Let $A$ be a Dedekind domain, $S$ a multiplicatively closed subset of $A$. Show that $S^{-1}A$ is either a Dedekind domain or the field of fractions of $A$.
Attempt: $A$ is an integrally closed Noetherian domain of dimension one. Then we know that $S^{-1}A$ is integrally closed and noetherian. $S^{-1}A$ a domain since it is contained in the field of fractions $K$ of $A$. After that how to proceed?
The localization $S^{−1}A$ is normal because $A$ is. Also, any nonzero localization of a noetherian domain is a noetherian domain.
It remains to show that $S ^{−1}A$ is either a field or of dimension $1$. Suppose it is not a field. Then there exists $0\neq f ∈ A\setminus S$. Thus ( $\frac{f}{1}) ⊂ S^{−1}A$ is a non-zero proper ideal and thus contained in some maximal ideal of $S^{−1}A$. This shows that $\dim(S^{−1}A) > 1$. On the other hand, the one-to-one correspondence of prime ideals of $S^{−1}A$ and prime ideals of $A$ which have empty intersection with $S$ is inclusion-preserving; hence $\dim(S^{−1}A)\leq \dim A = 1$