Regular ring
In commutative algebra, a regular ring is a commutative noetherian ring, such that the 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 (that is every point is 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.
Examples of regular rings include fields (of dimension zero) and Dedekind domains. If A is regular then so is A[X], with dimension one greater than that of A.
In particular if k is a field, the polynomial ring is regular. This is Hilbert's syzygy theorem.
![{\displaystyle k[X_{1},\ldots ,X_{n}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff472a050c05d0bbc73bd23888d3d654df28fbfb)
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