If $K \subset L$ are number fields and $I$ and $J$ are ideals of $\mathcal{O}_K$, then I'm trying to prove the statement $$I\mathcal{O}_L | J\mathcal{O}_L \implies I|J,$$
where the first division occurs in the ring $\mathcal{O}_L,$ while the second division occurs in the ring $\mathcal{O}_K$.
My initial thought was to try and factor $I = P_1^{a_1}\cdots P_n^{a_n}$ as a product of prime ideals. Then each of these primes will potentially split in the ideal $I\mathcal{O}_L$ to give the factorization
$$I\mathcal{O}_L = (Q_{11}^{e_11}Q_{12}^{e_{12}} \cdots Q_{1m}^{e_{1m}})^{a_1} \cdots (Q_{n1}^{e_{n1}}\cdots Q_{nr}^{e_{nr}})^{a_n} $$ where the $Q_{ij}$'s denote the splitting of the $P_i's$. Now since $I\mathcal{O}_L | J\mathcal{O}_L$, all of the $Q_{ij}'s$ must appear in the factorization of $J \mathcal{O}_L$. Since each $Q_{ij}$ lies above a unique prime $P_i$, is this enough to imply that each $P_i^{a_i}$ must divide $J$ and hence $I|J$?
If this isn't correct, is there any hint that could be given to send me on the right track to completing this proof?