Some questions about the factorization of an ideal and its extension in a ring of integers.

Question

Let $K\subset L$ be two number fields and let $I,J\subset \mathcal{O}_K,$ two ideals. I know that the ring of integers of a field is a Dedekind domain, and so I have the unique factorization in prime ideale for its ideals. My questions are: how are related the factorizations of $I$ in $ \mathcal{O}_K,$ and $I\mathcal{O}_L,$ in $\mathcal{O}_L?$ If $I\mathcal{O}_L,$ lies over $J\mathcal{O}_L,$ how can I prove that $I$ lies over $J?$ I can factor them in $\mathcal{O}_K,$ say $$I=P_1^{a_1}...P_r^{a_r}\quad J=Q_1^{b_1}...Q_s^{b_s}$$ I also know that the primes which figure out in the factorizations of the two ideal extensions are those who lie over the ideals. Probably the answer is quite simple, but I'm stuck..