Localization

Summary

The localization $S^{-1}R$ of a commutative ring $R$ at a multiplicative set $S$ is the ring of fractions $a/s$ with $a \in R$ and $s \in S$. It is the most general ring in which the elements of $S$ become units. It generalizes the construction of the field of fractions of an integral domain. Localization lets one study a ring "near" a prime ideal, which is the algebraic form of looking at a space near a point.