# 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

- 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 62$. Factorization in an integral domain