Definition:Divisor (Algebra)/Ring with Unity

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 of $y$
 * $y$ is a multiple of $x$
 * $y$ is divisible by $x$.

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

Also known as
A divisor is also known as a factor.