I wanted to come up with a few examples of rings of fractions $S^{-1}R$. Can you tell me if these are correct:
1.Let $R = \mathbb Z$, $S = (2 \mathbb Z \setminus \{0\}) \cup \{1\}$. Then every $[x] = \frac{r}{s} \in S^{-1}R$ consists of the elements: $[x] = \{ 2x, \frac12 x\}$. The ring homomorphism $f: R \to S^{-1}R$ is injective since if $f(r) = [\frac{r}{1}] = [\frac{r^\prime}{1}] = f(r^\prime)$ we have $2k r = 2k r^\prime$ for $2k \in \mathbb Z$ and since $\mathbb Z$ is an integral domain, $r=r^{\prime}$.
2.Now let $R=\mathbb Z / 12 \mathbb Z$ and $S = \{1,2,4,6,8,10\} = (2 R \setminus \{0\}) \cup \{1\}$. Then $f: R \to S^{-1}R$ is not injective since $f(6) = f(3)$.
Are there more interesting examples where $f: R \to S^{-1}R$ is not injective?