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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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closed as off topic by Mariano Suárez-Alvarez, Mark Sapir, Bruce Westbury, Qiaochu Yuan, Andreas Blass Jul 14 '11 at 18:14

Questions on MathOverflow are expected to relate to research level mathematics within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here.If this question can be reworded to fit the rules in the help center, please edit the question.

Reading the FAQ will explain why this question is not really well-suited to this site (in short, it is not a question about "research math"). The FAQ also suggests other places, like, where your question will be much more at home. Good luck! – Mariano Suárez-Alvarez Jul 14 '11 at 16:36
What notion of isomorphism of ideals is everyone implicitly working with here? – Qiaochu Yuan Jul 14 '11 at 18:03
@Qiaochu: I had a similar question, as I think of ideals as things that can be equal to each other, not isomorphic. One possibility is to use "isomorphic as $R$-modules"; this is Neil's interpretation below. But an ideal is really an $R$-module with a monomorphism to the rank-$1$ free module, and the category thereof is a poset. – Theo Johnson-Freyd Jul 14 '11 at 18:28
@Theo: the category thereof is that if you want it to be. In some conexts, you don't, though. For example, the ideal class group of a Dedekind domain is usefully seen as the set of isomorphism classes of ideals (as modules). Fractional ideals are, from that point of view, a kludge one uses to contruct the operation in the group. – Mariano Suárez-Alvarez Jul 14 '11 at 23:47

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.

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