TITLE: If ideal quotients of a ring are isomorphic, are these ideals isomorphic?
QUESTION [7 upvotes]: 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?

REPLY [6 votes]: 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.