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, ..., a_n is a minimal set of generators of m. Then by Krull's principal ideal theorem n ≥ dim A, and A is defined to be regular if n = dim A.

The appellation regular is justified by the geometric meaning. A point x on an algebraic variety X is nonsingular if and only if the local ring ${\displaystyle {\mathcal {O}}_{X,x}}$ of germs at x is regular. (See also: regular scheme.) Regular local rings are not related to von Neumann regular rings.

For Noetherian local rings, there is the following chain of inclusions:

Universally catenary rings ⊃ Cohen–Macaulay rings ⊃ Gorenstein rings ⊃ complete intersection rings ⊃ regular local rings