# Bezout domain

From Commalg

This article defines a property of integral domains, viz., a property that, given any integral domain, is either true or false for that.

View other properties of integral domains | 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

An integral domain is termed a **Bezout domain** if every finitely generated ideal in it is principal.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Euclidean domain | admits a Euclidean norm | click here | ||

Principal ideal domain | every ideal is a principal ideal | click here | ||

Multi-stage Euclidean domain | click here |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

gcd domain | any two elements have a well-defined gcd | Bezout implies gcd | gcd not implies Bezout | click here |

Bezout ring | click here |

### Conjunction with other properties

- Principal ideal domain is the conjunction with the property of being a Noetherian ring.