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

## 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