Noetherian ring

This article defines a property of commutative rings

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

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Relation with other properties

Stronger properties

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

Metaproperties

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.