# 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,[1] 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* in *R* and that *b* is a **right multiple** of *a*.[2] 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*; in this case, it is not necessarily true that (using the previous notation) *x*=*y*, only that both some *x* and some *y* which each individually satisfy the previous equations in *R* exist in *R*.

When *R* is commutative, a left divisor, a right divisor and a two-sided divisor coincide, so in this context one says 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*, and hence divides *R* into disjoint equivalence classes.

Notes: These definitions make sense in any magma *R*, but 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 principle 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*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.[3]

## See also

## Notes

- In this article, rings are assumed to have a 1.
- Bourbaki, p. 97
- Bourbaki, p. 98

## References

- Bourbaki, N. (1989) [1970],
*Algebra I, Chapters 1–3*, Springer-Verlag, ISBN 9783540642435

*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.*