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

  • 1
    $\begingroup$ the proof should be very exactly the same. $\endgroup$ – mookid Mar 4 '14 at 22:38
  • $\begingroup$ @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. $\endgroup$ – 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.