Let $\phi:A \to B $ an injective finite ring map between two noetherian integral domains $A,B$. Let $ C \subset B$ a subring of $B$ and assume that there exist a prime ideal $\mathfrak{p} \subset A$ , such that for the induced map on localizations $\phi_p:A_{\mathfrak{p} } \to B_{\mathfrak{p} }:= B \otimes_A A_{\mathfrak{p} }$ the ring $\widetilde{C}_{\mathfrak{p} }$ - which we define as the subring of $ B_{\mathfrak{p} }$ generated by $ C$ - is contained in the image $\phi_p(A_{\mathfrak{p}})$.
Question: Does in that case always exist a $ t \in A- \mathfrak{p}$ with $\phi(t) \in C $ such that $C_t$ - that means $C$ localized at $t$ - is contained in the image of $A_t$ under $\phi$?
The problem: even if the prime ideal $\mathfrak{p}$ is finitely generated since the ring A is noetherian, this not neccessary holds for the multiplicative system $ A- \mathfrak{p}$, so I see no direct way how to construct this $ t$ I'm looking for.
Rmk #1: the finiteness assumpion on $ \phi$ is neccessary, i.e. that $B$ is finite A-module, otherwise there is an obvious counterexample: let $ \phi$ the localization $A \to A_{\mathfrak{p} } =: B$ and let $C=B$.
Rmk #2: If the question is still too broad or the answer is negative, then one could also ask a 'baby' version of the problem by additionally assuming that $C$ is morevover a $A$- algebra such that $\widetilde{C}_{\mathfrak{p} }$ is given as $C_{\mathfrak{p} }=C \otimes_A A_{\mathfrak{p} }$.
I asked an identical question without getting an satisfying answer.
