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 $\map \ker \phi = \set {0_R}$ or $R$.

If $\map \ker \phi = \set {0_R}$, then $\phi$ is injective by Kernel is Trivial iff Monomorphism.

If $\map \ker \phi = R$, $\phi$ is the zero homomorphism by definition.