In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds.
Specifically, a UFD is an integral domain (a nontrivial commutative ring in which the product of any two non-zero elements is non-zero) in which every non-zero non-unit element can be written as a product of irreducible elements, uniquely up to order and units.
Unique factorization domains appear in the following chain of class inclusions: Formally, a unique factorization domain is defined to be an integral domain R in which every non-zero element x of R which is not a unit can be written as a finite product of irreducible elements pi of R: and this representation is unique in the following sense: If q1, ..., qm are irreducible elements of R such that then m = n, and there exists a bijective map φ : {1, ..., n} → {1, ..., m} such that pi is associated to qφ(i) for i ∈ {1, ..., n}.
Also, a Dedekind domain is a UFD if and only if its ideal class group is trivial.
For another example, consider a Noetherian integral domain in which every height one prime ideal is principal.