Let $R$ be a Dedekind Domain with fraction field $K$ with nonzero fractional ideals $A$ and $B$ (i.e., $A=d^{-1}I$ for some ideal $I$ of $R$ and $d\in R$).
$\mathbf{Problem}$. Show that there are elements $\alpha,\beta\in K$ such that $\alpha A$ and $\beta B$ are nonzero integral ideals of R that are relatively prime.
I am only trying to understand the problem. If $A=d^{-1}I$ then it seems to me that the only way $\alpha A$ can be an ideal of $R$ is when $d$ divides $\alpha$. But this seems to be a problem for getting $\alpha A+\beta B=R$.
In a Dedekind Domain every ideal is generated by two elements. The only thing I can think of is if $I=(a,b)$ where $a$ and $b$ have no common factors, then maybe this implies that $Ra+Rb=R$, and so we can take $\alpha =\beta =d$.