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**AlexanderW** Let $\displaystyle A$ be a commutative ring and $\displaystyle S \subset A$ a multiplicatively closed subset and $\displaystyle M, N\ $ $\displaystyle A-$modules.

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.