Unique Factorization Domain

A unique factorization domain, called UFD for short, is any integral domain in which every nonzero noninvertible element has a unique factorization, i.e., an essentially unique decomposition as the product of prime elements or irreducible elements. In this context, the two notions coincide, since in a unique factorization domain, every irreducible element is prime, whereas the opposite implication is true in every domain.

This definition arises as an application of the fundamental theorem of arithmetic, which is true in the ring of integers , to more abstract rings. Other examples of unique factorization domains are the polynomial ring , where is a field, and the ring of Gaussian integers . In general, every principal ideal domain is a unique factorization domain, but the converse is not true, since every polynomial ring is a unique factorization domain, but it is not a principal ideal domain if .