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

The hypothesis that the module $M$ be finitely generated is really necessary. For example, the field of $p$-adic numbers $\mathbb{Q}_{p}$ is not 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 no proper sub module which is also maximal. Thus $\mathbb{Q}_{p}$ has no non-zero simple quotient. This explains why the following Proof of Nakayama’s Lemma (http://planetmath.org/ProofOfNakayamasLemma2) does not work for non-finitely generated modules.

Title counter example to Nakayama’s lemma for non-finitely generated modules CounterExampleToNakayamasLemmaForNonfinitelyGeneratedModules 2013-03-22 18:03:55 2013-03-22 18:03:55 sjm (20613) sjm (20613) 9 sjm (20613) Example msc 13C99