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.)

Integer definition

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

Integral Domain definition

 * : $$\S 62$$