Editing Regular local ring (section)

==Regular ring==
{{for|the unrelated regular rings introduced by John von Neumann|von Neumann regular ring}}
In [[commutative algebra]], a '''regular ring''' is a commutative [[Noetherian ring]], such that the [[localization of a ring|localization]] at every [[prime ideal]] is a regular local ring: that is, every such localization has the property that the minimal number of generators of its maximal ideal is equal to its [[Krull dimension]].

The origin of the term ''regular ring'' lies in the fact that an [[affine variety]] is [[nonsingular variety|nonsingular]] (that is every point is [[regular point of an algebraic variety|regular]]) if and only if its [[ring of regular functions]] is regular.

For regular rings, Krull dimension agrees with [[global homological dimension]].

[[Jean-Pierre Serre]] defined a regular ring as a commutative noetherian ring of ''finite'' global homological dimension. His definition is stronger than the definition above, which allows regular rings of infinite Krull dimension.

Examples of regular rings include fields (of dimension zero) and [[Dedekind domain]]s.  If ''A'' is regular then so is ''A''[''X''], with dimension one greater than that of ''A''. 

In particular if {{mvar|k}} is a field, the ring of integers, or a [[principal ideal domain]], then the [[polynomial ring]] <math>k[X_1, \ldots,X_n]</math> is regular. In the case of a field, this is [[Hilbert's syzygy theorem]].

Any localization of a regular ring is regular as well.

A regular ring is [[reduced ring|reduced]]{{efn|since a ring is reduced if and only if its localizations at prime ideals are.}} but need not be an integral domain. For example, the product of two regular integral domains is regular, but not an integral domain.<ref>[https://math.stackexchange.com/q/18657 Is a regular ring a domain]</ref>