Book:M.F. Atiyah/Introduction to Commutative Algebra

Subject Matter

 * Commutative Algebra

Contents

 * Chapter 1: Rings and Ideals
 * Rings and ring homomorphisms
 * Ideals. Quotient rings
 * Zero-divisors. Nilpotent elements. Units
 * Prime ideals and maximal ideals
 * Nilradical and Jacobson radical
 * Operations on ideals
 * Extension and contraction
 * Exercises
 * Chapter 2: Modules
 * Modules and module homomorphisms
 * Submodules and quotient modules
 * Operations on submodules
 * Direct sum and product
 * Finitely generated modules
 * Exact sequences
 * Tensor product of modules
 * Restriction and extension of scalars
 * Exactness properties of the tensor product
 * Algebras
 * Tensor product of algebras
 * Exercises
 * Chapter 3: Rings and Modules of Fractions
 * Local properties
 * Extended and contracted ideals in rings of fractions
 * Exercises
 * Chapter 4: Primary Decomposition
 * Exercises
 * Chapter 5: Integral Dependence and Valuations
 * Integral dependence
 * The going-up theorem
 * Integrally closed integral domains. The going-down theorem
 * Valuation rings
 * Exercises
 * Chapter 6: Chain Conditions
 * Exercises
 * Chapter 7: Noetherian Rings
 * Primary decomposition in Noetherian rings
 * Exercises
 * Chapter 8: Artin Rings
 * Exercises
 * Chapter 9: Discrete Valuation Rings and Dedekind Domains
 * Discrete valuation rings
 * Dedekind domains
 * Fractional ideals
 * Exercises
 * Chapter 10: Completions
 * Topologies and completions
 * Filtrations
 * Graded rings and modules
 * The associated graded ring
 * Exercises
 * Chapter 11: Dimension Theory
 * Hilbert functions
 * Dimension theory of Noetherian local rings
 * Regular local rings
 * Transcendental dimension
 * Exercises
 * Index