localization of a ring as ring
1. Proposition
Let \(R\) be a commutative ring and \(S \subseteq R\) a multiplicatively closed set. Then \(S^{-1}R\) is a commutative ring with
\begin{align*} \frac{r_1}{s_1} + \frac{r_2}{s_2} =& \frac{r_1s_2 + r_2s_1}{s_1s_2} \\ \frac{r_1}{s_1} \cdot \frac{r_2}{s_2} =& \frac{r_1 r_2}{s_1 s_2} \end{align*}2. Proof
2.1. neutral element
2.1.1. additive
for \(\frac{0}{1}\) we conclude
\begin{align*} \frac{0}{1} + \frac{r_1}{s_1} =& \frac{r_1 \cdot 1 + 0 \cdot s_1}{s_1 \cdot 1} \\ =& \frac{r_1}{s_1} \end{align*}By cancelling it also follows:
\begin{align*} \frac{0}{1} =& \frac{s_1 \cdot 0}{s_1 \cdot 1} \\ =& \frac{0}{s_1} \end{align*}2.1.2. multiplicative
for \(\frac{1}{1}\) we conclude
\begin{align*} \frac{r_1}{s_1} \cdot \frac{1}{1} =& \frac{r_1 \cdot 1}{s_1 \cdot 1} \\ =& \frac{r_1}{s_1} \end{align*}2.2. associativity
2.2.1. additive
\begin{align*}
\frac{r_1}{s_1} + (\frac{r_2}{s_2} + \frac{r_3}{s_3}) =& \frac{r_1}{s_1} + \frac{r_2s_3 + r_3 s_2}{s_2s_3} \\
=& \frac{r_1s_2s_3 + r_2s_3s_1 + r_3s_2s_1}{s_1s_2s_3} \\
=& \frac{r_1s_2 + r_2s_1}{s_1s_2} + \frac{r_3}{s_3} \\
=& (\frac{r_1}{s_1} + \frac{r_2}{s_2}) + \frac{r_3}{s_3} \\
\end{align*}
2.2.2. multiplicative
\begin{align*}
\frac{r_1}{s_1} \cdot (\frac{r_2}{s_2} \cdot \frac{r_3}{s_3} ) =& \frac{r_1}{s_1} \cdot \frac{r_2r_3}{s_2s_3} \\
=& \frac{r_1r_2r_3}{s_1s_2s_3} \\
=& \frac{r_1r_2}{s_1s_2} \cdot \frac{r_3}{s_3} \\
=& ( \frac{r_1}{s_1} \cdot \frac{r_2}{s_2}) \cdot \frac{r_3}{s_3}
\end{align*}
2.3. commutativity
2.3.1. additive
\begin{align*}
\frac{r_1}{s_1} + \frac{r_2}{s_2} =& \frac{r_1s_2 + r_2s_1}{s_1s_2} \\
=& \frac{r_2s_1 + r_1s_2 }{s_2s_1} \\
=& \frac{r_2}{s_2} + \frac{r_1}{s_1}
\end{align*}
2.3.2. multiplicative
\begin{align*}
\frac{r_1}{s_1} \cdot \frac{r_2}{s_2} =& \frac{r_1r_2}{s_1s_2}\\
=& \frac{r_2r_1}{s_2s_1} \\
=& \frac{r_2}{s_2} \cdot \frac{r_1}{s_1}
\end{align*}
2.4. distributivity
\begin{align*}
\frac{r_1}{s_1} \cdot \left(\frac{r_2}{s_2} + \frac{r_3}{s_3}\right) =& \frac{r_1}{s_1} \cdot \frac{r_2s_3 + r_3s_2}{s_2s_3} \\
=& \frac{r_1(r_2s_3 + r_3s_2)}{s_1s_2s_3} \\
=& \frac{r_1r_2s_3 + r_1r_3s_2}{s_1s_2s_3} \\
=& \frac{s_1 \cdot \left(r_1r_2s_3 + r_1r_3s_2 \right) }{s_3 s_1s_2s_1 } \\
=& \frac{s_1r_1r_2s_3 + s_1r_1r_3s_2 }{s_3 s_1s_2s_1 } \\
=& \frac{r_1r_2(s_1s_3) + r_1r_3(s_1s_2)}{(s_1s_3)(s_1s_2)} \\
=& \frac{r_1r_2}{s_1s_2} + \frac{r_1r_3}{s_1s_3} \\
=& \frac{r_1}{s_1} \cdot \frac{r_2}{s_2} + \frac{r_1}{s_1} \cdot \frac{r_3}{s_3}
\end{align*}
2.5. additive inverse element
\begin{align*}
\frac{r_1}{s_1} + \frac{-r_1}{s_1} =& \frac{r_1s_1 + (-r_1s_1)}{s_1s_1} \\
=& \frac{r_1s_1 - r_1s_1}{s_1s_1} \\
=& \frac{0}{s_1s_1} \\
=& \frac{0}{1}
\end{align*}