Homomorphism of rings, inverses map to inverse?

Hi,

If $\displaystyle \phi: R \to S $ is a homomorphism between two rings $\displaystyle R$ and $\displaystyle S$.

Is it true that:

If $\displaystyle a \in R $ is a unit then $\displaystyle \phi(a^{-1})=\phi(a)^{-1}$, or more generally do units have to map to units?

Thanks

Re: Homomorphism of rings, inverses map to inverse?

Sorry, just got it:

$\displaystyle \phi(1)=1 \implies \phi(a \cdot a^{-1}) = \phi(a)\cdot \phi(a^{-1}) = 1 \implies \phi(a)^{-1} = \phi(a^{-1}) $