Difference between revisions of "Integral domain in which any two irreducible factorizations are equal upto ordering and associates"

From Commalg
Jump to: navigation, search
(New page: {{integral domain property}} ==Definition== An '''integral domain in which any two irreducible factorizations are equal upto ordering and associates''' is an integral domain satisfyi...)
 
(Relation with other properties)
 
Line 9: Line 9:
 
===Stronger properties===
 
===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::Unique factorization domain]]
+
|-
* [[Weaker than::Bezout domain]]
+
| [[Weaker than::Euclidean domain]] || || || || {{intermediate notions short|integral domain in which any two irreducible factorizations are equal upto ordering and associates|Euclidean domain}}
* [[Weaker than::Integral domain in which every irreducible is prime]]: {{proofat|[[Every irreducible is prime implies any two irreducible factorizations are equal upto ordering and associates]]}}
+
|-
 +
| [[Weaker than::principal ideal domain]] || every [[ideal]] is [[principal ideal|principal]] || || || {{intermediate notions short|integral domain in which any two irreducible factorizations are equal upto ordering and associates|principal ideal domain}}
 +
|-
 +
| [[Weaker than::unique factorization domain]] || || || || {{intermediate notions short|integral domain in which any two irreducible factorizations are equal upto ordering and associates|unique factorization domain}}
 +
|-
 +
| [[Weaker than::Bezout domain]] || every [[finitely generated ideal]] in it is [[principal ideal|principal]] || || || {{intermediate notions short|integral domain in which any two irreducible factorizations are equal upto ordering and associates|Bezout domain}}
 +
|-
 +
| [[Weaker than::integral domain in which every irreducible is prime]] || ||[[Every irreducible is prime implies any two irreducible factorizations are equal upto ordering and associates]] || || {{intermediate notions short|integral domain in which any two irreducible factorizations are equal upto ordering and associates|integral domain in which every irreducible is prime}}
 +
|}

Latest revision as of 05:00, 18 July 2013

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

An integral domain in which any two irreducible factorizations are equal upto ordering and associates is an integral domain satisfying the property that for any element, any two factorizations of that element as products of irreducible elements must be equal upto ordering and associates.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Euclidean domain click here
principal ideal domain every ideal is principal click here
unique factorization domain click here
Bezout domain every finitely generated ideal in it is principal click here
integral domain in which every irreducible is prime Every irreducible is prime implies any two irreducible factorizations are equal upto ordering and associates click here