Let $\mathbb{K}$ be an algebraic number field.
Let $I$ and $J$ be two fractional ideals of $\mathbb{K}$ and $q \in \mathbb{N}$ a positive integer.
Is it true that $I/qI $ isomorphic to $J/qJ$? If yes, what is the isomorphism?
If no, is there a condition on $q$ that make it True?