Artinian ring

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