Divisibility (ring theory)


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.

Definition

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 in R 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; in this case, it is not necessarily true that x=y, only that both some x and some y which each individually satisfy the previous equations in R exist in R.
When R is commutative, a left divisor, a right divisor and a two-sided divisor coincide, so in this context one says 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, and hence divides R into disjoint equivalence classes.
Notes: These definitions make sense in any magma R, but they are used primarily when this magma is the multiplicative monoid of a ring.

Properties

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.

Zero as a divisor, and zero divisors