Let $R$ be a ring with identity such that the quotient ring $R/S\_r$ is abelian, i. e., all idempotents of the quotient are central. Here, $S\_r$ means the right socle of the ring $R$. Do the nilpotent elements of $R$ belong to $S\_r$?
I have sent before, a post in reverse. 
Thanks for any answer!



Nilpotent Elements and the Socle
Let $R=K[x]/(x^3)$. Then the socle is $S\_r=(x^2)/(x^3)$ and the quotient is abelian because $R$ is abelian.
The element $x mod x^3$ is nilpotent but not in $S\_r$.


