In the book "Algebraic Geometry and Commutative Algebra" from S.Bosch, on page 37 there is a Corollary (n°11) of the Nakayama's Lemma:
"Let M be a finitely generated $R$-module and $Q\subset J(R)$ an ideal. Then if $N\subset M$ is an $R$-submodule satisfying $M = N+QM$, we must have $M=N$"
The proof is clear to me, generally speaking: it applies directly the Nakayama's Lemma to $M/N$ in order to say that $M/N = \{0\} \Rightarrow M=N$. It's not fully clear the first statement i.e $M/N = Q(M/N)$ (a). Indeed with the third isomorphism theorem, I would say that
$\dfrac{M}{N} = \dfrac{N+QM}{N} \sim \dfrac{QM}{QM\cap N}$
I think this is the wrong way to prove (a), mostly because I got isomorphic and not equal as required by Nakayama's Lemma. Plus I didn't prove the statement If I don't prove that $QM \cap N = Q(M\cap N) = QN$, so my idea has quite a lot of problems