Then there exists a uniquely determined isomorphism of \(\displaystyle S^{-1}A-\)modules:

\(\displaystyle \phi: S^{-1}M \otimes _{S^{-1}A}S^{-1}N \to S^{-1}(M \otimes_A N) \\\)

with

\(\displaystyle \phi\left ( \frac{m}{s} \otimes \frac{n}{t} \right ) = \frac{m \otimes n}{st}\)

Is it the right way, to proof that \(\displaystyle \widetilde{\phi}: S^{-1}M \times S^{-1}N \to S^{-1}(M \otimes_A N)\) is \(\displaystyle S^{-1}A-\)bilinear and induces the homomorphism \(\displaystyle \phi\)? Then I have to proof that \(\displaystyle \phi\) is injective and surjective.