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

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

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

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