given a ring $R$, a nilpotent ideal $I$ and a morphism $\phi$ of $R$-modules $M \to N$, such that $M/IM \to N/IN$ is an isomorphism.
It is easy to see that this implies $\phi$ surjective, but what about injectivity?
given a ring $R$, a nilpotent ideal $I$ and a morphism $\phi$ of $R$-modules $M \to N$, such that $M/IM \to N/IN$ is an isomorphism.
It is easy to see that this implies $\phi$ surjective, but what about injectivity?
The answer is no in general. E.g. if $M = R$ and $N = R/I$, then the natural surjection $M \to N$ will induce an isomorphism $M/I \to N/I$, which won't be an isomorphism unless $I = 0$.