| Preface | 7 |
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| Preface to the Second Edition | 9 |
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| 1 Basic Concepts | 15 |
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| 1.1 Zorn’s Lemma, Ordinal and Cardinal Numbers | 15 |
| 1.2 Modules | 22 |
| 1.3 Tensor Products of Modules and Nakayama Lemma | 28 |
| 1.4 Categories and Functors | 33 |
| 1.5 Complexes of Modules and Homology | 41 |
| 1.6 Direct and Inverse Limits | 47 |
| 1.7 I-adic Topology and Completions | 52 |
| 2 Flat Modules, Chain Conditions and Prime Ideals | 56 |
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| 2.1 Flat Modules | 56 |
| 2.2 Localization | 60 |
| 2.3 Chain Conditions | 63 |
| 2.4 Prime Ideals and Primary Decomposition | 68 |
| 2.5 Artin-Rees Lemma and Zariski Rings | 77 |
| 3 Injective and Flat Modules | 85 |
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| 3.1 Injective Modules | 85 |
| 3.2 Natural Identities, Flat Modules, and Injective Modules | 92 |
| 3.3 Injective Modules over Commutative Noetherian Rings | 101 |
| 3.4 Matlis Duality | 107 |
| 4 Torsion Free Covering Modules | 112 |
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| 4.1 Existence of Torsion Free Precovers | 112 |
| 4.2 Existence of Torsion Free Covers | 114 |
| 4.3 Examples | 116 |
| 4.4 Direct Sums and Products | 120 |
| 5 Covers | 124 |
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| 5.1 F-precovers and covers | 124 |
| 5.2 Existence of Precovers and Covers | 126 |
| 5.3 Projective and Flat Covers | 129 |
| 5.4 Injective Covers | 139 |
| 5.5 Direct Sums and T-nilpotency | 145 |
| 6 Envelopes | 149 |
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| 6.1 F-preenvelopes and Envelopes | 149 |
| 6.2 Existence of Preenvelopes | 150 |
| 6.3 Existence of Envelopes | 152 |
| 6.4 Direct Sums of Envelopes | 154 |
| 6.5 Flat Envelopes | 156 |
| 6.6 Existence of Envelopes for Injective Structures | 159 |
| 6.7 Pure Injective Envelopes | 164 |
| 7 Covers, Envelopes, and Cotorsion Theories | 172 |
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| 7.1 Definitions and Basic Results | 172 |
| 7.2 Fibrations, Cofibrations and Wakamatsu Lemmas | 174 |
| 7.3 Set Theoretic Homological Algebra | 180 |
| 7.4 Cotorsion Theories with Enough Injectives and Projectives | 182 |
| 8 Relative Homological Algebra and Balance | 187 |
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| 8.1 Left and Right F-resolutions | 187 |
| 8.2 Derived Functors and Balance | 189 |
| 8.3 Applications to Modules | 198 |
| 8.4 F-dimensions | 201 |
| 8.5 Minimal Pure Injective Resolutions of Flat Modules | 215 |
| 8.6 . and µ-dimensions | 225 |
| 9 Iwanaga-Gorenstein and Cohen-Macaulay Rings and Their Modules | 233 |
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| 9.1 Iwanaga-Gorenstein Rings | 233 |
| 9.2 The Minimal Injective Resolution of R | 237 |
| 9.3 More on Flat and Injective Modules | 246 |
| 9.4 Torsion Products of Injective Modules | 249 |
| 9.5 Local Cohomology and the Dualizing Module | 252 |
| 10 Gorenstein Modules | 263 |
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| 10.1 Gorenstein Injective Modules | 263 |
| 10.2 Gorenstein Projective Modules | 270 |
| 10.3 Gorenstein Flat Modules | 277 |
| 10.4 Foxby Classes | 282 |
| 11 Gorenstein Covers and Envelopes | 294 |
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| 11.1 Gorenstein Injective Precovers and Covers | 294 |
| 11.2 Gorenstein Injective Preenvelopes | 295 |
| 11.3 Gorenstein Injective Envelopes | 299 |
| 11.4 Gorenstein Essential Extensions | 302 |
| 11.5 Gorenstein Projective Precovers and Covers | 304 |
| 11.6 Auslander’s Last Theorem (Gorenstein Projective Covers) | 309 |
| 11.7 Gorenstein Flat Covers | 314 |
| 11.8 Gorenstein Flat and Projective Preenvelopes | 318 |
| 11.9 Kaplansky Classes | 319 |
| 12 Balance over Gorenstein and Cohen-Macaulay Rings | 325 |
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| 12.1 Balance of Hom(–, –) | 325 |
| 12.2 Balance of – . – | 329 |
| 12.3 Dimensions over n-Gorenstein Rings | 332 |
| 12.4 Dimensions over Cohen-Macaulay Rings | 337 |
| 12.5 O-Gorenstein Modules | 339 |
| Bibliographical Notes | 351 |
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| Bibliography | 355 |
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| Index | 365 |