Definition:Trivial Factorization

Definition

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

Let $\struct {U_D, \circ}$ be the group of units of $\struct {D, +, \circ}$.

A factorization in $\struct {D, +, \circ}$ of the form $x = u \circ y$, where $u \in U_D$ (that is, where $x$ is an associate of $y$) is called a trivial factorization.

A factorization in $\struct {D, +, \circ}$ of the form $x = z \circ y$, where neither $y$ nor $z$ is a unit of $D$, is called a non-trivial factorization.