Two ideals in R

Question

I am supposed to show that $R/J=(R/I)/(J/I)$ for two ideals $I\subseteq J\subset R$. How can I do this? I tried to construct an explicit isomorphism, but that did not work out (for me). Is there a more elegant solution?

Thank you in advance,

Answer 1

If you know it for groups, you can show that this works at the group level and then just verify that the resulting map is a homomorphism of rings.

As for an elegant method, there is an obvious map $\frac{R}{I}\to\frac{R}{J}$ given by $a+I \longmapsto a+J$ (which is well defined since $I\subseteq J$). Verify that the kernel is precisely $\frac{J}{I}$, and apply the First Isomorphism Theorem.