In commutative algebra

Commutative algebra

Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...

, a regular ring is a commutative noetherian ring

Noetherian ring

In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element...

, such that the localization

Localization of a ring

In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. Given a ring R and a subset S, one wants to construct some ring R* and ring homomorphism from R to R*, such that the image of S consists of units in R*...

 at every prime ideal

Prime ideal

In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...

 is a regular local ring

Regular local ring

In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let A be a Noetherian local ring with maximal ideal m, and suppose a1, ..., an is a minimal set of...

: that is, every such localization has the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension

Krull dimension

In commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull , is the supremum of the number of strict inclusions in a chain of prime ideals. The Krull dimension need not be finite even for a Noetherian ring....

.

Jean-Pierre Serre

Jean-Pierre Serre

Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...

 defines a regular ring as a commutative noetherian ring of finite global homological dimension and shows that this is equivalent to the definition above. For regular rings, Krull dimension agrees with global homological dimension.

Examples of regular rings include fields (of dimension zero) and Dedekind domain

Dedekind domain

In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors...

s. If A is regular then so is A[X], with dimension one greater than that of A.