The \(t\)-class group of a semigroup ring. (Q2773012)

Let \(S\) be an integrally closed torsion-free grading monoid with quotient group \(G\), and \(R\) an integral domain with quotient field \(K\). Recall that a non-zero element of \(G\) is of type \((0,0,\dots,1\) iff \(S\) satisfies the ascending chain condition on cyclic submonoids. We show that if each non-zero element of \(G\) is of type \((0,0,\dots)\), then every \(t\)-ideal \(J\) of \(K[S]\) is of the form \(J=hK[I]\) for some \(h\in K[G]\) and \(t\)-ideal \(I\) of \(S\). Using this, we also show that if the semigroup ring \(R[S]\) is a PVMD and each non-zero element of \(G\) is of type \((0,0,\dots)\), then Cl\(_t(R[S])\cong \text{Cl}_t (R)\oplus \text{Cl}_t(S)\).