Let $I$ be a nilpotent ideal in a commutative ring $R$, let $M$ and $N$ be $R$-modules and let $\phi : M \to N$ be an $R$-module homomorphism. Show that if the induced map $\overline\phi: M/IM \to N/IN$ is surjective, then $\phi$ is surjective.
I have proceeded in this way $\overline\phi: M/IM \to N/IN \Rightarrow \hat \phi:(M/IM)^n=M^n/(IM)^n \to (N/IN)^n=N^n/(IN)^n$ is surjective. Now from here how do I conclude the claim?
Do I have to show that $M/I^nM \to N/I^nN$ is surjective?
I am completely stuck here. Need help.