I would like to know what is a correct analogue of Noether normalization theorem for rings finitely generated over $\mathbb Z$. Obviously, Noether normalization can not hold "literately" in this case since, for example the ring $\mathbb Z_2[X]$ does not contain a polynomial subring with coefficients in $\mathbb Z$ over which it is finite.
I am asking this question to better understand the second part of the answer of Qing Liu to the question given here: https://mathoverflow.net/questions/57515/one-point-in-the-post-of-terence-tao-on-ax-grothendieck-theorem