Assume that $R$ is a f.g. $\mathbb{Z}$-algebra which is an integral domain. I want to show that the intersection of all the maximal ideals of $R$ is $0$.
I know that by the generalized version of Nullstellensatz that the f.g. algebra $S$ over Jacobson ring $R$ is Jacobson, but the proof of the theorem is quite non-trivial and long. Does someone have a shorter answer of this special case over $\mathbb{Z}$?
The previous question of this problem is to show the residue field at each maximal ideal $\mathfrak{m}$ of $R$ is a finite field, that may helps.
