Regular local ring
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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.[1] In symbols, let A be a Noetherian local ring with maximal ideal m, and suppose a1, ..., an 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 of germs at x is regular. (See also: regular scheme.) Regular local rings are not related to von Neumann regular rings.[lower-alpha 1]
For Noetherian local rings, there is the following chain of inclusions: