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.
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 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; 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 . Elements a and b of an integral domain are associates if both and . The associate relationship is an equivalence relation on R, so it divides R into disjointequivalence 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 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 principal 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 nonzerox such that ax = 0.[3]
divisibility, ring, theory, mathematics, notion, divisor, originally, arose, within, context, arithmetic, whole, numbers, with, development, abstract, rings, which, integers, archetype, original, notion, divisor, found, natural, extension, divisibility, useful. 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 ReferencesDefinition EditLet 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 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 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 displaystyle a mid b Elements a and b of an integral domain are associates if both a b displaystyle a mid b and b a 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 EditStatements about divisibility in a commutative ring R displaystyle R can be translated into statements about principal ideals For instance One has a b displaystyle a mid b if and only if b a displaystyle b subseteq a Elements a and b are associates if and only if a b 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 u R displaystyle u R If a b u 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 a displaystyle a denotes the principal ideal of R displaystyle R generated by the element a displaystyle a Zero as a divisor and zero divisors EditSome 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 EditDivisor Zero divisor GCD domainNotes Edit In this article rings are assumed to have a 1 Bourbaki p 97 Bourbaki p 98References EditBourbaki N 1989 1970 Algebra I Chapters 1 3 Springer Verlag ISBN 9783540642435This 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 Retrieved from https en wikipedia org w index php title Divisibility ring theory amp oldid 1008625667, wikipedia, wiki, book, books, library,
