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

Theorem
Let $\left({R, +_R, \circ}\right)$ and $\left({S, +_S, *}\right)$ be rings whose zeros are $0_R$ and $0_S$ respectively.

Let $\phi: R \to S$ be a ring homomorphism.

If $R$ is a division ring, then either:
 * $(1): \quad \phi$ is a monomorphism (that is, $\phi$ is injective)
 * $(2): \quad \phi$ is the zero homomorphism (that is, $\forall a \in R: \phi \left({a}\right) = 0_S$).

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.