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 \begin{equation*} S^{-1}R=\left\{s^{-1}x:x\in R,s\in S\right\} \end{equation*}(up to isomorphism). This is a subring of $k$ , and the following theorem states that localizations of Dedekind domains are again Dedekind domains.

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.