom;95
5.1.4;2.1.4 Natural Transforms of Functors;97
5.1.5;2.1.5 Torsionless Modules and Reflexive Modules;98
5.2;2.2 The Functor otimes;100
5.2.1;2.2.1 Bilinear Mappings and Tensor Products;100
5.2.2;2.2.2 Basic Properties of the Functor otimes;101
5.2.3;2.2.3 Change of Rings and Adjoint Isomorphism Theorem;103
5.2.4;2.2.4 Tensor Product and Localization;105
5.3;2.3 Projective Modules;106
5.3.1;2.3.1 Projective Modules;107
5.3.2;2.3.2 Kaplansky Theorem;110
5.4;2.4 Injective Modules;113
5.4.1;2.4.1 Injective Modules;113
5.4.2;2.4.2 Injective Envelope of a Module;117
5.5;2.5 Flat Modules;120
5.5.1;2.5.1 Flat Modules and Their Characterizations;120
5.5.2;2.5.2 Faithfully Flat Modules;126
5.5.3;2.5.3 Direct Limits;129
5.6;2.6 Finitely Presented Modules;134
5.6.1;2.6.1 Finitely Presented Modules;134
5.6.2;2.6.2 Isomorphism Theorems Related to Hom and otimes;139
5.7;2.7 Superfluous Submodules and Projective Covers;149
5.7.1;2.7.1 Jacobson Radical of a Module and Superfluous Submodules;149
5.7.2;2.7.2 Projective Cover of a Module;150
5.8;2.8 Noetherian Modules and Artinian Modules;153
5.8.1;2.8.1 Noetherian Modules and Noetherian Rings;153
5.8.2;2.8.2 Artinian Modules and Artinian Rings;155
5.9;2.9 Semisimple Modules and Composition Series;157
5.9.1;2.9.1 Semisimple Modules;157
5.9.2;2.9.2 Composition Series;158
5.10;2.10 Exercises;161
6;3 Homological Methods;167
6.1;3.1 Complexes and Homologies;167
6.1.1;3.1.1 Complexes and Complex Morphisms;167
6.1.2;3.1.2 Homology Modules;169
6.1.3;3.1.3 Homotopy;173
6.2;3.2 Derived Functors;175
6.2.1;3.2.1 Comparison Theorems;175
6.2.2;3.2.2 Left Derived Functors;177
6.2.3;3.2.3 Right Derived Functors;182
6.3;3.3 Derived Functor Ext;185
6.3.1;3.3.1 Properties of Ext;185
6.3.2;3.3.2 Dimension-Shifting Method and Isomorphism Theorems Related to Ext;189
6.4;3.4 Derived Functor Tor;191
6.4.1;3.4.1 Properties of Tor;191
6.4.2;3.4.2 Isomorphism Theorems Related to Tor;195
6.5;3.5 Projective Dimension and Injective Dimension of a Module ƒ;197
6.5.1;3.5.1 Projective Dimension of a Module;197
6.5.2;3.5.2 Injective Dimension of a Module;199
6.5.3;3.5.3 Global Dimension of a Ring and Semisimple Rings;200
6.6;3.6 Flat Dimension of a Module and Weak Global Dimension of a Ring;202
6.6.1;3.6.1 Flat Dimension of a Module;202
6.6.2;3.6.2 Weak Global Dimension of a Ring;204
6.6.3;3.6.3 von Neumann Regular Rings;205
6.7;3.7 Coherent Rings, Semihereditary Rings, and Hereditary Rings;206
6.7.1;3.7.1 Coherent Rings;206
6.7.2;3.7.2 Semihereditary Rings and Prüfer Domains;208
6.7.3;3.7.3 Valuation Domains;210
6.7.4;3.7.4 Hereditary Rings;212
6.8;3.8 Change of Rings Theorems;213
6.8.1;3.8.1 Several Dimension Inequalities;214
6.8.2;3.8.2 Rees Theorem and Homological Dimension of a Factor Ring;216
6.8.3;3.8.3 Homological Dimension of a Polynomial Ring;220
6.9;3.9 Homological Methods in Coherent Rings;223
6.10;3.10 Finitistic Dimension of a Ring and Perfect Rings;229
6.10.1;3.10.1 Finitistic Dimension and Small Finitistic Dimension of a Ring;229
6.10.2;3.10.2 Semiperfect Rings;232
6.10.3;3.10.3 Perfect Rings;233
6.11;3.11 Exercises;239
7;4 Basic Theory of Noetherian Rings;244
7.1;4.1 Artinian Rings;244
7.1.1;4.1.1 Semilocal Rings;244
7.1.2;4.1.2 Basic Properties of Artinian Rings;246
