# Localization of finitely generated algebra

Let $$R$$ be a reduced finitely generated algebra over $$\Bbb Z$$. Let $$T$$ be a finite set of prime ideals of $$R$$. Let $$S = \bigcap_{p \not\in T} R \setminus p$$.

1) Is it true that $$A := S^{-1}R$$ is finitely generated as an algebra over $$\Bbb Z$$?

2) Is it true that $$B := \bigcap_{p \not \in T} ((R \setminus p)^{-1}R)$$ is finitely generated as an algebra over $$\Bbb Z$$?

It seems to work if $$R$$ is the ring of integers of a number field, and we have $$A \cong B$$. I am wondering about the general case. Maybe I should add some assumptions to make this true (e.g. $$R$$ integral domain...).