\begin{definition}[Definition:Complete Factorization]
Let $\struct {D, +, \circ}$ be an integral domain.
Let $x$ be a non-zero non-unit element of $D$.
A '''complete factorization''' of $x$ in $D$ is a tidy factorization:
:$x = u \circ y_1 \circ y_2 \circ \cdots \circ y_n$
such that:
:$u$ is a unit of $D$
:all of $y_1, y_2, \ldots, y_n$ are irreducible in $D$.
\end{definition}