"descent" of $R$-algebras Not really sure what the best title for this question is... Anyway.
Suppose you have a finite type $\mathbb{C}$-algebra $A$ which is a Dedekind domain.
Let $R\subset\mathbb{C}$ be a subring, and let $B$ be a fppf $R$-subalgebra of $A$ such that the inclusion $B\subset A$ induces an isomorphism $B\otimes_R\mathbb{C} = A$.
Let $B'$ be another $R$-subalgebra of $A$ containing $B$ such that the inclusion $B'\subset A$ also induces an isomorphism $B'\otimes_R\mathbb{C} = A$.
I'm interested in conditions on $R$ and $B$ that would imply that $B = B'$.
If $R$ is a subfield, then this has gotta be true right? Can we find less restrictive conditions?
EDIT: Okay, this is basically my real situation. I've got a finite type $\mathbb{C}$-algebra $A$ which is a Dedekind domain, which is embedded in $\mathbb{C}((t))$. I've got an $R$-subalgebra $B$ of $A$ such that the map $B\otimes_R\mathbb{C} \rightarrow  A$ induced by the inclusion $B\subset A$ is an isomorphism, and such that the image of $B$ in $\mathbb{C}((t))$ is contained in $R((t))$. I want to conclude that $B = A\cap R((t))$.
 A: The algebra structures on $B, B', A$ are irrelevant: this is a question in module theory. You have a short exact sequence
$$0 \to B \to B' \to B/B' \to 0$$
of $R$-modules such that after tensoring with $\mathbb{C}$ over $R$ you have
$$B \otimes_R \mathbb{C} \cong B' \otimes_R \mathbb{C} \to B/B' \otimes_R \mathbb{C} \to 0$$
and you want to know whether this means the original map $B \to B'$ is an isomorphism. By right exactness we see that $B/B' \otimes_R \mathbb{C} = 0$, and hence if $R \to \mathbb{C}$ is faithful (by which I mean that if $M \otimes_R \mathbb{C} = 0$ then $M = 0$) then $B/B' = 0$ and we have the desired result. Otherwise it may happen that $B/B' \otimes_R \mathbb{C} = 0$ but that $B/B' \neq 0$. 
$R \to \mathbb{C}$ is faithful if $R$ is a field, but it already fails to be faithful if $R = \mathbb{Z}$, since $B/B'$ could be torsion and $(-) \otimes_{\mathbb{Z}} \mathbb{C}$ will kill torsion. For example, take $A = \mathbb{C}[x], B = \mathbb{Z}[2x], B' = \mathbb{Z}[x]$. So you'll need to be more specific about what kind of hypotheses you're asking for on $R$. Here $B$ is, again, as nice as it gets. 
