Difference between revisions of "Bezout domain"

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(Relation with other properties)
(Relation with other properties)
 
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===Stronger properties===
 
===Stronger properties===
  
* [[Weaker than::Euclidean domain]]
+
{| class="sortable" border="1"
* [[Weaker than::Principal ideal domain]]
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
* [[Weaker than::Multi-stage Euclidean domain]]
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|-
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| [[Weaker than::Euclidean domain]] || admits a [[Euclidean norm]]|| || || {{intermediate notions short|Bezout domain|Euclidean domain}}
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|-
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| [[Weaker than::Principal ideal domain]] || every [[ideal]] is a [[principal ideal]] || || || {{intermediate notions short|Bezout domain|principal ideal domain}}
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|-
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| [[Weaker than::Multi-stage Euclidean domain]] || || || || {{intermediate notions short|Bezout domain|multi-stage Euclidean domain}}
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|}
  
 
===Weaker properties===
 
===Weaker properties===
  
* [[Stronger than::gcd domain]]: {{proofofstrictimplicationat|[[Bezout implies gcd]]|[[gcd not implies Bezout]]}}
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{| class="sortable" border="1"
* [[Stronger than::Bezout ring]]
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
 +
|-
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| [[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}}
 +
|-
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| [[Stronger than::Bezout ring]] || || || || {{intermediate notions short|Bezout ring|Bezout domain}}
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|}
  
 
===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 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