Integral domain in which any two irreducible factorizations are equal upto ordering and associates

From Commalg
Revision as of 05:00, 18 July 2013 by Vipul (talk | contribs) (Relation with other properties)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search
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