TITLE: Ideals generated by idempotent elements
QUESTION [5 upvotes]: 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 $S$?

REPLY [2 votes]: I don't know if this helps. 
We readily obtain the following assertions:
(1) If the ideal $J=M_n(I)$ of $M_n(R)$ is generated by some idempotents,
then $I$ is an idempotent ideal, i.e., $I^2=I$.
(2) If $J=M_n(I)$ is generated by finitely many idempotents, then $I$
is a finitely generated ideal.
Thus in the case where $R$ is commutative and $J$ is generated by
finitely many idempotents, there exists an idempotent $e\in R$ such
that $I=Re$.