Ring Homomorphism from Division Ring is Monomorphism or Zero Homomorphism

Theorem
Let $\left({R, +_R, \cdot}\right)$ and $\left({S, +_S, *}\right)$ be rings.

Let $\varphi: R \to S$ a ring homomorphism.

Then if $R$ is a field, then $\varphi$ is injective or trivial (that is, $\forall a \in R: \varphi \left({a}\right) = 0_S$).

Proof
As the kernel of a homomorphism is an ideal of $R$, and the only ideals of a division ring are trivial, we have $\ker \left({\varphi}\right) = \left\{{0_R}\right\}$ or $R$.

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

If $\ker \left({\varphi}\right) = R$, $\varphi$ is trivial.

Alternative Proof
From Surjection by Restriction of Codomain, we can restrict the codomain of $\varphi$ and consider the mapping $\varphi': R \to \operatorname {Im} \left({R}\right)$

As $\varphi'$ is now a surjective homomorphism, it is by definition an epimorphism.

Then we invoke the result that an epimorphism from a division ring to a ring is either null or an isomorphism.

As an isomorphism is by definition injective, the result follows.