The map $M[f^{-1}]\to M[f^{-1}g^{-1}]$

Question

Let $M$ be a module over $R$. When one talks about the map $M[f^{-1}]\to M[f^{-1}g^{-1}]$, which map do they mean? My guess is that $M[f^{-1}][g^{-1}] \simeq M[f^{-1}g^{-1}]$, and then the above map is the composition $M[f^{-1}]\to M[f^{-1}][g^{-1}]\to M[f^{-1}g^{-1}]$ of the localization map $x\mapsto x/1$ with the isomorphism. If this is so, how to construct that isomorphism? If not, which map is that?

Answer 1

Probably $M[f^{-1}g^{-1}]$ is another notation for $M[(fg)^{-1}]$ (in general $M[h^{-1}]$ denotes the localization w.r.t. $\{h^i:i\ge 0\}$). The map $M[f^{-1}]\to M[(fg)^{-1}]$ is then given by $a/f^n\mapsto (ag^n)/(fg)^n$.