Ring Homomorphism from Field is Monomorphism or Zero Homomorphism/Proof 1
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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.
$\blacksquare$