Definition:Equivalent Factorizations

Definition
Let $\left({D, +, \circ}\right)$ 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 $\pi: \left\{{1, \ldots, m}\right\} \to \left\{{1, \ldots, n}\right\}$ such that $y_i$ and $z_{\pi \left({i}\right)}$ are associates of each other for each $i \in \left\{{1, \ldots, m}\right\}$.


 * $(2): \quad$ The multisets of principal ideals $\left\{{\left\{{\left({y_i}\right) : i = 1, \ldots, m}\right\}}\right\}$ and $\left\{{\left\{{\left({z_i}\right) : i = 1, \ldots, n}\right\}}\right\}$ are equal.

The equivalence of the definitions is shown by part $(3)$ of Principal Ideals in Integral Domain.