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, ..., an is a minimal set of generators of m. Then in general 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 a algebraic variety X is nonsingular if and only if the local ring of germs at x is regular. Regular local rings are not related to von Neumann regular rings.
Read more about Regular Local Ring: Characterizations, Examples, Basic Properties, Origin of Basic Notions
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