Let $\mathcal{O}_K$ be a Dedekind domain with field of fraction $K$, let $L|K$ be a finite separable extension of $K$ and $\mathcal{O}_L$ be the integral closure of $\mathcal{O}_K$ in $L$. Then show that $\mathcal{O}_L$ is again Noetherian.
My effort : Actually this is a part of theorem given in Algebraic Number Theory by NEUKRICH.
To show $\mathcal{O}_L$ noetherian I have to check that every ideal of $\mathcal{O}_L$ is finitely generated $\mathcal{O}_L$ module. Now I understood that $\mathcal{O}_L\subset M$, where $M$ is finitely generated free $\mathcal{O}_K$ module of rank $n=[L:K]$ and also for any ideal of $\mathcal{O}_L$ say $I$ then $I \subset \mathcal{O}_L \subset M$ and we know that $\mathcal{O}_K$ is noetherian and $I$ is $\mathcal{O}_K$ submodule but using this how I conclude that $I$ is finitely generated $\mathcal{O}_L$ module. Any help/hint in this regards would be highly appreciated. Thanks in advance!