Prime ideal in a Dedekind domain

Question

Let $\mathfrak p $ be a prime ideal in a Dedekind domain $O$ with field of fractions $K$. Define $$\mathfrak p^{-1}= \{x \in K: x\mathfrak p \subset O\}.$$ Let $\mathfrak a \subset \mathfrak p$ and $\mathfrak a \neq \mathfrak p$ be an ideal of $O$. How do I prove

i) $\mathfrak a\mathfrak p^{-1} \neq \mathfrak p\mathfrak p^{-1}$;

ii)$(\mathfrak a\mathfrak p^{-1})\mathfrak p = \mathfrak a$.

Answer 1

Massive hint: (1) and (2) follow immediately from a result in algebraic number theory that $pp^{-1} = \mathcal{O}$. I can add a proof of this result if you would like. For example in (1) if we had $ap^{-1} = pp^{-1} = \mathcal{O}$ then we get $a = p\mathcal{O} = p$ contradicting $a \neq p$.

Edit: If I recall correctly the result that $pp^{-1} = \mathcal{O}$ can be found in Neukirch.