A ring is called commutative if its multiplication is commutative. Commutative rings … See more

Noncommutative rings resemble rings of matrices in many respects. Following the model of algebraic geometry, attempts have been made recently at defining noncommutative geometry based on noncommu… See more

General
• Isomorphism theorems for rings
• Nakayama's lemma
Structure theorems
• … See more

In this section, R denotes a commutative ring. The Krull dimension of R is the supremum of the lengths n of all the chains of prime ideals . It turns out that the polynomial ring over a field k has dimension n. The fund… See more

If X is an affine algebraic variety, then the set of all regular functions on X forms a ring called the coordinate ring of X. For a projective variety, there is an analogous ring called the homogeneous coordinate ring. … See more

Commutative ring theory originated in algebraic number theory, algebraic geometry, and invariant theory. Central to the development of these subjects were the rings of integers in algebraic number fields and algebr… See more