# integrity characterized by places

###### Theorem.

Let $R$ be a subring of the field $K$, $1\in R$. An element $\alpha $ of the field is integral over $R$ if and only if all places (http://planetmath.org/PlaceOfField) $\phi $ of $K$ satisfy the implication^{}

$$\phi \mathrm{is}\mathrm{finite}\mathrm{in}R\Rightarrow \phi (\alpha )\mathrm{is}\mathrm{finite}.$$ |

1. Let $R$ be a subring of the field $K$, $1\in R$. The integral closure^{} of $R$ in $K$ is the intersection^{} of all valuation domains in $K$ which contain the ring $R$. The integral closure is integrally closed^{} in the field $K$.

2. Every valuation domain is integrally closed in its field of fractions^{}.

Title | integrity characterized by places |
---|---|

Canonical name | IntegrityCharacterizedByPlaces |

Date of creation | 2013-03-22 14:56:57 |

Last modified on | 2013-03-22 14:56:57 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 11 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 12E99 |

Classification | msc 13B21 |

Related topic | Integral |

Related topic | PlaceOfField |