7.2;4.2 Associated Prime Ideals and Primary Decompositions;248
7.2.1;4.2.1 Associated Prime Ideals;248
7.2.2;4.2.2 Primary Ideals and Primary Submodules;250
7.2.3;4.2.3 Primary Decomposition;252
7.3;4.3 Several Classical Theorems;255
7.3.1;4.3.1 Injective Modules over Noetherian Rings;255
7.3.2;4.3.2 Krull's Principal Ideal Theorem;257
7.3.3;4.3.3 Hilbert Basis Theorem;260
7.3.4;4.3.4 Krull--Akizuki Theorem;263
7.4;4.4 Systems of Parameters and Regular Sequences;267
7.4.1;4.4.1 Systems of Parameters;267
7.4.2;4.4.2 Regular Sequences;269
7.4.3;4.4.3 Auslander--Buchsbaum Theorem;273
7.5;4.5 Regular Local Rings;275
7.5.1;4.5.1 Definition and Properties of Regular Local Rings;275
7.5.2;4.5.2 Finite Free Resolutions;277
7.5.3;4.5.3 Characterizations of Regular Local Rings;280
7.6;4.6 Gorenstein Rings;281
7.6.1;4.6.1 QF Rings;281
7.6.2;4.6.2 n-Gorenstein Rings;285
7.7;4.7 Exercises;288
8;5 Extensions of Rings;291
8.1;5.1 Integral Dependence;291
8.1.1;5.1.1 Integral Extensions;291
8.1.2;5.1.2 GCD Domains and UFDs;295
8.1.3;5.1.3 Integrally Closed Domains;297
8.2;5.2 Dedekind Domains;300
8.2.1;5.2.1 Fractional Ideals;300
8.2.2;5.2.2 Discrete Valuation Rings;303
8.2.3;5.2.3 Characterizations of Dedekind Domains;305
8.3;5.3 Going Up Theorem and Going Down Theorem;307
8.3.1;5.3.1 Going Up Theorem;307
8.3.2;5.3.2 Flat Extensions;310
8.3.3;5.3.3 Going Down Theorem;312
8.4;5.4 Valuation Overrings and Valuative Dimension;315
8.4.1;5.4.1 Complete Integral Closure;315
8.4.2;5.4.2 Valuation Overrings;317
8.4.3;5.4.3 Valuative Dimension of a Ring;318
8.5;5.5 Quotient Rings R langleXrangle and R(X) of Polynomial Rings;320
8.5.1;5.5.1 Dimension of R langleXrangle and R(X);320
8.5.2;5.5.2 Stably Coherent Rings;324
8.6;5.6 Algebras;326
8.7;5.7 Valuation Methods in Rings with Zero-Divisors;330
8.7.1;5.7.1 Pseudo-Localization of Rings;330
8.7.2;5.7.2 Valuation Methods;331
8.7.3;5.7.3 Prüfer Rings;336
8.8;5.8 Trivial Extensions;339
8.9;5.9 Exercises;345
9;6 w-Modules over Commutative Rings;351
9.1;6.1 GV-Torsion-Free Modules and w-Modules;351
9.1.1;6.1.1 GV-Torsion Modules and GV-Torsion-Free Modules;351
9.1.2;6.1.2 w-Modules;355
9.2;6.2 w-Closure of Modules and Prime w-Ideals;358
9.2.1;6.2.1 w-Closure of Modules;358
9.2.2;6.2.2 Prime w-Ideals;360
9.3;6.3 w-Exact Sequences and DW-Rings;363
9.3.1;6.3.1 w-Exact Sequences;363
9.3.2;6.3.2 DW-Rings;366
9.4;6.4 Finite Type Modules and Finitely Presented Type Modules;367
9.4.1;6.4.1 Finite Type Modules;367
9.4.2;6.4.2 Finitely Presented Type Modules;369
9.5;6.5 w-Simple Modules and w-Semisimple Modules;372
9.5.1;6.5.1 w-Simple Modules;372
9.5.2;6.5.2 w-Semisimple Modules;373
9.5.3;6.5.3 w-Jacobson Radical;374
9.6;6.6 Quotient Ring R{X} of a Polynomial Ring R[X];376
9.6.1;6.6.1 GV-Ideals of a Polynomial Ring;376
9.6.2;6.6.2 Properties of R{X};380
9.7;6.7 w-Flat Modules and w-Projective Modules;384
9.7.1;6.7.1 w-Flat Modules;384
9.7.2;6.7.2 w-Projective Modules;386
9.7.3;6.7.3 Finite Type w-Projective Modules;390
9.8;6.8 w-Noetherian Modules and w-Noetherian Rings;398
9.8.1;6.8.1 Some Characterizations of w-Noetherian Rings;398
9.8.2;6.8.2 Associated Prime Ideals of a GV-Torsion-Free Module;401
9.8.3;6.8.3 Injective Modules over w-Noetherian Rings;404
9.8.4;6.8.4 Krull's Principal Ideal Theorem;407
