Artinian ring: Difference between revisions
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===Weaker properties=== | ===Weaker properties=== | ||
* [[Noetherian ring]] | * [[Noetherian ring]]: {{proofat|[[Artinian implies Noetherian]]}} | ||
* [[Zero-dimensional ring]] | * [[Zero-dimensional ring]]: {{proofat|[[Artinian implies zero-dimensional]]}} | ||
* [[Jacobson ring]] | * [[Jacobson ring]]: {{proofat|[[Artinian implies Jacobson]]}} | ||
Revision as of 01:18, 10 January 2008
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
Symbol-free definition
A commutative unital ring is termed Artinian if it satisfies the descending chain condition on ideals, that is, any descending chain of ideals stabilizes after a finite length.
Relation with other properties
Stronger properties
Weaker properties
- Noetherian ring: For full proof, refer: Artinian implies Noetherian
- Zero-dimensional ring: For full proof, refer: Artinian implies zero-dimensional
- Jacobson ring: For full proof, refer: Artinian implies Jacobson