Definition:Unique Factorization Domain

Definition
Let $\struct {D, +, \circ}$ be an integral domain.

If, for all $x \in D$ such that $x$ is non-zero and not a unit of $D$:


 * $(1): \quad x$ possesses a complete factorization in $D$
 * $(2): \quad$ Any two complete factorizations of $x$ in $D$ are equivalent

then $D$ is a unique factorization domain.