# Difference between revisions of "Bezout domain"

From Commalg

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===Stronger properties=== | ===Stronger properties=== | ||

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− | + | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |

− | + | |- | |

+ | | [[Weaker than::Euclidean domain]] || admits a [[Euclidean norm]]|| || || {{intermediate notions short|Bezout domain|Euclidean domain}} | ||

+ | |- | ||

+ | | [[Weaker than::Principal ideal domain]] || every [[ideal]] is a [[principal ideal]] || || || {{intermediate notions short|Bezout domain|principal ideal domain}} | ||

+ | |- | ||

+ | | [[Weaker than::Multi-stage Euclidean domain]] || || || || {{intermediate notions short|Bezout domain|multi-stage Euclidean domain}} | ||

+ | |} | ||

===Weaker properties=== | ===Weaker properties=== | ||

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− | + | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |

+ | |- | ||

+ | | [[Stronger than::gcd domain]] || any two elements have a well-defined gcd || [[Bezout implies gcd]] || [[gcd not implies Bezout]] || {{intermediate notions short|gcd domain|Bezout domain}} | ||

+ | |- | ||

+ | | [[Stronger than::Bezout ring]] || || || || {{intermediate notions short|Bezout ring|Bezout domain}} | ||

+ | |} | ||

===Conjunction with other properties=== | ===Conjunction with other properties=== | ||

* [[Principal ideal domain]] is the conjunction with the property of being a [[Noetherian ring]]. | * [[Principal ideal domain]] is the conjunction with the property of being a [[Noetherian ring]]. |

## Latest revision as of 20:23, 29 January 2014

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.