1. ## Localization - isomorphism

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

Then there exists a uniquely determined isomorphism of $S^{-1}A-$modules:

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

with

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

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

2. Originally Posted by AlexanderW
Let $A$ be a commutative ring and $S \subset A$ a multiplicatively closed subset and $M, N\$ $A-$modules.

Then there exists a uniquely determined isomorphism of $S^{-1}A-$modules:

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

with

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

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

$\cong M \otimes_A S^{-1}N \cong M \otimes_A (N \otimes_A S^{-1}A) \cong (M \otimes_A N) \otimes_A S^{-1}A \cong S^{-1}(M \otimes_A N).$