1. ## Localization - isomorphism

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.

2. Originally Posted by 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.
$\displaystyle S^{-1}M \otimes_{S^{-1}A} S^{-1}N \cong (M \otimes_A S^{-1}A) \otimes_{S^{-1}A} S^{-1}N \cong M \otimes_A (S^{-1}A \otimes_{S^{-1}A} S^{-1}N)$

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