localizations of Dedekind domains are Dedekind

If R is an integral domainMathworldPlanetmath with field of fractionsMathworldPlanetmath k and SR{0} is a multiplicative set, then the localizationMathworldPlanetmath at S is given by


(up to isomorphismPlanetmathPlanetmath). This is a subring of k, and the following theorem states that localizations of Dedekind domainsMathworldPlanetmath are again Dedekind domains.


Let R be a Dedekind domain and SR{0} be a multiplicative set. Then S-1R is a Dedekind domain.

A special case of this is the localization at a prime idealMathworldPlanetmathPlanetmath 𝔭, which is defined as R𝔭(R𝔭)-1R, and is therefore a Dedekind domain. In fact, if 𝔭 is nonzero then it can be shown that R𝔭 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