Integral domain question

SlipEternal

MHF Helper
Let $$\displaystyle R,S$$ be integral domains of characteristic zero. Let $$\displaystyle K$$ be the minimal field extension containing $$\displaystyle \text{Fr}(R)$$ and $$\displaystyle \text{Fr}(S)$$ as subfields. Does there exist an integral domain $$\displaystyle T$$ such that $$\displaystyle K = \text{Fr}(T)$$ with $$\displaystyle R$$ and $$\displaystyle S$$ are both subrings of $$\displaystyle T$$?

Deveno

MHF Hall of Honor
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 $K$.

1 person

SlipEternal

MHF Helper
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.