ON THE JACOBSON RADICAL OF SEMIGROUP GRADED RINGS

We study the Jacobson radical of semigroup graded rings. We show that the Jacobson radical of a ring graded by a (locally) finite semigroup is (locally) nilpotent if the same is true of each homogeneous compone nt corresponding to an idempotent semigroup element and that a ring gr aded by a finite semigroup is a Jacobson ring if each idempotent grade d component is a Jacobson ring. As an application of graded results we prove that a PI semigroup algebra is a Jacobson ring provided that al l homomorphic images of the semigroup have finite rank.