Table of Contents

I. Prime Ideals and Localization.- §1. Notation and definitions.- §2. Nakayama’s lemma.- §3. Localization.- §4. Noetherian rings and modules.- §5. Spectrum.- §6. The noetherian case.- §7. Associated prime ideals.- §8. Primary decompositions.- II. Tools.- A: Filtrations and Gradings.- B: Hilbert-Samuel Polynomials.- III. Dimension Theory.- A: Dimension of Integral Extensions.- B: Dimension in Noetherian Rings.- C: Normal Rings.- D: Polynomial Rings.- IV. Homological Dimension and Depth.- A: The Koszul Complex.- B: Cohen-Macaulay Modules.- C: Homological Dimension and Noetherian Modules.- D: Regular Rings.- Appendix I: Minimal Resolutions.- Appendix II: Positivity of Higher Euler-Poincaré Characteristics.- Appendix III: Graded-polynomial Algebras.- V. Multiplicities.- A: Multiplicity of a Module.- B: Intersection Multiplicity of Two Modules.- C: Connection with Algebraic Geometry.- Index of Notation.