Definition:Tidy Factorization

Definition
Let $\left({D, +, \circ}\right)$ be an integral domain whose unity is $1_D$.

Let $\left({U_D, \circ}\right)$ be the group of units of $\left({D, +, \circ}\right)$.

Any factorization of $x \in D$ can always be tidied into the form:


 * $x = u \circ y_1 \circ y_2 \circ \cdots \circ y_n$

where $u \in \left({U_D, \circ}\right)$, and may be $1_D$, and $y_1, y_2, \ldots, y_n$ are all non-zero and non-units.

This is done by forming the ring product of all units of a factorization into one unit, and rearranging all the remaining factors as necessary.

Such a factorization is called tidy.