In the book "The geometry of schemes" I ran into the notation $R[f^{-1}]$, where $R$ is a ring, and $f\in R$. I have to admit that I never quite got the precise meaning of $R[f^{-1}]$ and how it differs from $R_f$, the localisation of $R$ at $\{1,f,f^2,\dots\}$.
To me the notation $R[f^{-1}]$ is a bit unclear, especially since there is no clear "ambient" ring around in which $f^{-1}$ is defined. Unlike for example the case of adjoining a root to a field, where we can find the root in an algebraic closure. So without some bigger ring in which we have an element $f^{-1}$, what entity does the $f^{-1}$ in $R[f^{-1}]$ even refer to? To me the most logical interpretation would be to $f^{-1}\in R_f$, but then I would say that $R[f^{-1}]\cong R_f$, so that the notation is redundant.
Either way I feel like I must be overlooking something, so I would appreciate it if someone could shine some light on this.