nLab
unique factorization domain

Let $R$ be an integral domain. We say that an element $r\in R$ is unit if it is invertible. A non-unit is called irreducible if it can not be represented as a product of two non-units.

A commutative integral domain $R$ is a unique factorization domain if every non-unit has a factorization $u=r_1\cdots r_n$ as product of irreducible non-units and this decomposition is unique up to renumbering and rescaling the irreducibles by units.