Regular local ring: Difference between revisions
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===Symbol-free definition=== | ===Symbol-free definition=== | ||
A [[local ring|local]] [[commutative unital ring]] is said to be '''regular''' if its unique [[maximal ideal]] is generated (as a module) by as many elements as the [[Krull dimension]] of the ring. | A [[local ring|local]] [[commutative unital ring]] is said to be '''regular''' if its unique [[maximal ideal]] is generated (as a [[module]] over the ring) by as many elements as the [[Krull dimension]] of the ring. | ||
==Relation with other properties== | |||
===Stronger properties=== | |||
* [[Discrete valuation ring]] | |||
===Weaker properties=== | |||
* [[Local Cohen-Macaulay ring]] | |||
* [[Local domain]] | |||
* [[Unique factorization domain]] | |||
* [[Integral domain]]: {{proofat|[[Regular local ring implies integral domain]]}} | |||
Latest revision as of 16:34, 12 May 2008
This article defines a property that can be evaluated for a local ring
View other properties of local rings
Definition
Symbol-free definition
A local commutative unital ring is said to be regular if its unique maximal ideal is generated (as a module over the ring) by as many elements as the Krull dimension of the ring.