Artinian ring: Difference between revisions
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Revision as of 17:19, 11 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
Any integral domain satisfying this property of commutative unital rings, must be a field
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