On Semisimple Semigroup Rings

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.

Semisimple artinian graded rings

“…By Maschke's theorem the group algebras FG, are semisimple. Therefore FS is semisimple by [22], Theorem 1. Every semisimple finite dimensional algebra is semisimple Artinian.…”

“…A cornmutative band is called a semilattice. Semilattice-graded rings were considered in [2], [7], [ll], [22] and other papers. Then R = es,, R, is an S-graded ring, and Re = 0 for all idempotents e E S .…”