Localization of a module

Summary

Let $M$ be a module over a commutative ring $R$, and let $S$ be a multiplicative set. The localization of $M$ at $S$ is the module $S^{-1}M$ of fractions $m/s$ with $m \in M$ and $s \in S$. It is a module over the localization $S^{-1}R$. Localization of modules preserves exact sequences, so it is a basic tool to reduce statements about modules to the local case.