Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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,

share|improve this question
2  
I think you mean to write $(R/I)/(J/I)$. –  Dylan Moreland Nov 15 '11 at 22:02
4  
This is sometimes known as "the third isomorphism theorem." If you get stuck, just type that into a search engine (or look it up in a textbook) and see what comes up. –  Gerry Myerson Nov 15 '11 at 22:05
    
yes, thanks, now it actually makes more sense. that typo was in my homework. thanks also for the name of the theorem. –  Marie. P. Nov 15 '11 at 22:10
add comment

1 Answer

up vote 3 down vote accepted

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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.