Divisibility (ring theory)

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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[edit]

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.

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[edit]

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[edit]

  • 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[edit]

Notes[edit]

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

References[edit]

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.