Editing Regular local ring (section)

{{Short description|Type of ring in commutative algebra}}
In [[commutative algebra]], a '''regular local ring''' is a [[Noetherian ring|Noetherian]] [[local ring]] having the property that the minimal number of [[generating set of a module|generators]] of its [[maximal ideal]] is equal to its [[Krull dimension]].{{sfn|Atiyah|Macdonald|1969|p=123|loc=Theorem 11.22}} In symbols, let <math>A</math> be any Noetherian local ring with unique maximal ideal <math>\mathfrak{m}</math>, and suppose <math>a_1,\cdots,a_n</math> is a minimal set of generators of <math>\mathfrak{m}</math>. Then [[Krull's principal ideal theorem]] implies that <math>n\geq\dim A</math>, and <math>A</math> is regular whenever <math>n=\dim A</math>.

The concept is motivated by its geometric meaning. A point <math>x</math> on an [[algebraic variety]] <math>X</math> is [[Singular point of an algebraic variety|nonsingular]] (a [[Smooth scheme|smooth point]]) if and only if the local ring <math>\mathcal{O}_{X, x}</math> of [[germ (mathematics)|germs]] at <math>x</math> is regular. (See also: [[regular scheme]].) Regular local rings are ''not'' related to [[von Neumann regular ring]]s.{{efn|A local von Neumann regular ring is a division ring, so the two conditions are not very compatible.}}

For Noetherian local rings, there is the following chain of inclusions:
{{Commutative local ring classes}}