Ring_Homomorphism_from_Division_Ring_is_Monomorphism_or_Zero_Homomorphism

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, $\forall a \in R: \map \phi a = 0_S$).