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 iff:


 * $(1): \quad m = n$
 * $(2): \quad $ It is possible to pair off each of the $y$ and $z$ elements so that corresponding $y_i$ and $z_j$ are associates of each other.