# regular local ring

A local ring^{} $R$ of dimension^{} $n$ is *regular ^{}* if and only if its maximal ideal

^{}$\U0001d52a$ is generated by $n$ elements.

Equivalently, $R$ is regular if ${dim}_{R/\U0001d52a}\U0001d52a/{\U0001d52a}^{2}=dimR$, where the first dimension is that of a vector space^{}, and the latter is the Krull dimension, since by Nakayama’s lemma, elements generate $\U0001d52a$ if and only if their images under the projection^{} generate $\U0001d52a/{\U0001d52a}^{2}$.

By Krull’s principal ideal theorem, $\U0001d52a$ cannot be generated by fewer than $n$ elements, so the maximal ideals of regular local rings^{} have a minimal number of generators^{}.

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 |