On semisimple semigroup rings

Abstract:

Let $\pi$ be a property of rings that satisfies the conditions that (i) homomorphic images of $\pi$-rings are $\pi$-rings and (ii) ideals of $\pi$-rings are $\pi$-rings. Let S be a semilattice P of semigroups ${S_\alpha }$. If each semigroup ring $R[{S_\alpha }](\alpha \in P)$ is $\pi$-semisimple, then the semigroup ring $R[{S_\alpha }]$ is also $\pi$-semisimple. Conditions are found on P to insure that each $R[{S_\alpha }](\alpha \in P)$ is $\pi$-semisimple whenever S is a strong semilattice P of semigroups ${S_\alpha }$ and $R[S]$ is $\pi$-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.