# Artinian ring

From Commalg

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 ringsVIEW 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*