# proof of localizations of Dedekind domains are Dedekind

Let $R$ be a Dedekind domain^{} with field of fractions^{} $k$ and $S\subseteq R\setminus \{0\}$ be a multiplicative set. We show that the localization^{} at $S$,

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

is again a Dedekind domain.

We use the characterization of Dedekind domains as integral domains^{} in which every nonzero ideal is invertible (http://planetmath.org/FractionalIdeal) (see proof that a domain is Dedekind if its ideals are invertible).

Let $\U0001d51e$ be a nonzero integral ideal of ${S}^{-1}R$. Then $\U0001d51e\cap R$ is a nonzero ideal of the Dedekind domain $R$, so it has an inverse

$$\left(\U0001d51e\cap R\right)\U0001d51f=R.$$ |

Here, $\U0001d51f$ is a fractional ideal^{} of $R$. Also let ${S}^{-1}\U0001d51f$ be the fractional ideal of ${S}^{-1}R$ generated by $\U0001d51f$,

$${S}^{-1}\U0001d51f=\{{s}^{-1}x:x\in \U0001d51f,s\in S\}.$$ |

The equalities

$$\U0001d51e({S}^{-1}\U0001d51f)={S}^{-1}\left((\U0001d51e\cap R)\U0001d51f\right)={S}^{-1}R$$ |

show that $\U0001d51e$ is invertible, so ${S}^{-1}R$ is a Dedekind domain.

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

Canonical name | ProofOfLocalizationsOfDedekindDomainsAreDedekind |

Date of creation | 2013-03-22 18:35:16 |

Last modified on | 2013-03-22 18:35:16 |

Owner | gel (22282) |

Last modified by | gel (22282) |

Numerical id | 4 |

Author | gel (22282) |

Entry type | Proof |

Classification | msc 11R04 |

Classification | msc 13F05 |

Classification | msc 13H10 |