# 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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## 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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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

The quotient of a unique factorization domain by a prime ideal need not be a unique factorization domain. For instance, the ring $\mathbb{Z}[\sqrt{-5}]$, which is not a unique factorization domain, is the quotient $\mathbb{Z}[x]/(x^2 + 5)$ of a unique factorization domain by a prime ideal. Similarly, the ring of trigonometric polynomials $\R[x,y]/(x^2 + y^2 - 1)$ 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