Associated element

In mathematics, and specifically in abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain every nonzero element a has the cancellation property, that is, if a ≠ 0, an equality ab = ac implies b = c.

"Integral domain" is defined almost universally as above, but there is some variation. This article follows the convention that rings have a multiplicative identity, generally denoted 1, but some authors do not follow this, by not requiring integral domains to have a multiplicative identity. Noncommutative integral domains are sometimes admitted. This article, however, follows the much more usual convention of reserving the term "integral domain" for the commutative case and using "domain" for the general case including noncommutative rings.

Some sources, notably Lang, use the term entire ring for integral domain.

Some specific kinds of integral domains are given with the following chain of class inclusions:

There are a number of equivalent definitions of integral domain:

The following rings are not integral domains.

In this section, R is an integral domain.

Given elements a and b of R, we say that a divides b, or that a is a divisor of b, or that b is a multiple of a, if there exists an element x in R such that ax = b.

The elements that divide 1 are called the units of R; these are precisely the invertible elements in R. Units divide all other elements.

If a divides b and b divides a, then we say a and b are associated elements or associates. Equivalently, a and b are associates if a=ub for some unit u.

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