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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). 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). == 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.〔(proof )〕 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 , 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 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''. If ''R'' is a GCD domain, then the polynomial ring ''R''() is also a GCD domain.〔Robert W. Gilmer, ''Commutative semigroup rings'', University of Chicago Press, 1984, p. 172.〕 For a polynomial in ''X'' over a GCD domain, one can define its contents as the GCD of all its coefficients. Then the contents of a product of polynomials is the product of their contents, as expressed by Gauss's lemma, which is valid over GCD domains. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「GCD domain」の詳細全文を読む スポンサード リンク
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