PlanetMath: regular local ring

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regular local ring

(Definition)

A local ring of dimension is regular if and only if its maximal ideal $\mathfrak{m}$ is generated by elements.

Equivalently, is regular if $\dim_{R/\mathfrak{m}}\mathfrak{m}/\mathfrak{m}^2=\dim R$ , where the first dimension is that of a vector space, and the latter is the Krull dimension, since by Nakayama's lemma, elements generate $\mathfrak{m}$ if and only if their images under the projection generate $\mathfrak{m}/\mathfrak{m}^2$ .

By Krull's principal ideal theorem, $\mathfrak{m}$ cannot be generated by fewer than elements, so the maximal ideals of regular local rings have a minimal number of generators.

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Cross-references: generators, minimal number, Krull's principal ideal theorem, projection, images, generate, Nakayama's lemma, Krull dimension, vector space, generated by, maximal ideal, local ring
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This is version 3 of regular local ring, born on 2002-12-27, modified 2004-04-23.
Object id is 3851, canonical name is RegularLocalRing.
Accessed 2846 times total.

Classification:

AMS MSC:

13H05 (Commutative rings and algebras :: Local rings and semilocal rings :: Regular local rings)

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