A Note on the Jacobson Radical of a Graded Ring - Ilić-Georgijević, Emil (Sarajevo Journal of Mathematics, 2024)
We prove that $J(R_e)=R_e\cap J(R),$ where $S$ is a cancellative partial groupoid with idempotent $e,$ $R=\bigoplus_{s\in S}R_s$ an Artinian $S$-graded ring inducing $S,$ $J(R)$ the Jacobson radical of $R$ and $J(R_e)$ the Jacobson radical of $R_e.$ We also prove that $J(R)$ is nil if $J(R_e)$ is nil under certain assumptions.