9.9;6.9 w-Artinian Modules and w-Coherent Modules;408
9.9.1;6.9.1 w-Artinian Modules;408
9.9.2;6.9.2 w-Coherent Modules and w-Coherent Rings;411
9.10;6.10 Exercises;414
10;7 Multiplicative Ideal Theory over Integral Domains;420
10.1;7.1 Reflexive Modules and Determinants;420
10.1.1;7.1.1 Reflexive Modules over Integral Domains;420
10.1.2;7.1.2 Determinants of Torsion-Free Modules of Finite Rank;423
10.2;7.2 Star Operations;427
10.2.1;7.2.1 Basic Properties of Star Operations;427
10.2.2;7.2.2 *-Invertible Fractional Ideals;430
10.3;7.3 w-Operations and w-Ideals of a Polynomial Ring;433
10.3.1;7.3.1 w-Operations;433
10.3.2;7.3.2 w-Ideals of Polynomial Rings;435
10.3.3;7.3.3 Almost Principal Ideals;438
10.4;7.4 Mori Domains and Strong Mori Domains;442
10.4.1;7.4.1 H-Domains and TV-Domains;442
10.4.2;7.4.2 Mori Domains;444
10.4.3;7.4.3 Strong Mori Domains;445
10.5;7.5 Prüfer v-Multiplication Domains;447
10.5.1;7.5.1 Characterizations of PvMDs;447
10.5.2;7.5.2 Several (Other) Cases of Generalized Coherence;452
10.6;7.6 Finite Type Reflexive Modules over GCD Domains;454
10.6.1;7.6.1 GCD Domains;454
10.6.2;7.6.2 Finite Type Reflexive Modules over GCD Domains;456
10.7;7.7 w-Linked Extensions;459
10.7.1;7.7.1 w-Linked Extensions;459
10.7.2;7.7.2 w-Integral Dependence and w-Integral Closure;462
10.8;7.8 UMT-Domains;465
10.9;7.9 Krull Domains;470
10.10;7.10 Transforms of Multiplicative Systems of Ideals;473
10.10.1;7.10.1 Fractional Ideals of an mathscrS-Transform;474
10.10.2;7.10.2 Global Transforms and w-Global Transforms;476
10.10.3;7.10.3 Mori--Nagata Theorem;478
10.11;7.11 Exercises;481
11;8 Structural Theory of Milnor Squares;486
11.1;8.1 Basic Properties of Pullbacks;486
11.1.1;8.1.1 Pullbacks of Rings;486
11.1.2;8.1.2 Pullbacks of Modules;488
11.2;8.2 Homological Properties of Cartesian Squares;492
11.2.1;8.2.1 Pullbacks of Flat Modules;492
11.2.2;8.2.2 Pullbacks of Projective Modules;495
11.2.3;8.2.3 Finiteness Conditions and Coherence in Cartesian Squares;497
11.3;8.3 Basic Properties of Milnor Squares;499
11.3.1;8.3.1 Localization Methods in Milnor Squares;499
11.3.2;8.3.2 Star Operation Methods in Milnor Squares;502
11.3.3;8.3.3 Prime Ideals;506
11.3.4;8.3.4 Weak Finiteness Conditions in Milnor Squares;508
11.4;8.4 Chain Conditions of Rings in Milnor Squares;509
11.4.1;8.4.1 Pullbacks of Mori Domains;510
11.4.2;8.4.2 Pullbacks of Noetherian Domains and SM Domains;514
11.5;8.5 Coherence of Rings in Milnor Squares;517
11.5.1;8.5.1 Pullbacks of v-Coherent Domains;517
11.5.2;8.5.2 Pullbacks of Coherent Rings;522
11.5.3;8.5.3 Pullbacks of Quasi-Coherent Domains and FC Domains;525
11.5.4;8.5.4 Pullbacks of w-Coherent Domains, w-Quasi-Coherent Domains, and WFC Domains;527
11.6;8.6 Integrality and w-Invertibility in Milnor Squares;529
11.6.1;8.6.1 Pullbacks of Prüfer Domains and PvMDs;529
11.6.2;8.6.2 Integrally Closedness in Milnor Squares;531
11.6.3;8.6.3 Pullbacks of UMT-Domains;532
11.6.4;8.6.4 Basic Properties of D+M Constructions;534
11.6.5;8.6.5 Pullbacks of GCD Domains;535
11.7;8.7 Dimensions of Rings in Milnor Squares;537
11.7.1;8.7.1 Krull Dimensions of Rings in Milnor Squares;537
11.7.2;8.7.2 w-Dimensions of Rings in Milnor Squares;539
11.7.3;8.7.3 Valuative Dimensions in Milnor Squares;541
