Is faithful flatness implying $M \otimes_{S^{-1}R} S^{-1}N \cong M \otimes_R N$?

Question

Assume that $M$ is faithfully flat $S^{-1}R$-module, for $R$ commutative ring and $S \subset R$ a multiplicative subset. Furthermore, let $N$ denote an $R$-module.

Is it true that $M \otimes_{S^{-1}R} S^{-1}N \cong M \otimes_R N$?

Answer 1

The isomorphism is valid, but this has nothing to do with faithful flatness:

we have $\;S^{-1}N \cong S^{-1}R\otimes_RN,\;$ so $$ M \otimes_{S^{-1}R} S^{-1}N \cong M \otimes_{S^{-1}R}(S^{-1}R \otimes_R N)\cong (M \otimes_{S^{-1}R}S^{-1}R) \otimes_R N \cong M\otimes_R N. $$