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Let $S/R$ be an extension of rings where $R$ is a domain, and let $G$ be a finite group acting on $S$, fixing $R$. When do we have $S^G \subset \text{Frac}(R)$?

For instance, if $S$ is a domain, $G$ acts on $\text{Frac}(S)$, and $\text{Frac}(S) / \text{Frac}(R)$ is Galois, then $S^G \subset \text{Frac}(S)^G \subset \text{Frac}(R)$.

Ronald J. Zallman
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  • Is "$Frac(S)/Frac(R)$ is Galois" an assumption you're making, or can you prove it ? – Maxime Ramzi Jan 26 '19 at 21:01
  • It's definitely an assumption. And I mean for $G$ to be its Galois group. It's not a very general example, to be clear. – Ronald J. Zallman Jan 26 '19 at 21:25
  • Ok, I just wanted to make sure you weren't "hiding" some hypotheses about $G$ or $S/R$. What kind of characterization (/sufficient/necessary conditions) are you looking for ? – Maxime Ramzi Jan 26 '19 at 21:26
  • Let me give some context: I have seen that this is true in the context of the example for rings of integers inside Galois extensions. Note that, if also $R$ is integrally closed, then $R = S^G$. So if something like this holds, it would help me motivate the definition of integrally closed. I am aware of several other motivations for integrally closed in the context of "rings of integers", e.g. via valuations, but this line of interest appeals to me. – Ronald J. Zallman Jan 27 '19 at 15:37
  • I am hoping for conditions which are both sufficient and necessary. However, I realize that sometimes things don't just come out the way you want them, so either of these will suffice. – Ronald J. Zallman Jan 27 '19 at 15:39

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