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.

单词	RegularLocalRing
释义	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.
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