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