$R$ is a number ring, $K$ the field of fractions and $R$ is Dedekind.
$\Rightarrow$ exists a set of primes $S$ of $\mathscr{O}_{K}$ and: $R=\bigcap\limits_{\mathbb{p} \notin S}\mathscr{O}_{\mathbb{p}}=\{x\in K:ord_{\mathbb{p}}(x)\geq0\forall\mathbb{p}\notin S\}$
I do not know how to begin here. I started with what this set is: all the $\mathbb{p}$ that are not in $S$ are hence not integral, and thus of the form $\frac{r}{p}$ for some $p$ prime, and hence the inverse to certain primes?
We had a Theorem for number rings that are Dedekind:
$R$ as above, then $\exists \phi: \mathcal{I}(R)\rightarrow \oplus_{\mathbb{p}}$ $\mathbb{Z}$ with $I \mapsto (ord_{\mathbb{p}}(I))_{\mathbb{p}}$, and for all $I = \prod\limits_{\mathbb{p}}\mathbb{p}^{ord_{\mathbb{p}}(I)}$
I'd be very happy about any kind of help :) All the Best, Luca