# Finite-dimensional Noetherian ring

From Commalg

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 ringsVIEW 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

## Contents

## 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 isView other polynomial-closed properties of commutative unital rings

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