Localization of a module
= Localization of a module
{title2=$S^{-1}M$}
{wiki=Localization_of_a_module}
For a <multiplicative subset> $S\subseteq R$ and an $R$-module $M$, the localization $S^{-1}M$ consists of fractions $m/s$, with $m/s=m'/s'$ when some $u\in S$ satisfies $u(s'm-sm')=0$. It is an $S^{-1}R$-module, and the canonical map is $m\mapsto m/1$. If $M$ is finitely generated over a <Noetherian ring>, then $S^{-1}M$ is a <Noetherian module> over $S^{-1}R$.