Noetherian ring

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

Definition

Symbol-free definition

A commutative unital ring is termed Noetherian if it satisfies the following equivalent conditions:

• Ascending chain condition on ideals: Any ascending chain of ideals stabilizes after a finite length
• Every ideal is finitely generated

Definition with symbols

Fill this in later

Relation with other properties

Stronger properties

• Polynomial ring over a field
• Artinian ring
• Noetherian domain
• Principal ideal ring
• Dedekind domain

Metaproperties

Closure under taking the polynomial ring

This property of commutative unital rings is polynomial-closed: it is closed under the operation of taking the polynomial ring. In other words, if

$R$

is a commutative unital ring satisfying the property, so is

$R[x]$

View other polynomial-closed properties of commutative unital rings

The polynomial ring over a Noetherian ring is again Noetherian. This is a general formulation of the Hilbert basis theorem, which asserts in particular that the polynomial ring over a field is Noetherian. Further information: Noetherianness is polynomial-closed