# Two ideals in R

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?

I think you mean to write $(R/I)/(J/I)$. –  Dylan Moreland Nov 15 '11 at 22:02
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.