Definition:Divisor (Algebra)

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Definition

Integral Domain

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 $x \divides y$.


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 $\left({D, +, \times}\right)$ 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. GrahamDonald 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