counter example to Nakayama’s lemma for non-finitely generated modules

The hypothesis that the module $M$ be finitely generated is reallynecessary. For example, the field of $p$-adic numbers ${\mathbb{Q}}_p$ isnot finitely generated over its ring of integers ${\mathbb{Z}}_p$ and$(p){\mathbb{Q}}_p={\mathbb{Q}}_p$.

In one sense, the reason why ${\mathbb{Q}}_p$ is “bad” is that is has noproper sub module which is also maximal. Thus ${\mathbb{Q}}_p$ has no non-zero simplequotient. This explains why the followingProof of Nakayama’s Lemma (http://planetmath.org/ProofOfNakayamasLemma2)does not work for non-finitely generated modules.