Unique factorization domain

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.

Conjunction with other properties

 * Principal ideal domain is the conjunction with the property of being a Dedekind domain.
 * Principal ideal domain is the conjunction with the property of being a Bezout domain.

Stronger properties

 * Weaker than::Euclidean domain
 * Weaker than::Principal ideal domain:
 * Weaker than::Noetherian unique factorization domain

Weaker properties

 * Stronger than::Normal domain:
 * Stronger than::gcd domain:
 * Stronger than::Ring satisfying ACCP

Incomparable properties

 * Noetherian domain:

Metaproperties
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.

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.

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.