Regular local ring

It has been suggested that Geometrically regular ring be merged into this article. (Discuss) Proposed since July 2024.

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, ..., a_n 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 ${\displaystyle {\mathcal {O}}_{X,x}}$ of germs at x is regular. (See also: regular scheme.) Regular local rings are not related to von Neumann regular rings.^[a]

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

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