Let $f: S^{-1} M \to S^{-1}A \otimes_A M$ defined by $$m/s \to 1/s \otimes m$$ $g: S^{-1}A \otimes_A M \to S^{-1} M$ defined by $$a/s \otimes m \to am/s $$
Prove that $f$ and $g$ are well defined ?
How can we prove $f$ is an $S^{-1}A$ module homomorphism?
Here $A$ is commutative ring with identity and $M$ is module and $S$ multiplicative subset of $A$.