Divisibility (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.
Let R be a ring, 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. 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; in this case, it is not necessarily true that (using the previous notation) x=y, only that both some x and some y which each individually satisfy the previous equations in R exist in R.
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.
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
- 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.
- In this article, rings are assumed to have a 1.
- Bourbaki, p. 97
- Bourbaki, p. 98
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