Ring Homomorphism from Division Ring is Monomorphism or Zero Homomorphism/Proof 1

Proof
We have that:
 * The kernel of a homomorphism is an ideal of $R$
 * the only ideals of a division ring are trivial

So $\ker \left({\phi}\right) = \left\{{0_R}\right\}$ or $R$.

If $\ker \left({\phi}\right) = \left\{{0_R}\right\}$, then $\phi$ is injective by Kernel is Trivial iff Monomorphism.

If $\ker \left({\phi}\right) = R$, $\phi$ is the zero homomorphism by definition.