\begin{definition}[Definition:Equivalent Factorizations]
Let $\struct {D, +, \circ}$ be an integral domain.
Let $x$ be a non-zero non-unit element of $D$.
Let there be two tidy factorizations of $x$:
:$x = u_y \circ y_1 \circ y_2 \circ \cdots \circ y_m$
:$x = u_z \circ z_1 \circ z_2 \circ \cdots \circ z_n$
These two factorizations are '''equivalent''' if one of the following equivalent statements holds:  
:$(1): \quad$ There exists a bijection $\pi: \set {1, \ldots, m} \to \set {1, \ldots, n}$ such that $y_i$ and $z_{\map \pi i}$ are associates of each other for each $i \in \set {1, \ldots, m}$.
:$(2): \quad$ The multisets of principal ideals $\multiset {\ideal {y_i}: i = 1, \ldots, m}$ and $\multiset {\ideal {z_i}: i = 1, \ldots, n}$ are equal.
The equivalence of the definitions is shown by part $(3)$ of Principal Ideals in Integral Domain.
\end{definition}