Definition:Equivalent Factorizations

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) $$m = n$$;
 * 2) 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.