# 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 $S$,

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

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

 $\left(\mathfrak{a}\cap R\right)\mathfrak{b}=R.$

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

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

The equalities

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

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

Title proof of localizations of Dedekind domains are Dedekind ProofOfLocalizationsOfDedekindDomainsAreDedekind 2013-03-22 18:35:16 2013-03-22 18:35:16 gel (22282) gel (22282) 4 gel (22282) Proof msc 11R04 msc 13F05 msc 13H10