Assume all rings here are commutative with identity.
Let $\phi: R \longrightarrow A$ is a morphism of rings making $A$ into an $R$-algebra. Let $f \in R$. When an author uses the notation $A_{f}$, does this refer to the localization of $A$ at the image $\phi(f)$ $\textit{as a ring}$? Or does it refer to the localization of $A$ at $f$ as an $R$-module? Similarly, when they talk about the natural morphism $$ R_{f} \longrightarrow A_{f}, $$ is this a morphism of modules, or of rings? In particular my question refers to the tag 01SR in the stacks project, although the same notation seems to be used everywhere. I have tagged this as a soft question as well since I am not sure it really counts as a technical mathematics understanding question.
