Could you please show some hints about the proof

Question

I even don't know how to start.

Answer 1

Without loss of generality (can you see why?), assume that $1\in S$ so that $f(r)=r/1$ for any $r\in A$.

1) $f(s)=s/1$, which has the inverse $1/s$.

2) If $f(a)=a/1=0/1$, then we have $s(a\cdot 1-0\cdot 1)=0$ for some $s\in S$.

3) $a/s=a/1\cdot1/s$.