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:


 * $(1) \quad$ $m = n$, and there is a permutation $\pi$ of $S_n = \{1,\ldots,n\}$ such that $y_i$ and $z_{\pi(i)}$ are associates of each other for each $i \in S_n$.


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

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