# How to prove M/Ker(f) & Im(f) are isomomorphic

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

• the proof should be very exactly the same. – mookid Mar 4 '14 at 22:38
• @mookid, for vector spaces there are proofs in which you choose bases, and these do not work here. However, I agree that the “right” proof is the one that also works in this case. – Carsten S Mar 4 '14 at 22:54

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.