Unique factorization domain
This article defines a property of integral domains, viz., a property that, given any integral domain, is either true or false for that.
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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 unique factorization domain or factorial domain if every element can be expressed as a product of finite length of irreducible elements (possibly with multiplicity) in a manner that is unique upto the ordering of the elements.
Definition with symbols
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Relation with other properties
Conjunction with other properties
- Principal ideal domain is the conjunction with the property of being a Dedekind domain. For full proof, refer: Unique factorization and Dedekind iff principal ideal
- Principal ideal domain is the conjunction with the property of being a Bezout domain. For full proof, refer: Unique factorization and Bezout iff principal ideal
Stronger properties
- Euclidean domain
- Principal ideal domain: For proof of the implication, refer PID implies UFD and for proof of its strictness (i.e. the reverse implication being false) refer UFD not implies PID
- Noetherian unique factorization domain
Weaker properties
- Normal domain: For full proof, refer: Unique factorization implies normal
- gcd domain: For full proof, refer: UFD implies gcd
- Ring satisfying ACCP
Incomparable properties
- Noetherian domain: For full proof, refer: Noetherian not implies UFD
Metaproperties
Polynomial-closedness
This property of integral domains is closed under taking polynomials, i.e., whenever an integral domain has this property, so does the polynomial ring in one variable over it.
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For full proof, refer: Unique factorization is polynomial-closed
Closure under taking localizations
This property of integral domains is closed under taking localizations: the localization at a multiplicatively closed subset of a commutative unital ring with this property, also has this property. In particular, the localization at a prime ideal, and the localization at a maximal ideal, have the property.
View other localization-closed properties of integral domains | View other localization-closed properties of commutative unital rings
Those irreducibles which occur in the saturation of the multiplicatively closed subset that we invert, no longer remain irreducibles; the others continue to remain irreducible.
Further information: Unique factorization is localization-closed
Template:Not prime-quotient-closed idp
The quotient of a unique factorization domain by a prime ideal need not be a unique factorization domain. For instance, the ring , which is not a unique factorization domain, is the quotient
of a unique factorization domain by a prime ideal. Similarly, the ring of trigonometric polynomials
is not a unique factorization domain.
Further information: Unique factorization is not prime-quotient-closed
Ring of integer-valued polynomials
The ring of integer-valued polynomials over a unique factorization domain need not be a unique factorization domain. The simplest example is where the base ring is the ring of rational integers. Further information: Ring of integer-valued polynomials over ring of rational integers is not a UFD