Let $R$ be a Noetherian ring with finitely many maximal ideals. Let $I$ be its nilradical, i.e. the intersection of all prime ideals, and $J$ be the Jacobson radical i.e. the intersection, hence product (since there are finitely many), of maximal ideals. If $R/I$ is $J$-adically complete ( with the topology induced by the nbd base $\{(J^n+I)/I\}_{n\ge 0}$ ); then how to show that $R$ is $J$-adically complete ?
Since $R/I$ is Noetherian and $J$-adic complete, so for any Noetherian $R/I$-mod $M$, $M$ is $J$-adic complete , so in particular $\cap_{n\ge 1} J^n M=(0)$.
I don't know how to proceed further. Please help.