Let $A \subseteq B$ be integral domains with fields of fractions $Q(A) \subseteq Q(B)$. Assume that $Q(A) \cap B=A$.
What additional conditions are needed in order to guarantee that the following property holds: If $w \in B$ is algebraic over $A$, and $\sum \hat{a_i} w^i \in B$, for some $\hat{a_i} \in Q(A)$, then $\hat{a_i} \in A$.
Remarks: (1) Of course, if $\hat{a_i} \in B$, then $\hat{a_i} \in A$. (2) There is a nice theorem by H. Bass that appears in van den Essen's book about the JC and in Remark after Corollary 1.3, page 74; it presents conditions on two integral domains $A \subseteq B$ implying that $Q(A) \cap B=A$, but here I am assuming that we already know that this equality is satisfied.
Thank you very much for any hints!