I'm trying to solve the following problem:
Suppose $R$ is a Dedekind domain which contains (nonzero) ideals $\mathfrak{a}$ and $\mathfrak{b}$. By first dealing with the case where $R$ is a PID and then localising, show that $\mathfrak{a} \supset \mathfrak{b} \Longleftrightarrow \mathfrak{a} \, \vert \, \mathfrak{b}$.
Now the PID case was pretty easy to do, but I don't really understand what the question means or intends by "treating the case where $R$ is a PID and then localising"; specifically I don't think I have a good understanding of what 'localising' means.
To my understanding, the localisation of $R$ by $S \subset R$ is $S^{-1}R = \{\frac{r}{s}: r \in R,\,s \in S\} \subset K$, where $K$ is the field of fractions of $R$. However, I don't have a clue what I'm meant to do to 'localise' for the case of a Dedekind domain here: could anyone help? Since I don't have any experience working with localisations, I'd be very grateful for any detailed but relatively simple explanations you could provide; thanks in advance.