Is it true that $B=[\beta]$ when $B$ is and ideal of $\mathcal{O}$, $\beta \in B$ and $N(B)=|N(\beta)|$?

Question

Let $B$ be an ideal of $\mathcal{O}$ (Ring of integers), $\beta \in B$ and $N(B)=|N(\beta)|$. Does it follow that $B=[\beta]$?

I think that this isn't true but I'm struggling to find a counter example. Any tips how to find one?

I know that the converse is true: If $B=[\beta]$ then $N(B)=|N(\beta)|$.