Let R be a commutative ring with ideals I,J such that I ⊆J ⊆R. Show that J/I is an ideal of R/I.

I am not sure how to begin.(Thinking) Thanks in advance for any help.

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- Mar 8th 2011, 04:42 AMpage929Ideal
Let R be a commutative ring with ideals I,J such that I ⊆J ⊆R. Show that J/I is an ideal of R/I.

I am not sure how to begin.(Thinking) Thanks in advance for any help. - Mar 8th 2011, 05:37 AMtonio
- Mar 8th 2011, 07:12 AMtopspin1617
You do know what $\displaystyle J/I$ means, right? I've seen some people get stuck with this simply from not knowing what is meant by "an ideal mod an ideal".

- Mar 8th 2011, 08:59 AMpage929
I don't think that I do. Can you explain it to me?

- Mar 8th 2011, 12:50 PMtopspin1617
As a set, $\displaystyle J/I=\{j+I\mid j\in J\}$. They're the cosets in $\displaystyle R/I$ whose representatives come from $\displaystyle J$. Notice that it makes sense to discuss this since $\displaystyle I\subseteq J$.