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:

rngsringscommutative ringsintegral domainsintegrally closed domainsGCD domainsunique factorization domainsprincipal ideal domainsEuclidean domainsfieldsalgebraically closed fields

Definition

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

Examples

Non-examples

The following rings are not integral domains.

Neither nor is everywhere zero, but is.

Divisibility, prime elements, and irreducible elements

See also: Divisibility (ring theory)

In this section, R is an integral domain.

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.

Both notions of irreducible elements and prime elements generalize the ordinary definition of prime numbers in the ring if one considers as prime the negative primes.

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.

Properties

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 is the field of rational numbers 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) = xp is injective.

See also

Notes

  1. ^ Proof: First assume A is finitely generated as a k-algebra and pick a -basis of . Suppose (only finitely many are nonzero). For each maximal ideal of , consider the ring homomorphism . Then the image is and thus either or and, by linear independence, for all or for all . Since is arbitrary, we have the intersection of all maximal ideals where the last equality is by the Nullstellensatz. Since is a prime ideal, this implies either or is the zero ideal; i.e., either are all zero or are all zero. Finally, is an inductive limit of finitely generated k-algebras that are integral domains and thus, using the previous property, is an integral domain.
  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. McConnell 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, 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. Bibcode:1959PNAS...45..733A. doi:10.1073/pnas.45.5.733. PMC 222624. 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.

References