Prüfer v-multiplication domains and related domains of the form D+DS[Γ∗]

Let D be an integral domain, S be a saturated multiplicative subset of D with D⊊DS, and Γ be a nonzero torsion-free grading monoid with Γ∩−Γ={0}. Let DS[Γ] be the semigroup ring of Γ over DS, Γ∗=Γ−{0}, and D(S,Γ)=D+DS[Γ∗], i.e., D(S,Γ)={f∈DS[Γ]|f(0)∈D}. We show that D(S,Γ) is a PvMD (resp., GCD-domain, GGCD-domain) if and only if D is a PvMD (resp., GCD-domain, GGCD-domain), Γ is a valuation semigroup and S is a t-splitting (resp., splitting, d-splitting) set of D.