Integral domain

In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero.[1][2] 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.[3][4] Noncommutative integral domains are sometimes admitted.[5] 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.[6]

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

An integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Equivalently:

Given elements a and b of R, one says 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 units of R are the elements that divide 1; these are precisely the invertible elements in R. Units divide all other elements.

If a divides b and b divides a, then a and b are associated elements or associates.[9] Equivalently, a and b are associates if a = ub for some unit u.

An irreducible element is a nonzero non-unit that cannot be written as a product of two non-units.

A nonzero non-unit p is a prime element if, whenever p divides a product ab, then p divides a or p divides b. Equivalently, an element p is prime if and only if the principal ideal (p) is a nonzero prime ideal.

Integral domains are characterized by the condition that they are reduced (that is x2 = 0 implies x = 0) and irreducible (that is there is only one minimal prime ideal). The former condition ensures that the nilradical of the ring is zero, so that the intersection of all the ring's minimal primes is zero. The latter condition is that the ring have only one minimal prime. It follows that the unique minimal prime ideal of a reduced and irreducible ring is the zero ideal, so such rings are integral domains. The converse is clear: an integral domain has no nonzero nilpotent elements, and the zero ideal is the unique minimal prime ideal.

This translates, in algebraic geometry, into the fact that the coordinate ring of an affine algebraic set is an integral domain if and only if the algebraic set is an algebraic variety.

More generally, a commutative ring is an integral domain if and only if its spectrum is an integral affine scheme.

The characteristic of an integral domain is either 0 or a prime number.

If R is an integral domain of prime characteristic p, then the Frobenius endomorphism f(x) = xp is injective.