Definition:Divisor (Algebra)

Definition

Ring with Unity

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 $x \divides y$.

Natural Numbers

Let $\N$ be the natural numbers.

Let $n \in \N$ and $m \in \N_{>0}$.

Then $m$ divides $n$ is defined as:

$m \divides n \iff \exists p \in \N: m \times p = n$

Integers

As the set of integers form an integral domain, the concept divides is fully applicable to the integers.

Let $\struct {\Z, +, \times}$ be the ring of integers.

Let $x, y \in \Z$.

Then $x$ divides $y$ is defined as:

$x \divides y \iff \exists t \in \Z: y = t \times x$

Gaussian Integers

As the set of Gaussian integers form an integral domain, the concept divides is also fully applicable to the Gaussian integers.

Let $\struct {\Z \left[{i}\right], +, \times}$ be the ring of Gaussian integers.

Let $x, y \in \Z \left[{i}\right]$.

Then $x$ divides $y$ is defined as:

$x \divides y \iff \exists t \in \Z \left[{i}\right]: y = t \times x$

Real Numbers

The concept of divisibility can also be applied to the real numbers $\R$.

Let $\R$ be the set of real numbers.

Let $x, y \in \R$.

Then $x$ divides $y$ is defined as:

$x \divides y \iff \exists t \in \Z: y = t \times x$

where $\Z$ is the set of integers.

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 $x \nmid y$.

Factorization

Let $x, y \in D$ where $\struct {D, +, \times}$ is an integral domain.

Let $x$ be a divisor of $y$.

Then by definition it is possible to find some $t \in D$ such that $y = t \times x$.

The act of breaking down such a $y$ into the product $t \circ x$ is called factorization.

Also known as

A divisor can also be referred to as a factor.

If $x \divides y$, then $x$ may also be referred to as an aliquot part of $y$.

Some sources insist that $x$ must be a proper divisor of $y$ for this term to apply.

If $x \nmid y$, then $x$ may be referred to as an aliquant part.

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 see

• Results about divisibility can be found here.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics: aliquot part, aliquant part