I am trying to prove the following assertion:
Let $K$ be a number field, $R$ its ring of algebraic integers, and let $I$ be a nonzero ideal of $R$. Show that $\left |R/I\right |=\left |J/IJ\right |$ for all fractional ideals $J$ (first prove it for ideals $J$, then generalize)
I could only prove it in a very particular case: $J$ an ideal (not a fractional ideal) which is relatively prime to $I$. Indeed, the inclusion of $J$ into $R$ induces an homomorphism $J\to R/I$ whose kernel is $J/I\cap J=J/IJ$ and its image is $I+J=R$.
I then tried to factor each $I$ and $J$ into prime ideals and use the fact that $\left |R/AB\right |=\left |R/A\right |\left |R/B\right |$ for every pair of nonzero ideals $A,B$. But I could not conclude. Moreover, I still have to generalize to fractional ideals, and I do not know how.