Definition:Trivial Factorization

Let $$\left({D, +, \circ}\right)$$ be an integral domain.

Let $$\left({U_D, \circ}\right)$$ be the group of units of $$\left({D, +, \circ}\right)$$.

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

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