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.

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

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.
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In fact, it is true more generally, that the localization at a multiplicatively closed subset of a unique factorization domain, is a unique factorization domain. 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.

Closure under quotients by prime ideals

This property of integral domains is prime-quotient-closed: the quotient of any integral domain satisfying this property by a prime ideal also satisfies the property. Note that we need the ideal to be prime for the quotient to also be an integral domain.
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The quotient of a unique factorization domain by a prime ideal is again a unique factorization domain. The irreducibles in the new unique factorization domain are simply the irreducibles outside the prime ideal being quotiented.

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