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Commutative algebra
This article is about a branch of algebra. For algebras that are commutative, see Commutative algebra (structure).
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers Z ; and p-adic integers.
Commutative algebra is the main technical tool in the local study of schemes.
The study of rings that are not necessarily commutative is known as noncommutative algebra; it includes ring theory, representation theory, and the theory of Banach algebras.
Overview
Commutative algebra is essentially the study of the rings occurring in algebraic number theory and algebraic geometry.
In algebraic number theory, the rings of algebraic integers are Dedekind rings, which constitute therefore an important class of commutative rings.
Connections with algebraic geometry
Commutative algebra (in the form of polynomial rings and their quotients, used in the definition of algebraic varieties) has always been a part of algebraic geometry. However, in the late 1950s, algebraic varieties were subsumed into Alexander Grothendieck's concept of a scheme. Their local objects are affine schemes or prime spectra, which are locally ringed spaces, which form a category that is antiequivalent (dual) to the category of commutative unital rings, extending the duality between the category of affine algebraic varieties over a field k, and the category of finitely generated reduced k-algebras.
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