I am trying to show that the quotient of a Dedekind domain $A$ by an ideal $\mathfrak{a}$ is a PIR (principal ideal ring). Now by using the Chinese Remainder Theorem and the fact that a direct product of PIRs is a PIR it will suffice to prove the following:
Let $\mathfrak{p}$ be a prime ideal of a Dedekind domain $A$. Then for any $n \in \Bbb{N}$ we have that $A/\mathfrak{p}^n$ is a PIR.
Now it is proven in Proposition 9.2 of Atiyah - Macdonald that a Noetherian local domain of dimension $1$ is a PID. Applying this proposition mutadis mutandis I think it will suffice to prove that $A/\mathfrak{p}^n$ is isomorphic to some kind of localization. Now we know that $$(A/\mathfrak{p}^n)_{\mathfrak{p}} \cong A_\mathfrak{p} \otimes_A A/\mathfrak{p}^n.$$
If $A/\mathfrak{p}^n$ has the structure of an $A_\mathfrak{p}$ -module then the right hand side is in fact a tensor product over the localization and so $$A_\mathfrak{p} \otimes_A A/\mathfrak{p}^n \cong A_\mathfrak{p} \otimes_{A_\mathfrak{p} } A/\mathfrak{p}^n \cong A/\mathfrak{p}^n.$$
My Question is: Can we make $A/\mathfrak{p}^n$ into an $A_\mathfrak{p}$ - module?