Commutative Algebra (Elements of Mathematics)

Title To the reader Contents of The Elements of Mathematics Series Contents Introduction I. Flat Modules 1. Diagrams and exact sequences 2. Flat modules 3. Faithfully flat modules 4. Flat modules and "Tor" functors Exercises II. Localization 1. Prime ideals 2. Rings and modules of fractions 3. Local rings. Passage from the local to the global 4. Spectra of rings and supports of modules 5. Finitely generated projective modules. Invertible fractional ideals Exercises III. Graduations, Filtrations and Topologies 1. Finitely generated graded algebras 2. General results on filtered rings and modules 3. m-adic topologies on Noetherian rings 4. Lifting in complete rings 5. Flatness properties of filtered modules Exercises IV. Associated Prime Ideals and Primary Decomposition 1. Prime ideals associated with a module 2. Primary decomposition 3. Primary decomposition in graded modules Exercises V. Integers 1. Notion of an integral element 2. The lift of prime ideals 3. Finitely generated algebras over a field Exercises VI. Valuations 1. Valuation rings 2. Places 3. Valuations 4. The height of a valuation 5. The topology defined by a valuation 6. Absolute values 7. The approximation theorem 8. Extensions of a valuation to an algebraic extension 9. Application: Locally compact fields 10. Extensions of a valuation to a transcendental extension Exercises VII. Divisors 1. Krull domains 2. Dedekind domains 3. Factorial domains 4. Modules over integrally closed Noetherian domains Exercises Historical note Bibliography Index of notation Index of terminology Table of implications Table of invariances - I Table of invariances - II Ch. 1. Flat Models--ch. 2. Localization--ch. 3. Graduations, Filtrations, And Topologies--ch. 4 Associated Prime Ideas And Primary Decomposition--ch. 5 Integers--ch. 6 Valuations--ch. 7. Divisors. Translation Of Algebre Commutative. Includes Bibliographical References. Algebra is essentially concerned with calculating, that is, performing, on elements of a set, "algebraic operations", the most well-known example of which is provided by the "four rules" of elementary arithmetic.