Let $k$ be a field with characteristic 0. Let $R=k[x]$ consider the module and $U=\{1,x,x^2,\dots\}$. Compute $R[U^{-1}]$ and $R[(R\setminus U)^{-1}]$. After some calculations an interpretation I arrive that $R[U^{-1}]=K[x,x^{-1}]$.
I have some problems with $R[(R\setminus U)^{-1}]$, as far as I understand I'm inverting every element thats not of the of the $x^m$, hence I'm inverting the polynomials that are not zero at zero, is there a way to describe $R[(R\setminus U)^{-1}]$?