# Ring homomorphisms map units to units

Let $R$, $S$ be commutative rings, and let $\Phi:R\to S$ be a homomorphism of rings. Prove that if $a\in R$ is a unit, then $\Phi(a^{-1})=\Phi(a)^{-1}$. Deduce that $\Phi$ maps units of $R$ to units of $S$.

I could really use some help on this problem. I know how to prove $\Phi(-a)=-\Phi(a)$ by saying $\Phi(-a)+\Phi(a)=\Phi(-a+a)=\Phi(0)=0$, therefore $\Phi(-a)=-\Phi(a)$. I know how to do that, but I can't figure out how to do this one. Any help would be greatly appreciated. Thanks

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(The analogue of $\Phi(-a)+\Phi(a)$ would be $\Phi(a^{-1})\times\Phi(a)$)