Definition:Divisor/Integral Domain

Definition

Let $\left({D, +, \circ}\right)$ 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:

$x \mathop {\backslash_D} y \iff \exists t \in D: y = t \circ x$

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

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

If $x \mathop \backslash y$, then:

• $x$ is a divisor (or factor) of $y$
• $y$ is a multiple of $x$
• $y$ is divisible by $x$.

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

References

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

Sources

• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 6.26$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 62$