Let $M$ be a free $R$-module we knew that the endomorphism ring of $M$ is isomorphic to the ring of matrices over $R$. and also we know that $M_{n}(R)=I(M_{n}(R))$, where $I(M_{n}(R))$ is the subring of $M_{n}(R)$ generated by idempotents of $M_{n}(R)$. Does endomorphism ring of any free $R$-module equal to $I(M_{n}(R))$? or $M$ must be finitely generated free $R$-module?