Originally Posted by

**bleys** I need some help trying to prove an exercise in Introduction to Algebra by Cameron

Let R be a commutative ring with identity. Suppose I and J are ideals of R. Then there is a surjective R-module homomorphism from R/I to R/J if and only if I is a subset of J.

I'm able to prove if-part, but I'm having trouble with the 'only if'-part. I assume a surjective homomorphism exists, but I don't know how to use it.

well, it's very easy: let $\displaystyle f:R/I \longrightarrow R/J$ be a surjective R-module homomorphism. then $\displaystyle f(r+I)=1+J,$ for some $\displaystyle r \in R.$ now if $\displaystyle s \in I,$ then

$\displaystyle s+J=s(1+J)=sf(r+I)=srf(1+I)=rsf(1+I)=rf(s+I)=0$

and thus $\displaystyle s \in J,$ i.e. $\displaystyle I \subseteq J.$