Divisor (ring theory)

In mathematics, the notion of a divisor originally arose within the context of arithmetic of whole numbers. Please see the page about divisors for this simplest example. 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
A nonzero element b of a commutative ring R is said to divide an element a in R (notation: $$ a \mid b $$) if there exists an element x in R with $$ a = b x $$. We also say that b is a divisor of a, or that a is a multiple of b.

Notes: This definition makes sense when R is any commutative semigroup, but virtually the only time divisors are discussed is when this semi-group is the multiplicative monoid of a commutative ring with identity. Also, divisors are occasionally useful in non-commutative contexts, where one must then discuss left and right divisors.

Elements a and b of a commutative ring are said to be associates if both $$ a \mid b $$ and $$ b \mid a $$. The associate relationship is an equivalence relation on R, and hence divides R into disjoint equivalence classes each of which consists of all elements of R that are associates of any particular member of the class.

Properties
If R has an identity, then most statements about divisibility can be translated into statements about principal ideals. For instance,


 * $$ b \mid a $$ if and only if $$ (a) \subset (b) $$.
 * a and b are associates if and only if $$ (a) = (b) $$
 * u is a unit if and only if u is a divisor of every element of R
 * u is a unit if and only if $$ (u) = R $$.
 * If $$ a = b u $$ where u is a unit, then a and b are associates. If R is an integral domain, then the converse is true.