A Note on the Jacobson Radical of a Graded Ring

We prove that J (Rₑ) =Rₑ J (R), where S is a cancellative partial groupoid with idempotent e, R=ₒ ₒRₛ an Artinian S-graded ring inducing S, J (R) the Jacobson radical of R and J (Rₑ) the Jacobson radical of Rₑ. We also prove that J (R) is nil if J (Rₑ) is nil under certain assumptions.