Let $R$ be a commutative ring, $I$, $J$, two ideals in $R$ with $J \subset I \subset R$.

Suppose as quotient rings, $R/I$ is isomorphic to $(R/J) / (I/J)$.

Are there any conditions for which the two ideals $J$ and $I/J$ can be claimed to be isomorphic?

  • $\begingroup$ Ehr...under your conditions, $R/I$ is always isomorphic to $(R/J)/(I/J)$. This is one of the isomorphism theorems. No, you can't say anything about $J$ and $I/J$. $\endgroup$ – Arturo Magidin Sep 9 at 19:15

As @Arturo Magidin pointed out, for any commutative ring $R$ and ideals $I$, $J$ with $J \subset I \subset R$ we have

$$ R/I \simeq (R/J)/(I/J) $$

by the third isomorphism theorem (5th remark). So there is really nothing that can be deduced from $R/I \simeq (R/J)/(I/J)$.


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