Every divisor class of Krull monoid domains contains a prime ideal

Abstract

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 ) .