Let $R$ be a communicative unity ring, $I\subseteq R$ is a nilpotent ideal, $M$ is a $R$-module, $f:M\rightarrow M$ is a $R$-homomorphism. If the induced map $\bar{f}:M/IM\rightarrow M/IM$ is identity map, I want to show that $f$ is an isomorphism.
If $M$ is finite generated，by Nakayama‘s lemma，we have $f$ is surjective，then use Nakayama’s lemma again，the surjective endomorphism of finite generated module is isomorphic and we done
For general case，can someone give me a prove of this, or a counter example? Thank you.