This is a somewhat trivial question, but Atiyah-MacDonald doesn't quite specify and I can't find a reference at the moment.
Let $A$ be a ring with an action of a group $G$, and $S \subset A$ a multiplicative subset which is stable under the action of $G$. I can think of two reasonable definitions for a $G$-action on $S^{-1} A$: $$ g\left(\frac{a}{s} \right) = \frac{ga}{gs} \qquad g \left( \frac{a}{s} \right) = \frac{ga}{s} $$ Which one of these is the "natural" action?
EDIT: I now feel confident that the first is the "right" action, but I would still like to see someone explain why.