1. ## 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) $\neq$ 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?

2. The following up problem is related to this previous one:

Let M be an R-module, and let L,N be submodules. Show that $L/(L \cap N ) \cong (L+N)/N$

proof so far:

Define $p: L \rightarrow (L + N)/N$ by $p(x) = x + N$

My goal is to prove p is an onto homomorphism, then use the previous exercise.

Will this be a right start?