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.

fundamental theorem of arithmetic | unique factorization