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=s∈ SRₛ 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.