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Suppose that $R$ is a ring, $I$ and $J$ are ideals in $R,$ and $R/I\cong R/J$ as rings. When does $I\cong J$ as $R$-modules hold?

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And an example where they are not even isomorphic: $R=\mathbb Z_2\times\mathbb Z_4$ with $I=\mathbb (1)\times(2)$ and $J=(0)\times(1)$. – Henning Makholm Dec 15 '12 at 17:39
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In the case when $R$ is noetherian, and one of the ideals is contained in the other, we do have an equality. – Hmm. Dec 31 '15 at 18:55
up vote 4 down vote accepted

We can also consider $$\left(\prod_{n=1}^\infty\mathbb{Z}\right)/\mathbb{Z}\cong\left(\prod_{n=1}^\infty\mathbb{Z}\right)/(\mathbb{Z}\times\mathbb{Z}).$$ Clearly $\mathbb{Z}\not\cong\mathbb{Z}\times\mathbb{Z}$ since $\mathbb{Z}$ only has one generator while $\mathbb{Z}\times\mathbb{Z}$ has two generators.

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$\Bbb R[X,Y]/(X)\simeq\Bbb R[X,Y]/(Y)$ but $(X)\neq(Y)$. The question should be if $I$ and $J$ are isomorphic as $R$-modules.

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Now the question is as you wished. Unfortunately, the answer has became obsolete. – user26857 May 5 at 14:45

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