Definition:Divisor (Algebra)

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:
 * $x \backslash_D y \iff \exists t \in D: y = t \circ x$.

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

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

If $x \backslash y$, 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 $x \nmid y$.

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

Factorization
If $x \backslash y$, then by definition it is possible to find some $t \in D$ such that $y = t \circ x$.

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

(The UK English spelling is factorisation.)

Part
A more old-fashioned term for divisor is part:



... and again:

Parts


That is, when it is not a divisor of it, but is a multiple of some divisor of it.

The alert student will notice the circular definition: a part is defined in terms of measuring another quantity, and measurement is defined in terms of divisors, that is, parts.

Integer definition

 * : $\S 16, \ \S 24$
 * : $\S 0.1$
 * : $\S 3.10$
 * : $\S 2.2$
 * : $\S 22$
 * : Appendix $\text{A}.3$
 * : $\S 11$

Integral Domain definition

 * : $\S 5.26$
 * : $\S 62$