On a localized ring tensor with a module

Question

Let $A$ be a commutative ring, $S$ be a multiplicative subset of $A$ and $M$ be an $A$-module. The questions says to "describe a natural isomorphism $(S^{-1}A) \otimes_A M \cong S^{-1}M $ as $A$-modules". I manage to show these two are isomorphic using the universal property but what do they mean by a natural isomorphism and what is this isomorphism? Thanks!

"I manage to show these two are isomorphic using the universal property but [...] what is this isomorphism? " ... Well, the one which comes out of your proof! — Martin Brandenburg, Jul 25 '14 at 21:51

score 2 · Accepted Answer · answered Jul 25 '14 at 21:32

2

Use the map $(a/s)\otimes m\mapsto (am)/s$

answered Jul 25 '14 at 21:32

Hamou

6,745

On a localized ring tensor with a module

1 Answers1