Definition:Tidy Factorization

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.