[Math] A non-zero ring $R$ is a field if and only if for any non-zero ring $S$, any ring homomorphism from $R$ to $S$ is injective.

Question

Show that a non-zero ring $R$ is a field if and only if for any non-zero ring $S$, any unital ring homomorphism from $R$ to $S$ is injective.

I would like to verify my proof, especially the reverse implication.

$\Rightarrow$ Let $S$ be any ring, and $f:R\rightarrow S$ be a ring homomorphism. If $x\in \ker f$ where $x$ is non-zero, then $0= f(x)f(x^{-1}) = f(xx^{-1})=f(1) = 1$ contradiction. Thus $x=0$, so $f$ is injective.

$\Leftarrow$ Since any ring homomorphism is injective, the only ideals of $R$ are $\{0\}$ and $R$. Thus $R$ is a field.

Answer 1

Yes, this proof looks good to me!