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Abstract

Abstract. Let it be a property of rings that satisfies the conditions that (i) homomorphic images of w-rings are »r-rings and (ii) ideals of w-rings are w-rings. Let S be a semilattice P of semigroups Sa. If each semigroup ring R[Sa] (a e P) is v-semisimple, then the semigroup ring R[Sa) is also ir-semisimple. Conditions are found on P to insure that each R[Sa] (a G P) is w-semisimple whenever S is a strong semilattice P of semigroups Sa and R{S] is ir-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.Let it be a property of rings. A ring is called a ir-ring if it has property it. An ideal / of a ring is a it-ideal if / is a w-ring. A ring is m-semisimple if it has no nonzero w-ideals. Henceforth, we assume that the property m satisfies the conditions, (i) homomorphic images of w-rings are w-rings and (ii) ideals of w-rings are w-rings. For example, the properties of being nil, nilpotent, or left quasi-regular are such properties.Let A be a ring and let S be a semigroup.