we wrote in a comment in our script the following:
If $R$ is a ring: $\dim R=\sup\{\dim R_f\mid f\in R\}$.
We defined: $R_f:=(\{f^n|n\in\mathbb{N}\})^{-1}R$
We didn't prove this. Should this be obvious? Can someone explain why this is true? Thanks in advance.