Introduction Advice for the Beginner Information for the Expert Prerequisites Sources Courses A First Courses A Second Courses Acknowledgements 0 Elementary Definitions 0.1 Rings and Ideals 0.2 Unique Factorization 0.3 Modules �� Basic Constructions 1 Roots of Commutative Algebra 1.1 Number Theory 1.2 Algebraic Curves and Fhnction Theory 1.3 Invariant Theory 1.4 The Basis Theorem 1.4.1 Finite Generation of Invariants 1.5 Graded Rings 1.6 Algebra and Geometry�� The Nullstellensatz 1.7 Geometric Invariant Theory 1.8 Projective Varieties 1.9 Hilbert Functions and Polynomials 1.10 Free Resolutions and the Syzygy Theorem 1.11 Exercises Noetherian Rings and Modules An Analysis of Hilbert's Finiteness Argument Some Rings of Invariants Algebra and Geometry Graded Rings and Projective Geometry Hilbert Functions Free Resolutions Spec, max-Spec, and the Zariski Topology 2 Localization 2.1 Fractions 2.2 Horn and Tensor 2.3 The Construction of Primes 2.4 Rings and Modules of Finite Length 2.5 Products of Domains 2.6 Exercises Z-graded Rings and Their Localizations Partitions of Unity Gluing Constructing Primes Idempotents, Products, and Connected Components 3 Associated Primes and Primary Decomposition 3.1 Associated Primes 3.2 Prime Avoidance 3.3 Primary Decomposition 3.4 Primary Decomposition and Factoriality 3.5 Primary Decomposition in the Graded Case 3.6 Extracting Information from Primary Decomposition 3.7 Why Primary Decomposition Is Not Unique 3.8 Geometric Interpretation of Primary Decomposition 3.9 Symbolic Powers and Functions Vanishing to High Order 3.9.1 A Determinantal Example 3.10 Exercises General Graded Primary Decomposition Primary Decomposition of Monomial Ideals The Question of Uniqueness Determinantal Ideals ���� �� Dimension Theory �� Homological Methods Appendix 1 Field Theory Appendix 2 Multilinear Algebra Appendix 3 Homological Algebra Appendix 4 A Sketch of Local Cohomology Appendix 5 Category Theory Appendix 6 Limits and Colimits Appendix 7 Where Next Hints and Solutions for Selected Exercises References Index of Notation Index