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
Canonical name	CounterExampleToNakayamasLemmaForNonfinitelyGeneratedModules
Date of creation	2013-03-22 18:03:55
Last modified on	2013-03-22 18:03:55
Owner	sjm (20613)
Last modified by	sjm (20613)
Numerical id	9
Author	sjm (20613)
Entry type	Example
Classification	msc 13C99