Artinian ring: Difference between revisions
m (8 revisions) |
|||
| (6 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
==Definition | {{semibasicdef}} | ||
{{curing property}} | |||
{{domain-to-field property}} | |||
==Definition== | |||
===Symbol-free definition=== | ===Symbol-free definition=== | ||
A [[commutative unital ring]] | A [[commutative unital ring]] is termed '''Artinian''' if it satisfies the [[descending chain condition]] on [[ideal]]s, that is, any descending chain of ideals stabilizes after a finite length. | ||
==Relation with other properties== | |||
===Stronger properties=== | |||
* [[Field]] | |||
=== | ===Weaker properties=== | ||
{{ | * [[Noetherian ring]]: {{proofat|[[Artinian implies Noetherian]]}} | ||
* [[Zero-dimensional ring]]: {{proofat|[[Artinian implies zero-dimensional]]}} | |||
* [[Jacobson ring]]: {{proofat|[[Artinian implies Jacobson]]}} | |||
Latest revision as of 16:18, 12 May 2008
This article is about a standard (though not very rudimentary) definition in commutative algebra. The article text may, however, contain more than just the basic definition
View a complete list of semi-basic definitions on this wiki
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