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.

Examples

In the ring of integers  consider elements . 
  is a divisor of , because  with remainder  or, phrased differently and more appropriate for generalization, there is  such that .
  is not a divisor of , because  with remainder  resp. there is no  such that .

In the field of reals , an extension of the integers, we have 
  is still a divisor of , because  resp. there is  such that .
  becomes a divisor of , because  is now a real number in  resp. there is  such that .

In the ring of integer polynomials  we have that
  is a divisor of , because by polynomial division  with remainder  resp. there is  such that .
  is not a divisor of , because  with remainder  resp. there is no  such that . In fact one can show that  has no nontrivial divisors, which in this case means in particular that the quadratic polynomial  has no linear divisors like . Having no nontrivial divisors is called being irreducible. Expressed symbolically, there are no  of which not at least one is a unit of the ring  such that . 
  is not a divisor of , because  with remainder  resp. there is no  such that . As before, one can show that  has no nontrivial divisors.

In the ring of real polynomials , an extension of the integer polynomials, we have 
  is still a divisor of , because  resp. there is  such that .
  acquires nontrivial divisors, explicitly , because  resp. there is  such that . Note that  is still not a divisor of , because of the same reason as for  above.
  still has no nontrivial divisors in , i.e.  remains irreducible in  and acquires no nontrivial divisibility. 

These examples illustrate that the notion of divisibility does not only depend on the elements  and  themselves, but also on the context of the algebraic structure of a ring in which  and the multiplication operation are considered. In general divisibility is preserved under extensions and more divisibility may be acquired.

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 . Elements a and b of an integral domain are associates if both  and .  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  can be translated into statements about principal ideals.  For instance,

 One has  if and only if .
 Elements a and b are associates if and only if .
 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 .
 If  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,  denotes the principal ideal of  generated by the element .

Zero as a divisor, and zero divisors 
 Some authors require a to be nonzero in the definition of divisor, but this causes some of the properties above to fail.
 If one interprets the definition of divisor literally, every a is a divisor of 0, since one can take .  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 .

See also 
 Divisors of Integers
 Divisibility of polynomials
 Zero divisor
 GCD domain

Notes

References 

Ring theory