Divisor

In mathematics a divisor of an integer $n$ , also called a factor of $n$ , is an integer that can be multiplied by some other integer to produce $n$ . An integer $n$ is divisible by another integer $m$ if $m$ is a factor of $n$ , so that dividing $n$ by $m$ leaves no remainder.

Definition

Two versions of the definition of a divisor are commonplace:

For integers

m

and

n

, it is said that

m

divides

n

m

is a divisor of

n

, or

n

is a multiple of

m

, and this is written as

m \mid n,

As before, but with the additional constraint

m \neq 0

. Under this definition, the statement

0 \mid 0

does not hold.

In the remainder of this article, which definition is applied is indicated where this is significant.

General

Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4; they are 1, 2, 4, −1, −2, and −4, but only the positive ones (1, 2, and 4) would usually be mentioned.

1 and −1 divide (are divisors of) every integer. Every integer (and its negation) is a divisor of itself. Every integer is a divisor of 0. Integers divisible by 2 are called even, and numbers not divisible by 2 are called odd.

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Divisibility (ring theory)

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 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.

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 $a \mid b$ . Elements a and b of an integral domain are associates if both $a \mid b$ and $b \mid a$ . The associate relationship is an equivalence relation on R, and hence divides R into disjoint equivalence classes.