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

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

Weaker properties

• Normal domain: For full proof, refer: Unique factorization implies normal
• gcd domain: For full proof, refer: UFD implies gcd

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.
View other polynomial-closed properties of integral domains OR view polynomial-closed properties of commutative unital rings

This is essentially what Gauss's lemma states.