regular local ring
A local ring of dimension
is regular
if and only if its maximal ideal
is generated by elements.
Equivalently, is regular if , where the first dimension is that of a vector space, and the latter is the Krull dimension, since by Nakayama’s lemma, elements generate if and only if their images under the projection
generate .
By Krull’s principal ideal theorem, cannot be generated by fewer than elements, so the maximal ideals of regular local rings have a minimal number of generators
.