GCD ___domain: Difference between revisions

Revision as of 04:39, 22 June 2016

Revision as of 01:03, 23 June 2016

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___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]] [[semigroupmonoid 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