On Dedekind domains whose class groups are direct sums of cyclic groups

For a given family $(G_i)_{i \in \N}$ of finitely generated abelian groups, we construct a Dedekind domain $D$ having the following properties. \begin{enumerate} \item $\Pic(D) \cong \bigoplus_{i \in \N}G_i$. \item For each $i \in \N$, there exists a submonoid $S_i \subseteq D^{\bullet}$ with $\Pic (D_{S_i}) \cong G_i$. \item Each class of $\Pic (D)$ and of all $\Pic (D_{S_i})$ contains infinitely many prime ideals. \end{enumerate} Furthermore, we study orders as well as sets of lengths in the Dedekind domain $D$ and in all its localizations $D_{S_i}$.