| Preface | p. vii |
| Rings and Ideals | p. 1 |
| Recollection and Preliminaries | p. 1 |
| Prime and Maximal Ideals | p. 2 |
| Sums, Products and Colons | p. 6 |
| Radicals | p. 8 |
| Zariski Topology | p. 9 |
| Exercises | p. 10 |
| Modules and Algebras | p. 13 |
| Modules | p. 13 |
| Homomorphisms | p. 17 |
| Direct Products and Direct Sums | p. 19 |
| Free Modules | p. 23 |
| Exact Sequences | p. 25 |
| Algebras | p. 27 |
| Fractions | p. 30 |
| Graded Rings and Modules | p. 35 |
| Homogeneous Prime and Maximal Ideals | p. 38 |
| Exercises | p. 40 |
| Polynomial and Power Series Rings | p. 45 |
| Polynomial Rings | p. 45 |
| Power Series Rings | p. 47 |
| Exercises | p. 53 |
| Homological Tools I | p. 55 |
| Categories and Functors | p. 55 |
| Exact Functors | p. 58 |
| The Functor Hom | p. 61 |
| Tensor Product | p. 65 |
| Base Change | p. 74 |
| Direct and Inverse Limits | p. 76 |
| Injective, Projective and Flat Modules | p. 79 |
| Exercises | p. 85 |
| Tensor, Symmetric and Exterior Algebras | p. 89 |
| Tensor Product of Algebras | p. 89 |
| Tensor Algebras | p. 92 |
| Symmetric Algebras | p. 94 |
| Exterior Algebras | p. 97 |
| Anticommutative and Alternating Algebras | p. 101 |
| Determinants | p. 106 |
| Exercises | p. 109 |
| Finiteness Conditions | p. 111 |
| Modules of Finite Length | p. 111 |
| Noetherian Rings and Modules | p. 115 |
| Artinian Rings and Modules | p. 120 |
| Locally Free Modules | p. 123 |
| Exercises | p. 126 |
| Primary Decomposition | p. 129 |
| Primary Decomposition | p. 129 |
| Support of a Module | p. 135 |
| Dimension | p. 138 |
| Exercises | p. 139 |
| Filtrations and Completions | p. 143 |
| Filtrations and Associated Graded Rings and Modules | p. 143 |
| Linear Topologies and Completions | p. 147 |
| Ideal-adic Completions | p. 151 |
| Initial Submodules | p. 153 |
| Completion of a Local Ring | p. 154 |
| Exercises | p. 156 |
| Numerical Functions | p. 159 |
| Numerical Functions | p. 159 |
| Hilbert Function of a Graded Module | p. 162 |
| Hilbert-Samuel Function over a Local Ring | p. 163 |
| Exercises | p. 167 |
| Principal Ideal Theorem | p. 169 |
| Principal Ideal Theorem | p. 169 |
| Dimension of a Local Ring | p. 171 |
| Exercises | p. 172 |
| Integral Extensions | p. 175 |
| Integral Extensions | p. 175 |
| Prime Ideals in an Integral Extension | p. 178 |
| Integral Closure in a Finite Field Extension | p. 182 |
| Exercises | p. 184 |
| Normal Domains | p. 187 |
| Unique Factorization Domains | p. 187 |
| Discrete Valuation Rings and Normal Domains | p. 192 |
| Fractionary Ideals and Invertible Ideals | p. 198 |
| Dedekind Domains | p. 199 |
| Extensions of a Dedekind Domain | p. 203 |
| Exercises | p. 207 |
| Transcendental Extensions | p. 209 |
| Transcendental Extensions | p. 209 |
| Separable Field Extensions | p. 212 |
| Lüroth's Theorem | p. 217 |
| Exercises | p. 220 |
| Affine Algebras | p. 223 |
| Noether's Normalization Lemma | p. 223 |
| Hilbert's Nullstellensatz | p. 226 |
| Dimension of an Affine Algebra | p. 230 |
| Dimension of a Graded Ring | p. 234 |
| Dimension of a Standard Graded Ring | p. 236 |
| Exercises | p. 239 |
| Derivations and Differentials | p. 241 |
| Derivations | p. 241 |
| Differentials | p. 247 |
| Exercises | p. 253 |
| Valuation Rings and Valuations | p. 255 |
| Valuations Rings | p. 255 |
| Valuations | p. 258 |
| Extensions of Valuations | p. 262 |
| Real Valuations and Completions | p. 265 |
| Hensel's Lemma | p. 274 |
| Discrete Valuations | p. 276 |
| Exercises | p. 280 |
| Homological Tools II | p. 283 |
| Derived Functors | p. 283 |
| Uniqueness of Derived Functors | p. 286 |
| Complexes and Homology | p. 291 |
| Resolutions of a Module | p. 296 |
| Resolutions of a Short Exact Sequence | p. 300 |
| Construction of Derived Functors | p. 303 |
| The Functors Ext | p. 308 |
| The Functors Tor | p. 312 |
| Local Cohomology | p. 314 |
| Homology and Cohomology of Groups | p. 315 |
| Exercises | p. 320 |
| Homological Dimensions | p. 323 |
| Injective Dimension | p. 323 |
| Projective Dimension | p. 325 |
| Global Dimension | p. 327 |
| Projective Dimension over a Local Ring | p. 328 |
| Exercises | p. 330 |
| Depth | p. 331 |
| Regular Sequences and Depth | p. 331 |
| Depth and Projective Dimension | p. 336 |
| Cohen-Macaulay Modules over a Local Ring | p. 338 |
| Cohen-Macaulay Rings and Modules | p. 344 |
| Exercises | p. 346 |
| Regular Rings | p. 347 |
| Regular Local Rings | p. 347 |
| A Differential Criterion for Regularity | p. 350 |
| A Homological Criterion for Regularity | p. 352 |
| Regular Rings | p. 353 |
| A Regular Local Ring is a UFD | p. 354 |
| The Jacobian Criterion for Geometric Regularity | p. 356 |
| Exercises | p. 362 |
| Divisor Class Groups | p. 365 |
| Divisor Class Groups | p. 365 |
| The Case of Fractions | p. 369 |
| The Case of Polynomial Extensions | p. 371 |
| The Case of Galois Descent | p. 373 |
| Galois Descent in the Local Case | p. 377 |
| Exercises | p. 381 |
| Bibliography | p. 383 |
| Index | p. 385 |
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