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}.

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