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" 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 formula_1. Elements "a" and "b" of an integral domain are associates if both formula_1 and formula_3. 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.
Properties.
Statements about divisibility in a commutative ring formula_4 can be translated into statements about principal ideals. For instance,
In the above, formula_10 denotes the principal ideal of formula_4 generated by the element formula_12.