I am a bit confused by a step of a solution in commutative algebra.
So the context is that $a$ is an ideal of $A$, contained in the Jacobson Ideal of $A$, $M$ is an $A$ module and $N$ is a finitely generated $A$ module, $f$ is a homomorphism from $M$ to $N$, and the question asks us to prove that if the induced map from $M/aM$ to $N/aN$ is surjective, then so is $f$.
I understand the second part of the solution where we use Nakayama's Lemma, but the first part of the proof says $(N/{\rm Im}(f)/a(N/{\rm Im}(f))$ is a zero module, which I don't see why.
I am not super fluent with group homomorphisms' induced results, so I would really appreciate it if you could indicate which theorems/facts/intuitions were used here or can be generally helpful when dealing with homomorphisms and quotient (of quotient) groups.
Thank you.