I have been trying to show that if a ring $A$ is absolutely flat then so is the localisation $S^{-1}A$ by any multiplicative set. Now while trying to do this, I asked myself the following: Is there a description for the general form of an $S^{-1}A$ - module?
What I am trying to do to prove my original statement is this: If $A$ is absolutely flat, consider the exact sequence
$0 \stackrel{f'}{\longrightarrow} M' \stackrel{f}{\longrightarrow} M$
of $S^{-1}A$ modules. Then suppose we tensor this with some $S^{-1}A$ module $N$ and consider the map
$f \otimes 1: M'\otimes_{S^{-1}A } N \longrightarrow M \otimes_{S^{-1}A} N.$
Now we know that if $N$ is flat as an $S^{-1}A$ - module, then it is flat as an $A$ - module so that the map $f|_A \otimes 1 : M' \otimes_A N \longrightarrow M \otimes_A N$ is injective. We write $f|_A$ for $f$ viewed as an injective $A$ - module homomorphism, rather than an injective $S^{-1}A$ - module homomorphism. Since localisation preserves exactness, we have that the map
$S^{-1}(f|_A \otimes 1) : S^{-1}(M' \otimes_A N) \longrightarrow S^{-1}(M \otimes_A N)$ is injective. If I can produce some kind of $S^{-1}A$ isomorphism between $M' \otimes_{S^{-1}A} N$ and $S^{-1}(M'\otimes_A N)$, I should be done. This is what I am trying to do, which is why I asked for a general description for what an $S^{-1}A$ - module looks like.
Thanks.