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Suppose I have two domains, $A\subset B$, where $A$ is Dedekind and $\operatorname{Frac}(A)=\operatorname{Frac}(B)$. I also know that $B$ is both integrally closed and has height $1$. Is $B$ necessarily Dedekind? If not, I'd love to see a counterexample.

(Note: I'm adding the homework tag since this is motivated by (and would finish off) a homework problem about global function fields, although I now know another way, with slightly stronger hypotheses, to get the necessary result with Riemann-Roch.)

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    @Keenan You might find of interest the following results: for a domain D: every overring of D is Noetherian iff D is 1-dimensional Noetherian; every overring of D is integrally closed iff D is a Prufer domain.2012-12-12

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In the multiplicative theory of ideals it is well known that the overrings of Dedekind domains are also Dedekind. See, for instance, Larsen and McCarthy, Multiplicative Theory of Ideals, Theorem 6.21 or these notes, Proposition 22.2.

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    @Mariano That definition of overring is used frequently by rings theorists studying factorization / divisibility theory and related topics, e.g. see these [various characterizations of Prufer domains.](http://math.stackexchange.com/a/119552/242)2012-12-11