Show that if $F$ is a field and $R$ a non-zero ring, then any homomorphism $\varphi:F\to R$ is injective

Question

This is what I have so far:

Suppose $0\neq\alpha\in\ker(\varphi)$.

Then $$1=f(1)=f(\alpha\alpha^{-1})=f(\alpha)f(\alpha^{-1})=0$$

I'm pretty much there I think, how do I finish it off?

Answer 1

HINT: $\ker(\varphi) $ is an ideal of $F$ but $F$ is a field then $\ker(\varphi)=0$ or $F$