# Finite-dimensional 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

This property of commutative unital rings is completely determined by the spectrum, viewed as an abstract topological space. The corresponding property of topological spaces is: finite bound on length of ascending chains of irreducible closed subsetsView other properties of commutative unital rings determined by the spectrum

## Contents

## Definition

### Symbol-free definition

A **finite-dimensional ring** is a ring whose Krull dimension is finite.

## Relation with other properties

### Stronger properties

- Artinian ring
- Zero-dimensional ring
- One-dimensional ring
- Local Noetherian ring
- Multivariate polynomial ring over a field

### Conjunction with other properties

## Metaproperties

### Polynomial-closedness

In general, a polynomial ring over a finite-dimensional ring need not be finite-dimensional; when the ring is Noetherian, however, the polynomial ring is finite-dimensional, with dimension exactly one more than that of the original ring.

### Closure under taking quotient rings

This property of commutative unital rings is quotient-closed: the quotient ring of any ring with this property, by any ideal in it, also has this propertyView other quotient-closed properties of commutative unital rings