Do we have $(R/I)/(J/I) \cong (R/J)/(I/J)$? 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.
 A: You can only speak of $I/J$ if $J\subseteq I$, so your question of whether
$$(R/I)/(J/I) \cong (R/J)/(I/J)$$
only makes sense if both $I\subseteq J$ and $J\subseteq I$, or equivalently $I=J$. Of course, when that is the case, then the above statement is trivially true.
Let's use your specific example, where $R=\mathbb{Z}$, $I=3\mathbb{Z}$, and $J=5\mathbb{Z}$. There is no such thing as $I/J$, but you can talk about $$I/(I\cap J)=3\mathbb{Z}/15\mathbb{Z}=\{\bar{0},\bar{3},\bar{6},\bar{9},\bar{12}\}\cong \mathbb{Z}/5\mathbb{Z},$$
and
$$J/(I\cap J)=5\mathbb{Z}/15\mathbb{Z}=\{\bar{0},\bar{5},\bar{10}\}\cong \mathbb{Z}/3\mathbb{Z}.$$
However, note that (by the third isomorphism theorem)
$$(R/(I\cap J))/(I/(I\cap J))\cong R/I$$
$$(R/(I\cap J))/(J/(I\cap J))\cong R/J$$
and there is no reason for those two rings to be isomorphic in general.
A: As written, what you have does not make sense. $J/I$ requires $I\subseteq J$, which is not met in your example. If both $J/I$ and $I/J$ make sense as written, then this requires $I=J$, and so in both cases we have $$(R/I)/(J/I) = (R/I)/(I/I) \cong R/I =R/J \cong (R/J)/(J/J) = (R/J)/(I/J),$$ 
and we are fine, but I doubt this is what you meant.
Instead, we can consider $(R/I)/((J+I)/I)$ and $(R/J)/((I+J)/J)$, since $I+J$ is the smallest ideal that contains both $I$ and $J$.
If so, then the answer is "yes", and it follows from the Isomorphism Theorems. In both cases we have that the quotient is isomorphic to $R/(I+J)$. 
(However, as noted by Zev and as I failed to notice, this would not agree with your claim that "$5\mathbb{Z}/3\mathbb{Z} = \{\overline{0}, \overline{5},\overline{10}\}$", since my interpretation would be to set it equal to $\mathbb{Z}/3\mathbb{Z}$; while it also has $3$ elements, since $(I+J)/I \cong J/(I\cap J)$, it is not the "same" quotient.)
