## A counterexample regarding quotients of an algebraic integer ring

Let $\mathcal{O}$ be an algebraic integer ring with ideals $A$ and $B$. If $\mathcal{O}/A = \{a_i+A\}_{i=1}^n$ and $\mathcal{O}/B = \{b_i+B\}_{i=1}^m$, is it necessarily the case that $\mathcal{O}/AB = \{a_ib_j\}_{i=1,j=1}^{n,m}$?

No. If $A = B$ is nontrivial, then the set pairwise products of representatives of $\mathcal{O}/A$ is not large enough.