# localizations of Dedekind domains are Dedekind

If $R$ is an integral domain^{} with field of fractions^{} $k$ and $S\subseteq R\setminus \{0\}$ is a multiplicative set, then the localization^{} at $S$ is given by

$${S}^{-1}R=\{{s}^{-1}x:x\in R,s\in S\}$$ |

(up to isomorphism^{}). This is a subring of $k$, and the following theorem states that localizations of Dedekind domains^{} are again Dedekind domains.

###### Theorem.

Let $R$ be a Dedekind domain and $S\mathrm{\subseteq}R\mathrm{\setminus}\mathrm{\{}\mathrm{0}\mathrm{\}}$ be a multiplicative set. Then ${S}^{\mathrm{-}\mathrm{1}}\mathit{}R$ is a Dedekind domain.

A special case of this is the localization at a prime ideal^{} $\U0001d52d$, which is defined as ${R}_{\U0001d52d}\equiv {(R\setminus \U0001d52d)}^{-1}R$, and is therefore a Dedekind domain. In fact, if $\U0001d52d$ is nonzero then it can be shown that ${R}_{\U0001d52d}$ is a discrete valuation ring.

Title | localizations of Dedekind domains are Dedekind |
---|---|

Canonical name | LocalizationsOfDedekindDomainsAreDedekind |

Date of creation | 2013-03-22 18:35:13 |

Last modified on | 2013-03-22 18:35:13 |

Owner | gel (22282) |

Last modified by | gel (22282) |

Numerical id | 4 |

Author | gel (22282) |

Entry type | Theorem |

Classification | msc 11R04 |

Classification | msc 13F05 |

Classification | msc 13H10 |

Related topic | Localization |