Quasilocal ring: Difference between revisions
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* [[Local ring]] | * [[Local ring]] | ||
* [[Semilocal ring]] is a [[Noetherian ring|Noetherian]] quasilocal ring | * [[Semilocal ring]] is a [[Noetherian ring|Noetherian]] quasilocal ring | ||
* [[Artinian ring]]: {proofat|Artinian implies quasilocal}} | * [[Artinian ring]]: {{proofat|Artinian implies quasilocal}} | ||
===Weaker properties=== | ===Weaker properties=== | ||
Revision as of 17:59, 17 December 2007
Definition for commmutative rings
Symbol-free definition
A commutative unital ring is termed a quasilocal ring if it has only finitely many maximal ideals.
Definition for noncommutative rings
Further information: Quasilocal ring (noncommutative rings)
Relation with other properties
Stronger properties
- Local ring
- Semilocal ring is a Noetherian quasilocal ring
- Artinian ring: For full proof, refer: Artinian implies quasilocal
Weaker properties
Conjunction with other properties
- Semilocal ring is a Noetherian quasilocal ring