Unique Factorization Domain

Unique Factorization Domain

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

Note that unique factorization domains appear in the following chain of class inclusions:

Commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields