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

From Commalg

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− | + | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |

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− | + | | [[Weaker than::Euclidean domain]] || || || || {{intermediate notions short|integral domain in which any two irreducible factorizations are equal upto ordering and associates|Euclidean domain}} | |

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+ | | [[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}} | ||

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+ | | [[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}} | ||

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+ | | [[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}} | ||

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+ | | [[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}} | ||

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## 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 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 |