Back to the index. Or to the chambers

Planetmath Browser (2008—2009)

All articles taken from PlanetMath.org snapshot under CC-BY-SA licence.

→ The original article on PlanetMath.org

Other Formats: LaTeX

New! You can click on formulas to copy the LaTeX source to your clipboard. | Math Videos

Regular Local Ring

A local ring of dimension is regular if and only if its maximal ideal $\mathfrak{m}$ is generated by elements.

Equivalently, 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 elements, so the maximal ideals of regular local rings have a minimal number of generators.

1	Vector Space
1	Direct Image
1	Generating Set Of A Group
2	Minimal And Maximal Number
3	Involutary Ring
3	Maximal Ideal
3	Presentationgroup
5	Local Ring
7	Krull Dimension
14	Nakayamas Lemma
18	Krulls Principal Ideal Theorem