For a ring homomorphism prove that $f(a^{-1})=[f(a)]^{-1}$

Question

$R$ and $S$ are commutative rings and f is a ring homomorphism $f:R\rightarrow S.\ $ For $a\in R$ prove that $$f(a^{-1})=[f(a)]^{-1}$$ Please tell me how can I start this proof.

Answer 1

I assume that $f(1)=1$ (which is a standard assumption for unital ring homomorphisms), otherwise this is not true.

In that case if $a$ is invertible then $aa^{-1}=1$. And thus

$$1=f(1)=f(aa^{-1})=f(a)f(a^{-1})$$

Analogously $f(a^{-1})f(a)=1$ meaning $f(a^{-1})$ is the inverse of $f(a)$. Or in symbols $f(a)^{-1}=f(a^{-1})$.