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, and let a and b be elements of R. If there exists an element x in R with , one says that a is a left divisor of b and that b is a right multiple of a. Similarly, if there exists an element y in R with , 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,
In the above, denotes the principal ideal of generated by the element .