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Localization (of a Ring)

The localization of a commutative ring \((R,+,\cdot)\) by a multiplicatively closed subset \(S\) (i.e. \(\forall s,t\in S:s\cdot t\in S\)) is the ring \(S^{-1}R\) of all formal fractions \(\left\{\frac{r}{s}\middle|r\in R,\:s\in S\right\}\), subject to the standard rules of addition and multiplication of fractions, i.e. \(\frac{r_1}{s_1}+\frac{r_2}{s_2}:=\frac{r_1\cdot s_2+r_2\cdot s_1}{s_1\cdot s_2},\:\: \frac{r_1}{s_1}\cdot\frac{r_2}{s_2}:=\frac{r_1\cdot r_2}{s_1\cdot s_2}\) - in this way, the rational numbers \(\mathbb{Q}\) are simply the localization of the integers \(\mathbb{Z}\) by the set of non-zero integers. If \(\mathfrak{p}\) is a prime ideal of \(R\), its complement is (by definition) a multiplicatively closed subset, so one often refers to '\(R\) localised at \(\mathfrak{p}\)', meaning the localization \((R\backslash\mathfrak{p})^{-1}R\).