----- > [!proposition] Proposition. ([[module of fractions as an extension of scalars]]) > Let $M$ be an $R$-[[module]], $S \subset R$ a [[multiplicative subset of a ring|multiplicative set]]. Then there is an [[module isomorphism|isomorphism]] of $S^{-1} R$-[[module|modules]] $\begin{align} > S ^{-1}R \otimes_{R} M &\xrightarrow{{\sim}} S ^{-1} M \\ > \frac{r}{s} \otimes m & \mapsto \frac{rm}{s} > \end{align}$ > between the [[extension of scalars]] $S ^{-1} R \otimes_{R} M$ and the [[localization|module of fractions]] $S ^{-1} M$. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```