Regular ring

For the unrelated regular rings introduced by John von Neumann, see von Neumann 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 $k[X_{1},\ldots ,X_{n}]$ is regular. This is Hilbert's syzygy theorem.

Any localization of a regular ring is regular as well.

A regular ring is reduced but need not be an integral domain. For example, the product of two regular integral domains is regular, but not an integral domain.

See also

• Geometrically regular ring

References

• Tsit-Yuen Lam, Lectures on Modules and Rings, Springer-Verlag, 1999, ISBN 978-1-4612-0525-8. Chap.5.G.
• Jean-Pierre Serre, Local algebra, Springer-Verlag, 2000, ISBN 3-540-66641-9. Chap.IV.D.
• Regular rings at The Stacks Project