# When are kernels isomorphic?

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?

• 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 $I/J$. – 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 $$R/I \simeq (R/J)/(I/J)$$.