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'In mathematics, a '''GCD domain''' is an [[integral domain]] ''R'' with the property that any two non-zero elements have a [[greatest common divisor]] (GCD). Equivalently, any two non-zero elements of ''R'' have a [[least common multiple]] (LCM).<ref>{{cite book|author=Scott T. Chapman, Sarah Glaz (ed.)|title=Non-Noetherian Commutative Ring Theory|publisher=Springer|year=2000|series=Mathematics and Its Applications|isbn=0-7923-6492-9|page=479}}</ref> A GCD domain generalizes a [[unique factorization domain]] to the 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 [[subclass (set theory)|class inclusions]]: {{Commutative ring classes}} {{Algebraic structures |Ring}} == Properties == Every irreducible element of a GCD domain is prime (however irreducible elements need not exist, even if the GCD domain is not a field). A GCD domain is [[integrally closed]], and every nonzero element is [[primal element|primal]].<ref>[http://planetmath.org/proofthatagcddomainisintegrallyclosed proof]</ref> 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 a LCM ''m'' of ''x'' and ''y'' can be chosen such that {{nowrap|''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 a LCM of ''x'' and ''y'', and vice versa. It [[distributive lattice#Characteristic properties|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 lattice]]s, as the quotient ''R''/~ need not be a complete lattice for a GCD domain ''R''.{{Citation needed|date=March 2015}} If ''R'' is a GCD domain, then the polynomial ring ''R''[''X''₁,...,''X''_''n''] is also a GCD domain.<ref>Robert W. Gilmer, ''Commutative semigroup rings'', University of Chicago Press, 1984, p. 172.</ref> R is a GCD domain if and only if finite intersections of its [[principal ideal]]s are principal. In particular, <math>(a) \cap (b) = (c)</math>, where <math>c</math> is the LCM of <math>a</math> and <math>b</math>. 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 (polynomial)|Gauss's lemma]], which is valid over GCD domains. ==Examples== *A [[unique factorization domain]] is a GCD domain. Among the GCD domains, the unique factorization domains are precisely those that are also [[atomic domain]]s (which means that at least one factorization into irreducible elements exists for any nonzero nonunit). *A [[Bézout domain]] (i.e., an integral domain where every finitely generated ideal is principal) is a GCD domain. Unlike [[principal ideal domain]]s (where ''every'' ideal is principal), a Bézout domain need not be a unique factorization domain; for instance the ring of [[entire function]]s is a non-atomic Bézout domain, and there are many other examples. An integral domain is a [[Prüfer domain|Prüfer]] GCD domain if and only if it is a Bézout domain.<ref>{{citation | last1 = Ali | first1 = Majid M. | last2 = Smith | first2 = David J. | issue = 1 | journal = Beiträge zur Algebra und Geometrie | mr = 1990985 | pages = 75–98 | title = Generalized GCD rings. II | url = http://www.emis.de/journals/BAG/vol.44/no.1/6.html | volume = 44 | year = 2003}}. P.&nbsp;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.".</ref> *If ''R'' is a non-atomic GCD domain, then ''R''[''X''] is an example of a GCD domain that is neither a unique factorization domain (since it is non-atomic) nor a Bézout domain (since ''X'' and a non-invertible and non-zero element ''a'' of ''R'' generate an ideal not containing 1, but 1 is nevertheless a GCD of ''X'' and ''a''); more generally any ring ''R''[''X''₁,...,''X''_''n''] has these properties. *A [[Commutative ring|commutative]] [[monoid ring]] <math>R[X; S]</math> is a GCD domain iff <math>R</math> is a GCD domain and <math>S</math> is a [[Torsion-free group|torsion-free]] [[Cancellative semigroup|cancellative]] GCD-semigroup. A GCD-semigroup is a semigroup with the additional property that for any <math>a</math> and <math>b</math> in the semigroup <math>S</math>, there exists a <math>c</math> such that <math>(a + S) \cap (b + S) = c + S</math>. In particular, if <math>G</math> is an [[abelian group]], then <math>R[X;G]</math> is a GCD domain iff <math>R</math> is a GCD domain and <math>G</math> is torsion-free.<ref>{{citation | last1 = Gilmer | first1 = Robert | last2 = Parker | first2 = Tom | volume = 22 | issue = 1 | journal = Michigan Mathematical Journal | mr = 342635 | pages = 65–86 | title = Divisibility Properties in Semigroup Rings | url = http://projecteuclid.org/euclid.mmj/1029001210 | year = 1973}}.</ref> == References == <references /> [[Category:Commutative algebra]] [[Category:Ring theory]]'