Let $k$ be a field of characteristic zero and let $A \subseteq B \subseteq C$ be three integral domains (= commutative rings without zero divisors) which are $k$-algebras. Denote the field of fractions of $C$ by $F$.
Let $a \in A-k$, so $A[a^{-1}] \subseteq B[a^{-1}] \subseteq C[a^{-1}] \subseteq F$.
Assume that $B[a^{-1}] = C[a^{-1}]$.
Is there a 'mild' additional condition that guarantees that $B=C$?
Ideas for 'plausible' additional conditions: $A \subseteq C$ is an algebraic ring extension; $\dim(A)=\dim(B)=\dim(C)$ (Krull/GK dimension of rings or $A$-modules, etc.); $\{A,B,C\}$ are UFD's (perhaps helpful; please see the following example in which $B$ is not a UFD, but still $B=C$).
Example: $k[x^2] \subset k[x^2,x^3] \subset k[x]$ with $a=x$.
Remark: If I am not wrong, I have once showed that if $\{A,B,C\}$ are UFD's, then $B=C$. But I think I have also assumed flatness.
Any hints and comments are welcome!