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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 any Noetherian local ring with unique maximal ideal m, and suppose a1, ..., an is a minimal set of generators of m. Then Krull's principal ideal theorem implies that n ≥ dim A, and A is regular whenever n = dim A.
The concept is motivated by its geometric meaning. A point x on an algebraic variety X is nonsingular (a smooth point) 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.[a]
For Noetherian local rings, there is the following chain of inclusions:
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