Do we have $(R/I)/(J/I) \cong (R/J)/(I/J)$ where $R$ is a ring and $I,J$ are ideals?
If $I \subset J$ then it follows from the third isomorphism theorem.
In $R= \mathbb Z$ with $I = 3 \mathbb Z$ and $J = 5 \mathbb Z$ we have $I/J = \{ \bar{0}, \bar{3}, \bar{6}, \bar{9}, \bar{12} \} \cong R/J$ and $J/I = \{ \bar{0}, \bar{5}, \bar{10} \} \cong R/I$. Then $(R/I) / (J/I) = 0 = (R/J) / (I/J)$ so it seems to hold also.
I assume it doesn't hold in general. Or does it? Does it hold for principal ideal domains? Thanks.