Finite-dimensional 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
A finite-dimensional Noetherian ring is a commutative unital ring that is both finite-dimensional (i.e. its Krull dimension is finite) and Noetherian (i.e. it satisfies the ascending chain condition on all ideals).
Relation with other properties
Stronger properties
- Multivariate polynomial ring over a field
- Finite-dimensional Noetherian domain
- Dedekind domain
- Principal ideal 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 is a commutative unital ring satisfying the property, so is
View other polynomial-closed properties of commutative unital rings
![{\displaystyle R[x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ce54622cb380383ab3a42441b056626ea0d2440)
The polynomial ring over a finite-dimensional Noetherian ring has dimension one more than the original ring.