# Principal ideal 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

### Symbol-free definition

A commutative unital ring is termed a **principal ideal ring** if every ideal in it is principal, that is, if every ideal is generated by a single element.

### Definition with symbols

*Fill this in later*

## Relation with other properties

### Conjunction with other properties

- Principal ideal domain is a principal ideal ring which is also an integral domain

### Weaker properties

- Bezout ring
- Noetherian ring
- One-dimensional ring:
*For full proof, refer: Principal ideal ring implies one-dimensional*

## Metaproperties

### 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

### Closure under taking localizations

This property of commutative unital rings is closed under taking localizations: the localization at a multiplicatively closed subset of a commutative unital ring with this property, also has this property. In particular, the localization at a prime ideal, and the localization at a maximal ideal, have the property.

View other localization-closed properties of commutative unital rings

*For full proof, refer: Principal ideal ring is localization-closed*