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.

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