Divisibility (ring theory)

Short description: Concept in mathematical 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.

## Contents

* 1 Definition
* 2 Properties
* 3 Zero as a divisor, and zero divisors
* 4 See also
* 5 Notes
* 6 Citations
* 7 References

## Definition

Let R be a ring,[lower-alpha 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 and that b is a right multiple of a.[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. 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∣b. Elements a and b of an integral domain are associates if both a∣b and b∣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∣b if and only if (b)⊆(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=bu 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.[2]
* 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.

## See also

* Divisor – divisibility in integers
* Polynomial § Divisibility – divisibility in polynomials
* Quasigroup – an otherwise generic magma with divisibility
* Zero divisor
* GCD domain

## Notes

1. ↑ In this article, rings are assumed to have a 1.

## Citations

1. ↑ Bourbaki 1989, p. 97
2. ↑ Bourbaki 1989, p. 98

## References

* Bourbaki, N. (1989), Algebra I, Chapters 1–3, Springer-Verlag, ISBN 9783540642435, https://books.google.com/books?id=STS9aZ6F204C

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