Let $A, B, C$ be ideals in a Dedekind domain $R$. Showing $A \cap (B+C) = (A \cap B) + (A \cap C)$

Question

Let $A, B, C$ be Ideals in a Dedekind domain $R$. I want to show that $A \cap (B+C) = (A \cap B) + (A \cap C)$, where $(A \cap B) + (A \cap C) \subset A \cap (B+C)$ is obviously true. Now I would like to show the opposite, but I don't really know how I should apply the characteristics of the Dedekind domain and where to start from here... As far as my script goes, I know that each ideal is a product of prime/maximal ideals and (to prove said product) we just introduced the fractional ideals