Definition:Tidy Factorization

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

Let $\struct {U_D, \circ}$ be the group of units of $\struct {D, +, \circ}$.

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 $u \in \struct {U_D, \circ}$, 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.