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 ax = b, 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 ya = b, 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 $$ a \mid b $$. Elements a and b of an integral domain are associates if both $$ a \mid b $$ and $$ b \mid a $$. 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 $$R$$ can be translated into statements about principal ideals. For instance,
 * One has $$ a \mid b $$ if and only if $$ (b) \subseteq (a) $$.
 * Elements a and b are associates if and only if $$ (a) = (b) $$.
 * An element u is a unit if and only if u is a divisor of every element of R.
 * An element u is a unit if and only if $$ (u) = R $$.
 * If $$ a = b u $$ for some unit u, then a and b are associates. If R is an integral domain, then the converse is true.
 * Let R be an integral domain. If the elements in R are totally ordered by divisibility, then R is called a valuation ring.

In the above, $$(a)$$ denotes the principal ideal of $$R$$ generated by the element $$a$$.

Zero as a divisor, and zero divisors

 * If one interprets the definition of divisor literally, every a is a divisor of 0, since one can take x = 0. Because of this, it is traditional to abuse terminology by making an exception for zero divisors: one calls an element a in a commutative ring a zero divisor if there exists a nonzero x such that ax = 0.
 * Some texts apply the term 'zero divisor' to a nonzero element x where the multiplier a is additionally required to be nonzero where x solves the expression ax = 0, but such a definition is both more complicated and lacks some of the above properties.