## Over an algebraic integer ring, if (a)|(b) then a|b

Let $\mathcal{O}$ be the ring of integers in an algebraic number field $K$, and let $\alpha,\beta \in \mathcal{O}$. Prove that if $(\alpha)|(\beta)$, then $\alpha|\beta$ in $\mathcal{O}$.

Recall that $(\alpha)|(\beta)$ means that $(\beta) = C(\alpha)$ for some ideal $C$. Now $(\beta) \subseteq (\alpha)$, so that $\beta \in (\alpha)$, and thus $\beta = \alpha\gamma$ for some $\gamma$. Thus $\alpha|\beta$ in $\mathcal{O}$.