regular local ring

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

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

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