Prove: $N$ is a submodule of $M \iff N$ is a kernel of some homomorphism.

Question

Prove: $N$ is a submodule of $M \iff N$ is a kernel of some homomorphism.

$(\implies)$ Fix $N<M$, $\kappa:M \rightarrow M/N. \kappa$ is the canonical epimorphism. Then $N=\ker\kappa.$

How to prove the other direction?

Answer 1

If $n,n'\in\ker\kappa$, then $\kappa(n+n')=\kappa(n)+\kappa(n')=0$, and therefore $n+n'\in\ker\kappa$. And if $r$ belongs to the ring that you are working with, then $\kappa(rn)=r\kappa(n)=0$, and therefore $rn\in\ker\kappa$.