Isomorphism in module
Let A,B be R-modules, and let f: A -> B be an onto module homomorphism and let C be the kernel of f. Prove that A/C is isomorphic.
Proof so far:
Let g : A/C -> B be defined by g(a+C) = f(a)
My claim is that g is an isomorphism.
Well-defined: Pick an element in A/C, say x, then x is in A but not in C. meaning f(x) 0, since f is well-defined, g also is well defined.
Linear: Let x,y be in A/C, and let s be a scalar. Again, f(x) and f(y) are not equal to 0.
g(x+C) = f(x) = u , g(y+C) = f(y) = v , for elements u,v in B.
g(x+y+C) = f(x+y) = f(x) + f(y) = u + v since f is an homomorphism.
Let s be a scalar, g(s(x+C)) = f(sx) = sf(x) since f is module homomorphism.
So g is also linear.
Am I doing this right so far?
The following up problem is related to this previous one:
Let M be an R-module, and let L,N be submodules. Show that
proof so far:
My goal is to prove p is an onto homomorphism, then use the previous exercise.
Will this be a right start?