The Jacobson radical of commutative semigroup rings (Q1204541)

Let \(R[S]\) be a commutative semigroup ring. The author gives another description of the Jacobson radical of \(R[S]\) which yields necessary and sufficient conditions on \(R\) and \(S\) for the Jacobson radical and prime radical of \(R[S]\) to coincide. This in turn yields necessary and sufficient conditions for \(R[S]\) to be semiprimitive. These results recover several special cases of W. D. Munn.