The hypothesis that the module $M$ be finitely generated is really necessary. For example, the field of $p$ -adic numbers $\Q_p$ is not finitely generated over its ring of integers $\Z_p$ and $(p)\Q_p = \Q_p$ .

In one sense, the reason why $\Q_p$ is ``bad'' is that is has no proper sub module which is also maximal. Thus $\Q_p$ has no non-zero simple quotient. This explains why the following Proof of Nakayama's Lemma does not work for non-finitely generated modules.