Generalized Krull domains and the composite semigroup ring D+E[Γ⁎]

Let D⊆E be an extension of integral domains, Γ be a nonzero torsion-free (additive) grading monoid with quotient group G such that Γ∩−Γ={0}. Set Γ⁎=Γ∖{0} and R=D+E[Γ⁎]. In this paper, we show that if G satisfies the ascending chain condition on cyclic subgroups, then R is a generalized Krull domain (resp., generalized unique factorization domain) if and only if D=E, D is a generalized Krull domain (resp., generalized unique factorization domain) and Γ is a generalized Krull semigroup (resp., weakly factorial GCD-semigroup).