Let $I=[x,y]$ be the prime ideal generated by the polynomials $x,y$ with real coefficients and let $R_I$ be the localization of the ring $R=\mathbb{R}[x,y]$ in $I$. Can someone help my to visualize/to interpret the elements of $R_I$ as rational functions on $\mathbb{R}^2$ which are defined in 0 and to determine the domain of these rational functions?
In particular I don't understand the part with defined in $0$. Isn't it the case, that all elements of $R_I$ just sets of rational functions, where the numerator is some polynomial of $R=\mathbb{R}[x,y]$ and the denominator just a constant polynomial ? And isn't the domain just $\mathbb{R}$?