localization of a ring

Let \(R\) be a commutative ring and \(S \subseteq R\) a multiplicatively closed set. Then the localization of \(R\) is defined as

\begin{align*} S^{-1} R \coloneqq R \times S/ ~ \end{align*}

with

\begin{align*} (a,s_1) \sim (b,s_2) :\Leftrightarrow \exists s : s(a s_2 - b s_1) = 0 \end{align*}

• localization of a ring as ring
• equivalence relation of a localization of a ring
• universal property of a ring localization