I want to show that if $S$ consists of units then $S^{-1}R \cong R$.
Can you tell me if my proof is correct?
Since $S$ consists of units, $S$ is zero-divisor free and hence $f: R \to S^{-1}R$, $r \mapsto \frac{r}{1}$ is injective. So we have an isomorphism $h: R \to f(R)$.
Now we construct an isomorphism $S^{-1}R \to f(R)$: For this we pick any $s_0$ in $S$ and denote by $u \in R$ its inverse, i.e. $s_0 u = us_0 = 1$. Then the map $g_u : S^{-1}R \to f(R)$, $\frac{r}{s} \mapsto \frac{ru}{1}$ is an isomorphism. It is injective since if $\frac{ru}{1} = \frac{r^\prime u}{1}$ then $ru = r^\prime u$ and since $u$ is a unit, $r = r^\prime$. And it is surjective since for $\frac{r}{1}$ in $f(R)$, $g(\frac{rs}{s}) = \frac{rsu}{1} = \frac{r}{1}$. Hence $R \cong f(A) \cong S^{-1}R$.