Let $R$ be a commutative ring with a $1\neq 0$.
Show that if $\{$ $0$ $\}$ is a maximal ideal then every non-zero homomorphism $f:R\rightarrow S$ of rings is injective.
My attempt:
Suppose $\{$ $0$ $\}$ is maximal. Let $\phi:R\rightarrow S$ be any non-zero homomorphism. I must show that it is injective. Since $\{$ $0$ $\}$ $\subseteq ker\phi$, it follows from maximality of $\{$ $0$ $\}$ and the fact that the kernel is an ideal that $ker\phi=$ $\{$ $0$ $\}$ or $ker\phi=R$. If $ker\phi=R$ then by the first isomorphism theorem, $im\phi$ is $0$ , contradiction. So $ker\phi$ is the zero element.
Is this correct? How else could I deduce that the kernel is a proper ideal?