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.

[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.

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

This article incorporates material from the Citizendium article "Divisibility (ring theory)", which is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License but not under the GFDL.