Ring Homomorphism from Field is Monomorphism or Zero Homomorphism/Proof 2

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

Suppose $\phi$ is not a monomorphism.

By definition, $\phi$ is not an injection.

So there must exist $a, b \in F: \map \phi a = \map \phi b$.

Let $k = a +_F \paren {-b}$.

Then:

As $a \ne b$ then $k \ne 0$ and so has a product inverse $\exists k^{-1} \in F$.

So for any $x \in F$ we can write $x = k \circ \paren {k^{-1} \circ x}$ and so:

So if $\phi$ is not a monomorphism, it is the zero homomorphism.