Lemma 9.6.1. If $F$ is a field and $R$ is a nonzero ring, then any ring homomorphism $\varphi : F \to R$ is injective.

**Proof.**
Indeed, let $a \in \mathop{\mathrm{Ker}}(\varphi )$ be a nonzero element. Then we have $\varphi (1) = \varphi (a^{-1} a) = \varphi (a^{-1}) \varphi (a) = 0$. Thus $1 = \varphi (1) = 0$ and $R$ is the zero ring.
$\square$

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