11.7.4;8.7.4 t-Dimensions of Rings in Milnor Squares;545
11.8;8.8 Exercises;549
12;9 Coherent Rings with Finite Weak Global Dimension;551
12.1;9.1 Fitting Invariant Ideals and Coherent Regular Rings;551
12.1.1;9.1.1 Fitting Invariant Ideals;551
12.1.2;9.1.2 w-Ideals of Coherent Regular Rings;558
12.2;9.2 Super Coherent Regular Local Rings and Generalized Umbrella Rings;559
12.2.1;9.2.1 Super Coherent Regular Local Rings;559
12.2.2;9.2.2 Generalized Umbrella Rings;568
12.3;9.3 Domains with Weak Global Dimension 2;570
12.4;9.4 Umbrella Rings and U2-rings;574
12.4.1;9.4.1 Structural Characterizations of Umbrella Rings;574
12.4.2;9.4.2 Properties of U2-rings;577
12.5;9.5 GE Rings;579
12.6;9.6 Exercises;585
13;10 The Grothendieck Group of a Ring;588
13.1;10.1 Basic Properties of Grothendieck Groups;588
13.2;10.2 Picard Groups of Rings;594
13.2.1;10.2.1 Invertible Modules;594
13.2.2;10.2.2 Exterior Powers;596
13.3;10.3 Grothendieck Groups of Dedekind Domains;603
13.4;10.4 Grothendieck Groups of Polynomial Rings;613
13.4.1;10.4.1 Grothendieck Groups in the Category of Finitely Presented Modules;613
13.4.2;10.4.2 Grothendieck Groups of Polynomial Rings;614
13.5;10.5 The Bass--Quillen Problem;618
13.5.1;10.5.1 Gluing Theorem;618
13.5.2;10.5.2 Bass--Quillen Conjecture and Quillen's Method;623
13.5.3;10.5.3 Lequain--Simis Method;628
13.6;10.6 Exercises;632
14;11 Relative Homological Algebra;634
14.1;11.1 Gorenstein Projective Modules and Strongly Gorenstein Projective Modules;634
14.1.1;11.1.1 Gorenstein Projective Modules;634
14.1.2;11.1.2 Strongly Gorenstein Projective Modules;637
14.1.3;11.1.3 n-Strongly Gorenstein Projective Modules;643
14.2;11.2 Gorenstein Injective Modules and Strongly Gorenstein Injective Modules;647
14.2.1;11.2.1 Gorenstein Injective Modules;647
14.2.2;11.2.2 Strongly Gorenstein Injective Modules;648
14.2.3;11.2.3 n-Strongly Gorenstein Injective Modules;649
14.3;11.3 Gorenstein Projective Dimension and Gorenstein Injective Dimension of a Module;650
14.3.1;11.3.1 Gorenstein Projective Dimension of a Module;650
14.3.2;11.3.2 Gorenstein Injective Dimension of a Module;657
14.4;11.4 Gorenstein Global Dimension of a Ring;659
14.4.1;11.4.1 Basic Properties of the Gorenstein Global Dimension of a Ring;659
14.4.2;11.4.2 Rings of Gorenstein Global Dimension 0;662
14.5;11.5 Change of Rings Theorems for the Gorenstein Projective Dimension;664
14.5.1;11.5.1 Gorenstein Global Dimension of a Factor Ring;664
14.5.2;11.5.2 Gorenstein Global Dimension of a Polynomial Ring;666
14.6;11.6 Finitely Generated Gorenstein Projective Modules;668
14.6.1;11.6.1 Super Finitely Presented Modules;668
14.6.2;11.6.2 Finitely Generated Gorenstein Projective Modules;671
14.7;11.7 Gorenstein Hereditary Rings and Gorenstein Dedekind Domains;677
14.7.1;11.7.1 Gorenstein Hereditary Rings;677
14.7.2;11.7.2 Gorenstein Dedekind Domains;682
14.7.3;11.7.3 Noetherian Warfield Domains;684
14.8;11.8 Pseudo Valuation Rings and 2-Discrete Valuation Rings;686
14.8.1;11.8.1 Pseudo Valuation Rings;687
14.8.2;11.8.2 2-Discrete Valuation Rings;692
14.9;11.9 Exercises;694
15;12 Erratum to: Foundations of Commutative Rings and Their Modules;699
15.1;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
16;References;701
17;Index;708