# Definition:Divisor (Algebra)/Ring with Unity

## Definition

Let $\struct {R, +, \circ}$ be an ring with unity whose zero is $0_R$ and whose unity is $1_R$.

Let $x, y \in D$.

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

- $x \mathrel {\divides_R} y \iff \exists t \in R: y = t \circ x$

When no ambiguity results, the subscript is usually dropped, and **$x$ divides $y$ in $R$** 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.

From Ronald L. Graham, Donald E. Knuth and Oren Patashnik: *Concrete Mathematics: A Foundation for Computer Science* (2nd ed.):

*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**.

## Sources

- 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $6$: Polynomials and Euclidean Rings: $\S 26$. Divisibility - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 62$. Factorization in an integral domain