Let $R$ be a unital ring such that each right ideal of $R/J(R)$ is idempotent and the Jacobson radical $J(R)$ of $R$ is nil, and $I$ be a right ideal of $R$. Assume that $i\in I$ and $x\in RiR$. I search for a non-trivial idempotent right ideal $A$ of $R$ such that $ix\in A$.
Since $ix\in (iR)^2$, we may clearly choose $A=iR$ if for any $i\in I$ one has $iR=(iR)^2$.
Any help would be appreciated!