# regular local ring

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

Equivalently, $R$ 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 $n$ elements, so the maximal ideals of regular local rings have a minimal number of generators.

Title regular local ring RegularLocalRing 2013-03-22 13:20:14 2013-03-22 13:20:14 mps (409) mps (409) 6 mps (409) Definition msc 13H05