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单词 RegularLocalRing
释义

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.

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更新时间:2025/5/4 11:12:13