Ring Homomorphism from Field is Monomorphism or Zero Homomorphism

Theorem
Let $\struct {F, +_F, \circ}$ be a field whose zero is $0_F$.

Let $\struct {S, +_S, *}$ be a ring whose zero is $0_S$.

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

Then either:
 * $(1): \quad \phi$ is a monomorphism (that is, $\phi$ is injective)

or
 * $(2): \quad \phi$ is the zero homomorphism (that is, $\forall a \in F: \map \phi a = 0_S$).

Also see

 * Ring Homomorphism from Division Ring is Monomorphism or Zero Homomorphism