Localization at a prime ideal

Summary

The localization of a commutative ring $R$ at a prime ideal $\mathfrak p$ is the ring $R_{\mathfrak p}$ of fractions whose denominators lie outside $\mathfrak p$. It is a local ring with maximal ideal $\mathfrak p R_{\mathfrak p}$. It keeps the information about $R$ near the point $\mathfrak p$ of the spectrum and forgets the rest. Many questions in commutative algebra reduce to questions about these local rings.