proof of localizations of Dedekind domains are Dedekind

Let $R$ be a Dedekind domain with field of fractions $k$ and $S\subseteq R\setminus \{0\}$ be a multiplicative set. We show that the localization at $S$,

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

is again a Dedekind domain.

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

Let $𝔞$ be a nonzero integral ideal of $S^{-1}R$. Then $𝔞\cap R$ is a nonzero ideal of the Dedekind domain $R$, so it has an inverse

$$\left(𝔞\cap R\right)𝔟=R.$$

Here, $𝔟$ is a fractional ideal of $R$. Also let $S^{-1}𝔟$ be the fractional ideal of $S^{-1}R$ generated by $𝔟$,

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

The equalities

$$𝔞(S^{-1}𝔟)=S^{-1}\left((𝔞\cap R)𝔟\right)=S^{-1}R$$

show that $𝔞$ is invertible, so $S^{-1}R$ is a Dedekind domain.