regular local ring

A local ringMathworldPlanetmath R of dimensionPlanetmathPlanetmathPlanetmath n is regularPlanetmathPlanetmath if and only if its maximal idealMathworldPlanetmath 𝔪 is generated by n elements.

Equivalently, R is regular if dimR/𝔪𝔪/𝔪2=dimR, where the first dimension is that of a vector spaceMathworldPlanetmath, and the latter is the Krull dimension, since by Nakayama’s lemma, elements generate 𝔪 if and only if their images under the projectionPlanetmathPlanetmath generate 𝔪/𝔪2.

By Krull’s principal ideal theorem, 𝔪 cannot be generated by fewer than n elements, so the maximal ideals of regular local ringsMathworldPlanetmath have a minimal number of generatorsPlanetmathPlanetmathPlanetmath.

Title regular local ring
Canonical name RegularLocalRing
Date of creation 2013-03-22 13:20:14
Last modified on 2013-03-22 13:20:14
Owner mps (409)
Last modified by mps (409)
Numerical id 6
Author mps (409)
Entry type Definition
Classification msc 13H05