# 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 |