Integral domain

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.[1][2] Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain the cancellation property holds for multiplication by a nonzero element a, 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:

commutative ringsintegral domainsintegrally closed domainsGCD domainsunique factorization domainsprincipal ideal domainsEuclidean domainsfieldsfinite fields


There are a number of equivalent definitions of integral domain:



The following rings are not integral domains.

Divisibility, prime elements, and irreducible elements

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.[9] Equivalently, a and b are associates if a=ub for some unit u.

If q is a nonzero non-unit, we say that q is an irreducible element if q cannot be written as a product of two non-units.

If p is a nonzero non-unit, we say that 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. The notion of prime element generalizes the ordinary definition of prime number in the ring Z, except that it allows for negative prime elements.

Every prime element is irreducible. The converse is not true in general: for example, in the quadratic integer ring the element 3 is irreducible (if it factored nontrivially, the factors would each have to have norm 3, but there are no norm 3 elements since has no integer solutions), but not prime (since 3 divides without dividing either factor). In a unique factorization domain (or more generally, a GCD domain), an irreducible element is a prime element.

While unique factorization does not hold in , there is unique factorization of ideals. See Lasker–Noether theorem.


Field of fractions

Main article: Field of fractions

The field of fractions K of an integral domain R is the set of fractions a/b with a and b in R and b ≠ 0 modulo an appropriate equivalence relation, equipped with the usual addition and multiplication operations. It is "the smallest field containing R" in the sense that there is an injective ring homomorphism RK such that any injective ring homomorphism from R to a field factors through K. The field of fractions of the ring of integers Z is the field of rational numbers Q. The field of fractions of a field is isomorphic to the field itself.

Algebraic geometry

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.

Characteristic and homomorphisms

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) = x p is injective.

See also

The Wikibook Abstract algebra has a page on the topic of: Integral domains


  1. Bourbaki, p. 116.
  2. Dummit and Foote, p. 228.
  3. B.L. van der Waerden, Algebra Erster Teil, p. 36, Springer-Verlag, Berlin, Heidelberg 1966.
  4. I.N. Herstein, Topics in Algebra, p. 88-90, Blaisdell Publishing Company, London 1964.
  5. J.C. McConnel and J.C. Robson "Noncommutative Noetherian Rings" (Graduate Studies in Mathematics Vol. 30, AMS)
  6. Pages 91–92 of Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley Pub. Co., ISBN 978-0-201-55540-0, Zbl 0848.13001
  7. Auslander, Maurice; Buchsbaum, D. A. (1959). "Unique factorization in regular local rings". Proc. Natl. Acad. Sci. USA. 45 (5): 733–734. doi:10.1073/pnas.45.5.733. PMC 222624Freely accessible. PMID 16590434.
  8. Masayoshi Nagata (1958). "A general theory of algebraic geometry over Dedekind domains. II". Amer. J. Math. The Johns Hopkins University Press. 80 (2): 382–420. doi:10.2307/2372791. JSTOR 2372791.
  9. Durbin, John R. (1993). Modern Algebra: An Introduction (3rd ed.). John Wiley and Sons. p. 224. ISBN 0-471-51001-7. Elements a and b of [an integral domain] are called associates if a | b and b | a.


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