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If $ A $, $ B $ are subrings of a field $ K $ that are Dedekind domains, is it true that $ A \cap B $ is also a Dedekind domain?

Case in point: $ \mathbb{Z}_{(3) } , \mathbb{Z}_{(5) } \subset \mathbb{Q} $ and their intersection is again a localizattion (at the complement of $ (3) \cup (5) $) which is a Dedekind domain.

Hodge-Tate
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  • Essentially, what I need to know is whether $ \dim (A \cap B) \leq 1 $. In general, subrings can have larger dimension (i.e. subrings in fields). – Hodge-Tate Sep 06 '18 at 23:22

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