If I and J are isomorphic ideals of a ring R, does it follow that $R/I \simeq R/J$?

Question

The title pretty much sums it up. We know that $R/I \simeq R/J$ does not necessarily imply $ I \simeq J$. But does the converse hold? I can't find any counterexample and all my efforts in proving it seem to me rather sketchy.

Answer 1

All non-zero ideals of $R=\mathbb Z$ are isomorphic. Are the corresponding quotients isomorphic?

Answer 2

No. Take a field $F$ and consider the ring $R=F[X]$. The ideals $I=(X-1)$ and $J=(X^2-1)$ are both isomorphic to $R$, but $$R/I\sim F, \quad R/J\simeq F[X]/(X-1)\times F[X]/(X+1)\simeq F\times F,$$ which can't be isomorphic, since the latter has zero-divisors, while the former has not.