Every divisor class of Krull monoid domains contains a prime ideal

Let D be an integral domain, Γ be a torsion-free grading monoid, and D[Γ] be the monoid domain of Γ over D. Suppose that D[Γ] is a Krull domain, and let Cl(D[Γ]) be the divisor class group of D[Γ]. We show that every divisor class of D[Γ] contains a prime ideal. As a corollary, we have that D[Γ] is a half-factorial domain if and only if |Cl(D[Γ])|⩽2; hence in this case, either D or Γ is factorial. We also show that if T is the set of non-homogeneous prime elements of D[Γ], then D[Γ]T is a π-domain with Cl(D[Γ])=Cl(D[Γ]T).