proof of localizations of Dedekind domains are Dedekind

Let R be a Dedekind domainMathworldPlanetmath with field of fractionsMathworldPlanetmath k and SR{0} be a multiplicative set. We show that the localizationMathworldPlanetmath at S,


is again a Dedekind domain.

We use the characterization of Dedekind domains as integral domainsMathworldPlanetmath in which every nonzero ideal is invertible ( (see proof that a domain is Dedekind if its ideals are invertible).

Let 𝔞 be a nonzero integral ideal of S-1R. Then 𝔞R is a nonzero ideal of the Dedekind domain R, so it has an inverse


Here, 𝔟 is a fractional idealMathworldPlanetmath of R. Also let S-1𝔟 be the fractional ideal of S-1R generated by 𝔟,


The equalities


show that 𝔞 is invertible, so S-1R 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