Noetherian ring
This article defines a property of commutative unital rings; a property that can be evaluated for a commutative unital ring
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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
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Relation with other properties
Stronger properties
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 is a commutative unital ring satisfying the property, so is
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![{\displaystyle R[x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ce54622cb380383ab3a42441b056626ea0d2440)
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.