On radicals of semigroup algebras (Q1300540)

For any prime number \(p\) a relation \(\xi_p\) on a semigroup \(S\) is defined by \((x,y)\in\xi_p\) if there exists \(n\geq 1\) such that \(xs_1x\cdots xs_{p^n-1}x=ys_1y\cdots ys_{p^n-1}y\) for all \(s_1,\dots,s_{p^n-1}\in S\). It is shown that \(\xi_p\) is a congruence on \(S\). If \(R\) is a ring of characteristic \(p\), then the ideal \(I(\xi_p)\) of the semigroup ring \(R[S]\) determined by \(\xi_p\) is contained in the prime radical of \(R[S]\). This is applied to a study of the Jacobson radical of semigroup rings \(F[S]\) of certain semigroups \(S\) that are semilattices of groups, where \(F\) is a field. Reviewer's note: Theorem 2.4, and hence Theorem 4.1, are not proved in the form in which they are stated. However, the proofs are complete for the classes of semigroups to which these results are applied in Section 4.