Let be integral domains of characteristic zero. Let be the minimal field extension containing and as subfields. Does there exist an integral domain such that with and are both subrings of ?
Well, sure, but it's probably not the answer you're looking for: clearly $K$ fits the bill.
What you are probably looking for is the sub-ring generated by $R$ and $S$ (I think you can characterize this ring as $\{r+s+r's':r,r' \in R, s,s' \in S\}$). Since this ring is contained in $K$ it has no zero-divisors, and is thus an integral domain. Since its field of fractions is contained in any field that contains it, $\text{Fr}(\langle R\cup S\rangle) \subseteq K$.
Since this field of fractions also contains $R$ and $S$, it must also contain $\text{Fr}(R),\text{Fr}(S)$, and by the minimality of $K$, it must BE $K$.
Excellent! Thank you! I had been looking through old notes from my Ring Theory class, and remembering my professor writing so fast, I couldn't pay attention to him and copy what he was writing at the same time. I feel like I learned so little in that class. One of these years, I want to go back and try to relearn it all.