Bezout domain: Difference between revisions

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	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.

@@ Line 11: / Line 11: @@
 ===Stronger properties===
-* [[Weaker than::Euclidean domain]]
+{| class="sortable" border="1"
-* [[Weaker than::Principal ideal domain]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Weaker than::Multi-stage Euclidean domain]]
+|-
+| [[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===
-* [[Stronger than::gcd domain]]: {{proofofstrictimplicationat|[[Bezout implies gcd]]|[[gcd not implies Bezout]]}}
+{| class="sortable" border="1"
-* [[Stronger than::Bezout ring]]
+! 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===
 * [[Principal ideal domain]] is the conjunction with the property of being a [[Noetherian ring]].