Let $R$ be noncommutative unital ring such that each element of the quotient $R/Soc(R_R)$ is idempotent. If the nilpotent elements of $R$ form an ideal, is it true that the idempotents of $R/J(R)$ are central? Or, conversely, does this latter condition imply that the nilpotent elements of $R$ form an ideal (with the pre-assumption about $R/Soc(R_R)$ of course!)?
I know that if the nilpotent elements of $R$ form an ideal, then this ideal is just the Jacobson radical $J(R)$, because, the former would be equal to the prime radical which is always contained in $J(R)$. On the other hand, $J(R/Soc(R_R))=0$, so that $J(R)\subseteq Soc(R_R)$ hence $J(R)^2=0$ and so $J(R)$ falls into the set of nilpotent elements of $R$.
Thanks for any contribution!
