# How to show $M$ is a finitely generated $A$-module?

In Corollary 1.2.3, in order to apply Nakayama's lemma, how to show $$M$$ is a finitely generated $$A$$-module?

• $M$ is not necessarily f.g. as $A$-module! But if $M=mM$, then $M=nM$. – user26857 Sep 11 at 10:14