Given $A \subseteq C \subseteq B$ with $A \subseteq B$ flat, when $C \subseteq B$ is flat? This question appears here.
Now, I wonder if restricting the above question to to following special case would be easier to answer:
Let $R \subseteq S$ be two (commutative) integral domains, $I$ an ideal of $S$. Obviously, $R+I$ is a ring. We have, $R \subseteq R+I \subseteq S$. Assume that $R \subseteq S$ is flat.
Question: When $R+I \subseteq S$ is flat?
One plausible positive answer to this question is when all those three rings have the same field of fractions, see Richman Lemma 2(1), page 795.
Motivation: The following is a known result: Given $A \subseteq C \subseteq B$ with $A \subseteq B$ separable, implies that $C \subseteq B$ is separable.
Thank you very much!