Let $R$ be a unitary associative ring and $J$ be an ideal of $M_{n}(R)$. We know that there is an ideal $I$ of $R$ such that $J=M_{n}(I)$. Now there is a question.

Question: If $J$ is generated by a subset of idempotent elements of $M_{n}(R)$ say $S$, **is** $I$ generated by a subset of idempotent elements of $R$, which related to $S$?