## In an algebraic integer ring, two ideals whose intersection contains 3 may be comaximal

Let $K$ be an algebraic number field with ring of integers $\mathcal{O}$, and let $A,B \subseteq \mathcal{O}$ be nontrivial ideals. If $3 \in A \cap B$, must it be the case that $(A,B) \neq (1)$?

Consider $K = \mathbb{Q}(\sqrt{-2})$, so that $\mathcal{O} = \mathbb{Z}[\sqrt{-2}]$. Consider $A = (1+\sqrt{-2})$ and $B = (1-\sqrt{-2})$. Now $AB = (3)$, so that $3 \in A \cap B$. However, note that $1 = (1+\sqrt{-2})(1-\sqrt{-2}) - (1+\sqrt{-2}) - (1-\sqrt{-2}) = (A,B)$, so that $(A,B) = (1)$.