Regular ring: Difference between revisions
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==Definition== | ==Definition== | ||
A [[ | A [[Noetherian ring]] is termed a '''regular ring''' if its [[localization at a prime ideal|localization at any prime ideal]] is a [[regular local ring]]. | ||
==Relation with other properties== | ==Relation with other properties== | ||
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===Weaker properties=== | ===Weaker properties=== | ||
* [[Cohen-Macaulay ring]] | * [[Cohen-Macaulay ring]]:{{proofofstrictimplicationat|[[Regular implies Cohen-Macaulay]]|[[Cohen-Macaulay not implies regular]]}} | ||
* [[Universally catenary ring]] | * [[Universally catenary ring]] | ||
* [[Catenary ring]] | * [[Catenary ring]] | ||
* [[Noetherian ring]] | * [[Noetherian ring]] | ||
==Spectrum== | |||
The spectrum of a regular ring has the fairly strong property that every connected component is irreducible. Thus, any regular ring is a direct product of integral domains. If we localize at a point, it should be an integral domain. | |||
==External links== | ==External links== | ||
===Definition links=== | ===Definition links=== | ||
* {{mathworld|RegularRing}} | * {{mathworld|RegularRing}} | ||
Revision as of 02:13, 10 March 2008
This is not to be confused with von-Neumann regular ring
This article defines a property of commutative unital rings; a property that can be evaluated for a commutative unital ring
View all properties of commutative unital rings
VIEW RELATED: Commutative unital ring property implications | Commutative unital ring property non-implications |Commutative unital ring metaproperty satisfactions | Commutative unital ring metaproperty dissatisfactions | Commutative unital ring property satisfactions | Commutative unital ring property dissatisfactions
Definition
A Noetherian ring is termed a regular ring if its localization at any prime ideal is a regular local ring.
Relation with other properties
Stronger properties
Weaker properties
- Cohen-Macaulay ring:For proof of the implication, refer Regular implies Cohen-Macaulay and for proof of its strictness (i.e. the reverse implication being false) refer Cohen-Macaulay not implies regular
- Universally catenary ring
- Catenary ring
- Noetherian ring
Spectrum
The spectrum of a regular ring has the fairly strong property that every connected component is irreducible. Thus, any regular ring is a direct product of integral domains. If we localize at a point, it should be an integral domain.