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Concept
Regular local ring
Summary
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 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 \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:
Characterizations
There are a number of useful definitions of a regular local ring, one of which is mentioned above. In particular, if A is a Noetherian local ring with maximal ideal \mathfra
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