Localization of a Ring

Definition

The localization of a commutative ring $R$ by a multiplicatively closed set $S$ is a new ring $S^{-1}R$ whose elements are fractions with numerators in $R$ and denominators in $S$.

Definition

In particular, the complement $S=R\setminus \mathfrak{p}$ of a prime ideal $\mathfrak{p}$ in a commutative ring $R$ is a multiplicative set. In this case, the localization $S^{-1}R$ is commonly denoted $R_p$. The ring $R_p$ is a local ring, that is called the local ring of $R$ at $\mathfrak{p}$, where $\mathfrak{p}{R}_{\mathfrak{p}}=\mathfrak{p}{\otimes }_R{R}_{\mathfrak{p}}$ is the unique maximal ideal of the ring $R_{\mathfrak{p}}$.

Analogously one can define the localization of a module $M$ at a prime ideal $\mathfrak{p}$ of $R$. Again, the localization $S^{-1}M$ is commonly denoted $M_{\mathfrak{p}}$.