On Semisimple Semigroup Rings

Let be a property of rings that satisfies the conditions that (i) homomorphic images of -rings are -rings and (ii) ideals of -rings are -rings. Let S be a semilattice P of semigroups <!-- MATH ${S_\alpha }$ --> . If each semigroup ring <!-- MATH $R[{S_\alpha }](\alpha \in P)$ --> is -semisimple, then the semigroup ring <!-- MATH $R[{S_\alpha }]$ --> is also -semisimple. Conditions are found on P to insure that each <!-- MATH $R[{S_\alpha }](\alpha \in P)$ --> is -semisimple whenever S is a strong semilattice P of semigroups <!-- MATH ${S_\alpha }$ --> and is -semisimple. Examples are given to show that the conditions on P cannot be removed. These results and examples answer several questions raised by J. Weissglass.