Let $f$ be an idempotent element in a ring $S$ with Jacobson radical $J$ so that both $fJf$ and $(1-f)J(1-f)$ are nil. I guess that $J$ is nil too, but I am not sure.
I know that the former is the Jacobson radical of the ring $fSf$, and the latter is that of the ring $(1-f)S(1-f)$. I thank in advance any help or suggestion in this regard.