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.