Let $k=\mathbf Q[i]$ be the field of Gaussian numbers. I've proved the following easy lemma:
"If $x \in \mathcal{O}_k$ (the ring of integers of $k$) and $p$ is an odd rational prime dividing the norm $N_k(x)$, then there is a prime ideal $P$ in $\mathcal{O}_k$ such that $P$ divides both $p\mathcal{O}_k$ and $x\mathcal{O}_k$."
(The proof goes distinguishing if $p \equiv 1 \bmod 4$ or $p \equiv 3 \bmod 4$)
My question is if the same (or a similar) fact is true for other number fields $k$ (?)
Thanks for any idea/reference.