# Math Help - Homomorphism of rings, inverses map to inverse?

1. ## Homomorphism of rings, inverses map to inverse?

Hi,

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

Is it true that:

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

Thanks

2. ## Re: Homomorphism of rings, inverses map to inverse?

Sorry, just got it:

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