Characterization of $S^{-1}R$-modules

Question

Let $R $ be a commutative ring and $S $ a multiplicative subset of $R $.

Is any $S^{-1}R $-module of the form $S^{-1}M $ with $M $ an $R$-module ?

Answer 1

Every $S^{-1}R$-module $M$ can be regarded as an $R$-module via $$ rx=\frac{r}{1}x $$ (assuming $1\in S$, which is not restrictive). For better clarity, denote this $R$-module by $\check{M}$. Now, what's $S^{-1}\check{M}$?

Consider the map $f\colon M\to S^{-1}\check{M}$ defined by $f(x)=x/1$ and prove it is an isomorphism of $S^{-1}R$-modules.