Generalized Krull domains and the composite semigroup ring \(D+E[\Gamma^{\ast}]\)
Let \(D\subseteq E\) be an extension of integral domains, let \(\Gamma\) be a nonzero torsion free additive grading monoid with quotient group \(G\) such that \(\Gamma \cap {-\Gamma}=\{0\}\), and let \(R =D+E[{\Gamma}^*]\), where \({\Gamma}^*= {\Gamma} \backslash \{0\} \). Under the assumption that \(G \) satisfies the ascending chain condition on cyclic subgroups, the author proves the following results: (i) \(R\) is generalized Krull domain if and only if \(D=E\), \(D\) is a generalized Krull domain and \(\Gamma\) is a generalized Krull semigroup, and (ii) \(R\) is a generalized unique factorization domain if and only if \(D=E\), \(D\) is generalized unique factorization domain and \(\Gamma \) is a weakly factorial GCD-semigroup.
scientific article; zbMATH DE number 6104523
scientific article