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. I guess I can't assume the construction I used in the if-part (that is, f(I+r)=J+r)?
I know the R-modules R/I and R/J are cyclic but I don't think a homomorphism maps the generator of one to the other (f(I+1)=J+1)? Could you point me in the right direction?
good question! no, it's not true for noncommutative rings. for example let the ring of matrices with real entries.
let and see that are left ideals of . obviously is not contained in
now define in this way: for any we define it is easy to see that is a
well-defined -homomorphism. is surjective because if and then
because