# Finite-dimensional 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

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).

## 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 $R[x]$

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The polynomial ring over a finite-dimensional Noetherian ring has dimension one more than the original ring.