Definition:Tidy Factorization

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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.


Examples

Tidy Factorizations of $6$

Two examples of tidy factorizations of $6$ in the set of integers $\Z$ are:

\(\displaystyle 6\) \(=\) \(\displaystyle 1 \times 2 \times 3\)
\(\displaystyle 6\) \(=\) \(\displaystyle \paren {-1} \paren {-3} \times 2\)


Linguistic Note

The spelling factorization is the US English version.

The UK English spelling is factorisation.


Sources