Let $I$ be a nilpotent ideal in a commutative ring $R$, let $M$ and $N$ be $R-$modules and let $\phi$ : $M \rightarrow N$ be an $R-$module homomorphism. Show that if the induced map $\bar{\phi} : \frac{M}{IM} \rightarrow \frac{N}{IN}$ is surjective, then $\phi$ is surjective.
Let $I$ be a nilpotent ideal in a commutative ring $R$, let $M$ and $N$ be $R-$modules and let $\phi$ : $M \rightarrow N$ be an $R-$module homomorphism. Show that if the induced map $\bar{\phi} : \frac{M}{IM} \rightarrow \frac{N}{IN}$ is surjective, then $\phi$ is surjective.
so $I^k=0,$ for some $k \geq 1.$ since $\bar{\phi}$ is surjective, we have: $\frac{N}{IN}=\bar{\phi}\left(\frac{M}{IM}\right)=\ frac{\phi(M)+IN}{IN}.$ thus: $N=\phi(M)+IN,$ which gives us: $N=\phi(M)+I(\phi(M)+IN)=\phi(M)+I^2N,$ because
$I\phi(M) \subseteq \phi(M).$ repeating this we'll get: $N=\phi(M)+I^2(\phi(M) + IN)=\phi(M) + I^3N = \cdots = \phi(M) + I^kN=\phi(M). \ \Box$