A little doubt about norm of ideals in $O_K$

Question

If $I$ is an integral ideal of $O_K$ for $K=\mathbb{Q}(\theta)$, and $a \in O_K$ then can I say, $N(I)|N(\langle a \rangle) \implies I \ | \ a$

Answer 1

Not necessarily. For example, let $\theta = i$ and consider the two primes lying above $(5)$. $(5) = (2 + i)(2 - i)$, and it is easy enough to calculate the norms: $N((2 + i)) = N((2 - i)) = 5$, so the norms divide each other. However, these are distinct primes, and as such, neither divides the other.