On semisimple semigroup rings

Abstract

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