Let $R$ be a right Noetherian ring and $S=\{f\in R[x]\;|\;f\text{ monic}\}$. It is a result of Stafford that $S$ is a right denominator set in $R[x]$, so in particular we can localize $R[x]$ at any $f\in S$. My question is this, can we localize $R[x,x^{-1}]$ at any $f\in S$? I know,

$R[x,x^{-1}]_f\cong R[x]_{xf}\cong R[x]_{fx}$,

and of course $R[x]_f$ and $(R[x]_f)_x$ both exist, but is this enough?

I suppose the question is this, does the above guarantee $\{f^n\:|\:n\geq 0\}$ is a right denominator set in $R[x,\,x^{-1}]$? As $R[x,x^{-1}]$ is a Laurent polynomial ring over a Noetherian ring, then it is also right Noetherian, so I know that I only need to check $\{f^n\:|\:n\geq 0\}$ is right Ore, but does the above guarantee this?