Prove that there exist only finitely many integral ideals of $O_K$ to which $m$ belongs

Question

Let $K$ be an algebraic number field and $O_K$ be the ring of integers. Let $m \in \Bbb Z\setminus \{0\}$. Prove that there exist only finitely many integral ideals of $O_K$ to which $m$ belongs.

Edit: For each prime divisor $p$ of $m$ we will get a corresponding prime ideal $<p>$ in $\Bbb Z$ and by lifting it we will get a prime ideal $P$ in $O_K$. Now can I say something more?

Answer 1

Hint. If $I$ contains $m$, for all prime ideal $P$, what can you say about $v_P(I)$ ? can you find an upper bound ?