weak equivalence (awaiting review)

Let $R$ be an order in an étale $\mathbf{Q}$-algebra. Two fractional $R$-ideals $I$ and $J$ are weakly equivalent if $I_\mathfrak{p}\simeq J_\mathfrak{p}$ as $R_\mathfrak{p}$-modules for every maximal ideal $\mathfrak{p}$ of $R$. If $I$ and $J$ are weakly equivalent then $(I:I)=(J:J)$. Such ideals are also said to be locally isomorphic or in the same genus.