# Definition:Divisor (Algebra)/Integral Domain

## Definition

Let $\struct {D, +, \circ}$ be an integral domain whose zero is $0_D$ and whose unity is $1_D$.

Let $x, y \in D$.

We define the term $x$ divides $y$ in $D$ as follows:

$x \mathrel {\divides_D} y \iff \exists t \in D: y = t \circ x$

When no ambiguity results, the subscript is usually dropped, and $x$ divides $y$ in $D$ is just written $x \divides y$.

## Notation

The conventional notation for $x$ is a divisor of $y$ is "$x \mid y$", but there is a growing trend to follow the notation "$x \divides y$", as espoused by Knuth etc.

The notation '$m \mid n$' is actually much more common than '$m \divides n$' in current mathematics literature. But vertical lines are overused -- for absolute values, set delimiters, conditional probabilities, etc. -- and backward slashes are underused. Moreover, '$m \divides n$' gives an impression that $m$ is the denominator of an implied ratio. So we shall boldly let our divisibility symbol lean leftward.

An unfortunate unwelcome side-effect of this notational convention is that to indicate non-divisibility, the conventional technique of implementing $/$ through the notation looks awkward with $\divides$, so $\not \! \backslash$ is eschewed in favour of $\nmid$.

Some sources use $\ \vert \mkern -10mu {\raise 3pt -} \$ or similar to denote non-divisibility.

## Also known as

A divisor can also be referred to as a factor.

## Terminology

Let $x \divides y$ denote that $x$ divides $y$.

Then the following terminology can be used:

$x$ is a divisor of $y$
$y$ is a multiple of $x$
$y$ is divisible by $x$.

In the field of Euclidean geometry, in particular:

$x$ measures $y$.

To indicate that $x$ does not divide $y$, we write $x \nmid y$.

## Also see

• Results about divisibility can be found here.