Consider the ring $R = S^{-1} \mathbb{C}[x,y]$ where $S$ is the multiplicative set generated by $$\left\{x - a \mid a \in \mathbb{C} \right\} \cup \left\{ y - a \mid a \in \mathbb{C} \right\}. $$
I have shown that $R$ is actually a field, since multiplication is defined as (recall the elements are equivalence classes): $$ [m, s] \cdot [m^{'}, s^{'}] := \frac{m}{s} \cdot \frac{m^{'}}{s^{'}} = \frac{m m^{'}}{s s^{'}} $$ where $m \in \mathbb{C}[x,y]$ and $s \in S$. I have shown that $[s,m]$ is the multiplicative inverse of $[m,s]$.
Also, I have proven that any artinian integral domain is a field.
Problem: Now, I need to write down a maximal (with respect to inclusion) subfield of $R$. I know that $\mathbb{C}(x) \subset R \subset \mathbb{C}(x,y)$.
I don't know how to find this maximal subfield. If I let $K := \left\{x - a \mid a \in \mathbb{C} \right\}$, I was thinking of taking $R^{'} = K^{-1} \mathbb{C}[x,y]$. I think it is clear that $R^{'} \subset R$ because $K \subset S$. And $R^{'}$ is again a field.
Also, does $R$ contain a largest sub-field?