ring homomorphism for a localization
1. Proposotion
Let \(R\) be a commutative ring and \(S^{-1}R\) a localization of a ring for a multiplicatively closed set \(S \subseteq R\) Then
\begin{align*} \varphi: R \rightarrow& S^{-1} R \\ r \mapsto \frac{r}{1} \end{align*}is a homomorphism
2. Proof
2.1. additive
\begin{align*}
\varphi(r + r') = \frac{r + r'}{1} \\
=& \frac{r \cdot 1 + r' \cdot 1}{1 \cdot 1} \\
=& \frac{r}{1} + \frac{r'}{1} \\
=& \varphi(r) + \varphi(r')
\end{align*}
2.2. multiplicative
\begin{align*}
\varphi(r \cdot r') =& \frac{r \cdot r'}{1} \\
=& \frac{r \cdot r'}{1 \cdot 1'} \\
=& \frac{r}{1} \cdot \frac{r'}{1} \\
=& \varphi(r) \cdot \varphi(r')
\end{align*}