f:M->N us an A-module homomorphism. A is a commutative ring.
How to prove M/Ker(f) and Im(f) are isomomorphic
I can't prove this statement. But if A is a field,it's not hard to be proved.
For commutative ring, I have proved they are homomorphism and surjective.
could you show me some hint about proving injecitve
I guess you defined a map $g\colon M/\mathop{\mathrm{Ker}} f\to\mathop{\mathrm{Im}} f$, $g([m]):=f(m)$. The first step is to prove that this is well-defined and linear. Surjectivity is obvious. For injectivity, assume $g([m])=g([m'])$. You have to show that this implies $[m]=[m']$, and as soon as you write down what that means, this is also obvious.