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

From Commalg

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

## 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 |