Let $A$ be a ring and let $\mathcal{F}$ be the inductive system of subrings of $A$ which are of finite type over $\mathbb{Z}$: $ \mathcal{F} = \{ \mathbb{Z}[a_1,\dots,a_n] \subseteq A \mid n \geq 0, \ a_1, \dots, a_n \in A \}. $ I'd like to know whether the following statement is true: For every $B \in \mathcal{F}$, there exists $C \in \mathcal{F}$ such that $C \supseteq B$ and the map $\mathrm{Spec}(A) \to \mathrm{Spec}(C)$, induced by the inclusion $C \hookrightarrow A$, is surjective.
Maybe EGA IV.8.3.8.(i) is useful, but I have no ideas to prove or disprove the statement. Any hint will be welcome. Thanks to all!