# 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

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

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

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

Title localizations of Dedekind domains are Dedekind LocalizationsOfDedekindDomainsAreDedekind 2013-03-22 18:35:13 2013-03-22 18:35:13 gel (22282) gel (22282) 4 gel (22282) Theorem msc 11R04 msc 13F05 msc 13H10 Localization