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

Theorem

Let $\struct {R, +_R, \circ}$ and $\struct {S, +_S, *}$ 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, $\forall a \in R: \map \phi a = 0_S$).

Proof

We have that:

The kernel of a homomorphism is an ideal of $R$

the only ideals of a division ring are trivial.

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

$\blacksquare$