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

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

Proof

We have by definition that a field is a division ring.

The result can be seen to be an application of Ring Homomorphism from Division Ring is Monomorphism or Zero Homomorphism.

$\blacksquare$