Semigroup rings as almost Prüfer v-multiplication domains

Let D be an integral domain with quotient field K, Γ a nonzero torsion-free grading monoid and Γ ^∗= Γ \ { 0 }. In this paper, we characterize when the semigroup ring D[Γ] is an almost Prüfer v-multiplication domain or an almost Prüfer domain. We also give an equivalent condition for the composite semigroup ring D+ K[Γ ^∗] to be an almost Prüfer v-multiplication domain or an almost Prüfer domain when Γ ∩ - Γ = { 0 }.