In mathematics, a GCD domain is an integral domain R with the property that any two elements have a greatest common divisor (GCD); i.e., there is a unique minimal principal ideal containing the ideal generated by two given elements. Equivalently, any two elements of R have a least common multiple (LCM).[1]

A GCD domain generalizes a unique factorization domain (UFD) to a non-Noetherian setting in the following sense: an integral domain is a UFD if and only if it is a GCD domain satisfying the ascending chain condition on principal ideals (and in particular if it is Noetherian).

GCD domains appear in the following chain of class inclusions:

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


Every irreducible element of a GCD domain is prime. A GCD domain is integrally closed, and every nonzero element is primal. In other words, every GCD domain is a Schreier domain.

For every pair of elements x, y of a GCD domain R, a GCD d of x and y and an LCM m of x and y can be chosen such that dm = xy, or stated differently, if x and y are nonzero elements and d is any GCD d of x and y, then xy/d is an LCM of x and y, and vice versa. It follows that the operations of GCD and LCM make the quotient R/~ into a distributive lattice, where "~" denotes the equivalence relation of being associate elements. The equivalence between the existence of GCDs and the existence of LCMs is not a corollary of the similar result on complete lattices, as the quotient R/~ need not be a complete lattice for a GCD domain R.[citation needed]

If R is a GCD domain, then the polynomial ring R[X1,...,Xn] is also a GCD domain.[2]

R is a GCD domain if and only if finite intersections of its principal ideals are principal. In particular, , where is the LCM of and .

For a polynomial in X over a GCD domain, one can define its content as the GCD of all its coefficients. Then the content of a product of polynomials is the product of their contents, as expressed by Gauss's lemma, which is valid over GCD domains.


G-GCD domains

Many of the properties of GCD domain carry over to Generalized GCD domains,[6] where principal ideals are generalized to invertible ideals and where the intersection of two invertible ideals is invertible, so that the group of invertible ideals forms a lattice. In GCD rings, ideals are invertible if and only if they are principal, meaning the GCD and LCM operations can also be treated as operations on invertible ideals.

Examples of G-GCD domains include GCD domains, polynomial rings over GCD domains, Prüfer domains, and π-domains (domains where every principal ideal is the product of prime ideals), which generalizes the GCD property of Bézout domains and unique factorization domains.


  1. ^ Anderson, D. D. (2000). "GCD domains, Gauss' lemma, and contents of polynomials". In Chapman, Scott T.; Glaz, Sarah (eds.). Non-Noetherian Commutative Ring Theory. Mathematics and its Application. Vol. 520. Dordrecht: Kluwer Academic Publishers. pp. 1–31. doi:10.1007/978-1-4757-3180-4_1. MR 1858155.
  2. ^ Robert W. Gilmer, Commutative semigroup rings, University of Chicago Press, 1984, p. 172.
  3. ^ Ali, Majid M.; Smith, David J. (2003), "Generalized GCD rings. II", Beiträge zur Algebra und Geometrie, 44 (1): 75–98, MR 1990985. P. 84: "It is easy to see that an integral domain is a Prüfer GCD-domain if and only if it is a Bezout domain, and that a Prüfer domain need not be a GCD-domain".
  4. ^ Gilmer, Robert; Parker, Tom (1973), "Divisibility Properties in Semigroup Rings", Michigan Mathematical Journal, 22 (1): 65–86, MR 0342635.
  5. ^ Mihet, Dorel (2010), "A Note on Non-Unique Factorization Domains (UFD)", Resonance, 15 (8): 737–739.
  6. ^ Anderson, D. (1980), "Generalized GCD domains.", Commentarii Mathematici Universitatis Sancti Pauli., 28 (2): 219–233