# module-finite

Let $S$ be a ring with subring $R$.

We say that $S$ is module-finite over $R$ if $S$ is finitely generated^{} as an $R$-module.

We say that $S$ is ring-finite over $R$ if $S=R[{v}_{1},\mathrm{\dots},{v}_{n}]$ for some ${v}_{1},\mathrm{\dots},{v}_{n}\in S$.

Note that module-finite implies ring-finite, but the converse is false.

If $L$ is ring-finite over $K$, with $L,K$ fields, then $L$ is a finite extension^{} of $K$.

Title | module-finite |
---|---|

Canonical name | Modulefinite |

Date of creation | 2013-03-22 12:36:56 |

Last modified on | 2013-03-22 12:36:56 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 6 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 13B02 |

Classification | msc 13C05 |

Classification | msc 16D10 |

Related topic | FinitelyGeneratedRModule |

Defines | ring-finite |