Let $A$ be a fractional ideal of some number field extension $K:\Bbb Q$. Let $\omega_1, \dots ,\omega_n$ be a $\Bbb Z$ basis for $\mathcal O_K$ and let $\alpha_1, \dots ,\alpha_n$ be a $\Bbb Z$ basis for $A$. Define a matrix $\bar A = (a_{ij})$ by $\alpha_i = \sum_{j=1}^{n}a_{ij}\omega_j$.
Similarly, let $B$ be the inverse ideal of $A$ with $\Bbb Z$ basis $\beta_1,\dots,\beta_n$ and define $\bar B = (b_{ij})$ by $\beta_i = \sum_{j=1}^{n}b_{ij}\omega_j$.
Is it true that $\bar A\bar B = I_n$ or at the least that $|\det(A)\det(B)|=1$?
The motivation here is to define the norm of a fractional ideal directly as $N(A) = |det(\bar A)|$ without extending it from the normal definition for ideals using unique factorization.