Is my proof of proposition 6.2 on page 75 correct? (it's different from what they do in the book)
Proposition 6.2.: $M$ is a Noetherian $A$ module $\iff$ every submodule of $M$ is finitely generated.
My proof:
$\implies$ Assume $M$ has a submodule $N$ that is not finitely generated. Say, $N = \langle \bigcup_{i=1}^\infty \{ n_i \} \rangle$. Then the following is an increasing chain that is not stationary: $\langle n_1 \rangle \subset \langle n_1, n_2 \rangle \subset \langle n_1, n_2, n_3 \rangle \subset \dots$ Hence $M$ is not Noetherian.
$\Longleftarrow$ Assume $M$ is not Noetherian. Then there is a non-stationary increasing chain of submodules $N_1 \subset N_2 \subset \dots$. Then $N = \bigcup_{k=1}^\infty N_k$ is a submodule. If $N$ was finitely generated, $N_k$ would be stationary. Hence $N$ is a submodule that is not finitely generated.