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In [[mathematics]], the notion of a '''divisor''' originally arose within the context of arithmetic of whole numbers. See the article on [[divisor]]s for this simplest example. With the development of abstract [[Ring (mathematics)|rings]], of which the [[integer]]s 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 ring]]s because of its relationship with the [[ideal (ring theory)|ideal]] structure of such rings. | |||
==Definition== | |||
Let ''R'' be a commutative ring,<ref>In this article, rings are assumed to have a 1.</ref> and let ''a'' and ''b'' be elements of ''R''. | |||
One says that ''a'' '''divides''' ''b'' in ''R'' if there exists an element ''x'' in ''R'' with <math> ax=b </math>.<ref>Bourbaki, p. 97</ref> | |||
In this case, one writes <math> a \mid b </math>, and one says also that ''a'' is a '''divisor''' of ''b'', or that ''b'' is a '''[[multiple (mathematics)|multiple]]''' of ''a''. | |||
Notes: These definitions make sense in any commutative [[semigroup]] ''R'', but they are used primarily when this semigroup is the multiplicative [[monoid]] of a commutative ring. Divisors are useful also in non-commutative contexts; one must then discuss [[right divisor|left]] and [[right divisor]]s. | |||
Elements ''a'' and ''b'' of a commutative ring are '''associates''' if both <math> a \mid b </math> and <math> b \mid a </math>. The associate relationship is an [[equivalence relation]] on ''R'', and hence divides ''R'' into [[disjoint sets|disjoint]] [[equivalence class]]es. | |||
== Properties == | |||
Statements about divisibility can be translated into statements about [[principal ideal]]s. For instance, | |||
* One has <math> a \mid b </math> if and only if <math> (b) \subseteq (a) </math>. | |||
* Elements ''a'' and ''b'' are associates if and only if <math> (a) = (b) </math>. | |||
* 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 <math> (u) = R </math>. | |||
* If <math> a = b u </math> for some unit ''u'', then ''a'' and ''b'' are associates. If ''R'' is an [[integral domain]], then the converse is true. | |||
== 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.<ref>Bourbaki, p. 98</ref> | |||
== Notes == | |||
{{reflist}} | |||
== References == | |||
* {{citation |author= N. Bourbaki |title=Algebra I, Chapters 1–3 |publisher=Springer-Verlag |year=1989}}. | |||
{{Citizendium|Divisor (ring theory)}} | |||
== See also == | |||
* [[Divisor]] | |||
* [[Zero divisor]] | |||
[[Category:Ring theory]] | |||
Revision as of 11:13, 20 August 2013
In mathematics, the notion of a divisor originally arose within the context of arithmetic of whole numbers. See the article on divisors for this simplest example. 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 commutative ring,[1] and let a and b be elements of R. One says that a divides b in R if there exists an element x in R with .[2] In this case, one writes , and one says also that a is a divisor of b, or that b is a multiple of a.
Notes: These definitions make sense in any commutative semigroup R, but they are used primarily when this semigroup is the multiplicative monoid of a commutative ring. Divisors are useful also in non-commutative contexts; one must then discuss left and right divisors.
Elements a and b of a commutative ring are associates if both and . The associate relationship is an equivalence relation on R, and hence divides R into disjoint equivalence classes.
Properties
Statements about divisibility 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.
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]
Notes
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References
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