# Introduction to commutative algebra

@inproceedings{Atiyah1969IntroductionTC, title={Introduction to commutative algebra}, author={Michael Francis Atiyah and Ian G. MacDonald}, year={1969} }

* 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

## 4,223 Citations

### Noetherian Rings and Modules

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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.

### CLASSICAL PRIMARY SUBMODULES AND DECOMPOSITION THEORY OF MODULES

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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

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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

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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

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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

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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

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. 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

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: 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

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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,…