Let ${\Bbb C}$ be a complex number field. We shall define the two-variable polynomial ring $R$ as follows$\colon$ \begin{equation*} R \colon = {\Bbb C}[X,Y] = {\Bbb C}[X] \otimes_{\Bbb C}{\Bbb C}[Y] \quad \cdots\cdots \quad (\lozenge). \end{equation*} Let us denote by $R_{{\frak m}}$ the localisation of $R$ at the maximal ideal ${\frak m} \colon= (X, Y)$ of $R$. $R_{\frak m}$ is a two-dimensional regular local ring with the unique maximal ideal ${\frak m}$.
Now we consider the quotient ring $S \colon= R_{\frak m}/(XY)$, which is a one-dimensional local ring with the unique maximal ideal $\overline{\frak m}$ and two generic points $(X)$ and $(Y)$. The complement $U \colon= {\mathrm{Spec}}\,S \setminus {\overline{\frak m}}$ consists of two generic points.
Q. How is the affine ring ${\cal O}_U$ of $U$ written explicitly in terms of the presentations $(\lozenge)$?
This might be elementary commutative algebra, but I got confused when trying to write down the explicit product ring structure of ${\cal O}_U$ with respect to two presentations of $R$ at $(\lozenge)$.
You can think of $\mathcal O_U$ as the localization of $S$ by the multiplicative system $Q$ generated by $X+Y$, that is, $Q=\{1,X+Y,(X+Y)^2,...\}$. The point here is that $X+Y\in\mathfrak m$ but $X+Y\notin(X)\cup(Y)$. Instead of $X+Y$ you can use any other polynomial belonging to $\mathfrak m\setminus((X)\cup(Y))$; the resulting localization will be canonically isomorphic to $S_Q$. The space $Spec\ S_Q$ embeds naturally into $Spec\ S$ and its image is $(X)\cup(Y)$.
The ring $\mathcal O_U$ is isomorphic to the direct product of two fields: $\mathcal O_U\cong\mathbb C(X)\times\mathbb C(Y)$.