Let $A$ and $B$ be ideals. I want to show that there exists an element $c \in K$ (where $K$ is the quotient field of a Dedekind domain $O$) such that $cA$ is an ideal relatively prime to $B$.
As hinted in the question, I tried to apply Chinese Remainder Theorem using the fact that every prime ideal $P$ is maximal in a Dedekind domain $O$.
You may use any definition of Dedekind domains.