On the Jacobson Radical of Semigroup Graded Rings Original Research Article

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 component corresponding to an idempotent semigroup element and that a ring graded by a finite semigroup is a Jacobson ring if each idempotent graded component is a Jacobson ring. As an application of graded results we prove that a PI semigroup algebra is a Jacobson ring provided that all homomorphic images of the semigroup have finite rank.