Ring Homomorphism from Division Ring is Monomorphism or Zero Homomorphism

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, $\forall a \in R: \phi \left({a}\right) = 0_S$).

Also see

 * Ring Homomorphism from Field is Monomorphism or Zero Homomorphism