In mathematics, the notion of a **divisor** originally arose within the context of arithmetic of whole numbers. With the development of abstract rings, of which the integers are the archetype, the original notion of divisor found a natural extension.

Divisibility is a useful concept for the analysis of the structure of commutative rings because of its relationship with the ideal structure of such rings.

Let *R* be a ring,^{[a]} and let *a* and *b* be elements of *R*. If there exists an element *x* in *R* with *ax* = *b*, one says that *a* is a **left divisor** of *b* and that *b* is a **right multiple** of *a*.^{[1]} Similarly, if there exists an element *y* in *R* with *ya* = *b*, one says that *a* is a **right divisor** of *b* and that *b* is a **left multiple** of *a*. One says that *a* is a **two-sided divisor** of *b* if it is both a left divisor and a right divisor of *b*; the *x* and *y* above are not required to be equal.

When *R* is commutative, the notions of left divisor, right divisor, and two-sided divisor coincide, so one says simply that *a* is a **divisor** of *b*, or that *b* is a **multiple** of *a*, and one writes . Elements *a* and *b* of an integral domain are **associates** if both and . The associate relationship is an equivalence relation on *R*, so it divides *R* into disjoint equivalence classes.

Note: Although these definitions make sense in any magma, they are used primarily when this magma is the multiplicative monoid of a ring.

Statements about divisibility in a commutative ring can be translated into statements about principal ideals. For instance,

- One has if and only if .
- Elements
*a*and*b*are associates if and only if . - An element
*u*is a unit if and only if*u*is a divisor of every element of*R*. - An element
*u*is a unit if and only if . - If for some unit
*u*, then*a*and*b*are associates. If*R*is an integral domain, then the converse is true. - Let
*R*be an integral domain. If the elements in*R*are totally ordered by divisibility, then*R*is called a valuation ring.

In the above, denotes the principal ideal of generated by the element .

- If one interprets the definition of divisor literally, every
*a*is a divisor of 0, since one can take*x*= 0. Because of this, it is traditional to abuse terminology by making an exception for zero divisors: one calls an element*a*in a commutative ring a zero divisor if there exists a*nonzero**x*such that*ax*= 0.^{[2]} - Some texts apply the term 'zero divisor' to a nonzero element
*x*where the multiplier*a*is additionally required to be nonzero where*x*solves the expression*ax*= 0, but such a definition is both more complicated and lacks some of the above properties.