Isomorphism between quotient modules

Problem 22 of Chapter 4 from Steven Roman's Advanced Linear Algebra asks to prove this question in the affirmative when $R/I\simeq R/J$ as $R$-modules, and then asks about the case when $R/I\simeq R/J$ as rings.

The nice existing answers show it is not necessarily true that $I=J$ when the quotients are isomorphic as rings. However, suppose $R/I\simeq R/J$ as $R$-modules with the standard $R$-module structure.

Let $\tau\colon R/I\to R/J$ be an $R$-module isomorphism. Then for any $j\in J$, $$ \tau(j+I)=\tau(j\cdot 1+I)=j\cdot\tau(1+I)=0+J $$ since $\tau(1+I)\in R/J$, and any element of $R/J$ is annihilated by multiplication by elements of $J$. Thus $j+I\in\ker\tau=\{I\}$, the equality following since $\tau$ is injective. So $j+I=I$, thus $j\in I$, and $J\subseteq I$. The reverse containment follows similarly, by looking at $\tau^{-1}$ say. So $I=J$.

No, just take a polynomial ring $k[X]$ for some field $k$ (which is a commutative domain), and consider the ideal generated by $X$ and the ideal generated by $X-1$. The quotient rings are both (isomorphic to) $k$.

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(added later, by mixedmath)

This answer asserts that $R/I \simeq R/J$ (as rings) does not imply that $I = J$. However, the OP asks about $R/I \simeq R/J$ (as $R$-modules), which is covered in the other answer.