Divisibility (ring theory)

For divisibility in integers, see Divisor. For divisibility in polynomials, see Polynomial § Divisibility.

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,[a] 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 ${\displaystyle a\mid b}$. Elements a and b of an integral domain are associates if both ${\displaystyle a\mid b}$ and ${\displaystyle b\mid a}$. 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 ${\displaystyle R}$ can be translated into statements about principal ideals. For instance,

• One has ${\displaystyle a\mid b}$ if and only if ${\displaystyle (b)\subseteq (a)}$.
• Elements a and b are associates if and only if ${\displaystyle (a)=(b)}$.
• 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 ${\displaystyle (u)=R}$.
• If ${\displaystyle a=bu}$ 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, ${\displaystyle (a)}$ denotes the principal ideal of ${\displaystyle R}$ generated by the element ${\displaystyle a}$.

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) [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.