Introduction to commutative algebra

  title={Introduction to commutative algebra},
  author={Michael Francis Atiyah and Ian G. MacDonald},
* Introduction * Rings and Ideals * Modules * Rings and Modules of Fractions * Primary Decomposition * Integral Dependence and Valuations * Chain Conditions * Noetherian Rings * Artin Rings * Discrete Valuation Rings and Dedekind Domains * Completions * Dimension Theory 

Noetherian Rings and Modules

This chapter may serve as an introduction to the methods of algebraic geometry rooted in commutative algebra and the theory of modules, mostly over a Noetherian ring.


We introduce the notion of classical primary submodules that generalizes the concept of primary ideals of commutative rings to modules. Existence and uniqueness of classical primary decompositions in

A Theorem on Unique Factorization Domains Analogue for Modules

All rings are assumed to be commutative with identity. We define and study the properties of integral modules, the generalization of integral domains. We also generalize basic results concerning the

Basic Results on Ideals and Varieties in Finite Fields

The connection between ideals and varieties for polynomial rings over finite fields is investigated. An extension to Hilbert's NullstellenSatz is given for these ideals. Furthermore projections and

Integral Closure of a Ring Whose Regular Ideals Are Finitely Generated

Abstract We show that if R is a commutative ring with identity whose regular ideals are finitely generated, then the integral closure of R is a Krull ring. This is a generalization of the Mori–Nagata

Abstract Algebra: Theory and Applications

Preliminaries The integers Groups Cyclic groups Permutation groups Cosets and Lagrange's theorem Isomorphisms Algebraic coding theory Homomorphisms ad factor groups Matrix groups and symmetry The

Arithmetic Invariant Theory of Reductive Groups

. In this manuscript, we define the notion of linearly reductive groups over commutative unital rings and study the finiteness and the Cohen-Macaulay property of the ring of invariants under rational

The study of commutative rings in commutative algebra

: In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra.

Some Results Concerning Localization of Commutative Rings and Modules

In this paper some results that concerning localization of commutative rings and modules are proved. It also, studies the eect of localization on certain types of ideals and modules such as G ideals,

Chains of prime ideals in tensor products of algebras