# Ideal

• Mar 8th 2011, 04:42 AM
page929
Ideal
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 AM
tonio
Quote:

Originally Posted by page929
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.

Prove that $J/I$ is an additive subgroup of $R/I$ and that $(r+I)(j+I)\in J/I\,\,\,\forall\,r+I\in R/I$

Tonio
• Mar 8th 2011, 07:12 AM
topspin1617
You do know what $J/I$ 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 AM
page929
I don't think that I do. Can you explain it to me?
• Mar 8th 2011, 12:50 PM
topspin1617
As a set, $J/I=\{j+I\mid j\in J\}$. They're the cosets in $R/I$ whose representatives come from $J$. Notice that it makes sense to discuss this since $I\subseteq J$.