What is "unique factorization domain"

n. (context algebra ring theory English) A unique factorization ring which is also an integral domain.

In mathematics, a unique factorization domain (UFD) is a commutative ring, which is an integral domain, and in which every non-zero non-unit element can be written as a product of prime elements (or irreducible elements), uniquely up to order and units, analogous to the fundamental theorem of arithmetic for the integers. UFDs are sometimes called factorial rings, following the terminology of Bourbaki.

Unique factorization domains appear in the following chain of class inclusions: