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

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M.F. Atiyah and I.G. MacDonald: Introduction to Commutative Algebra

Published $1969$.


Subject Matter


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