Number of generators of localized ideal

Question

Let $A$ be a Dedekind domain and let $\mathfrak{p}\subset A$ be a prime. Assume that $\mathfrak{p}=(a,b)$ and suppose that $\mathfrak{p}A_\mathfrak{p}$ is principal.

Is it true that $\mathfrak{p}A_\mathfrak{p}$ must be generated by either $a/1$ or $b/1$?

Answer 1

Hint:

For each non-zero prime ideal $\mathfrak p$ of a Dedekind domain $A$, $A_\mathfrak p$ is a (discrete) valuation domain.