Suppose $J_1$ and $J_2$ are two ideals in a ring both containing another ideal $I$. If $J_1/I \cong J_2/I$ then is $J_1 \cong J_2$?
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Take $R=\mathbb{Z}[x,y]/(x^2,xy,y^2,4x,4y)$ and $J_1=(x)$ and $J_2=(2x,2y)$ and $I=(2x)$. Then $J_1/I\simeq J_2/I\simeq R/(2,x,y)$ as $R$modules, but $2J_2=0$ and $2J_1\neq 0$, so $J_1\not\simeq J_2$. On the other hand, if $R$ is a principal ideal domain, then the answer to your question is positive. 

