Bezout domain
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 rings
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
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.