Artinian ring: Difference between revisions

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

Field

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

@@ Line 1: / Line 1: @@
-==Definition for commutative rings==
+{{semibasicdef}}
+{{curing property}}
+{{domain-to-field property}}
+==Definition==
 ===Symbol-free definition===
-A [[commutative unital ring]] (or more generally a [[commutative 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.
+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.
-===Definition with symbols===
-{{fillin}}
+==Relation with other properties==
-==Definition for non-commutative rings==
+===Stronger properties===
-There are two notions:
+* [[Field]]
-* [[Left Artinian ring]] must satisfy the descending chain condition on [[left ideal]]s
+===Weaker properties===
-* [[Right Artinian ring]] must satisfy the descending chain condition on [[right ideal]]s
-[[Category: Properties of commutative rings]]
+* [[Noetherian ring]]: {{proofat|[[Artinian implies Noetherian]]}}
+* [[Zero-dimensional ring]]: {{proofat|[[Artinian implies zero-dimensional]]}}
+* [[Jacobson ring]]: {{proofat|[[Artinian implies Jacobson]]}}