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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

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Take a look at this: http://www.math.lsa.umich.edu/~hochster/615W10/supNoeth.pdf. It proves the generalized version of Noether Normalization, which is what you need (or rather what Qing Liu uses in his answer). In general I think Mel Hochster's notes are really good.

The above link was in the following answer https://mathoverflow.net/questions/42276, which was a link in Qing Liu's answer that you mention in your question.

Sorry, I should mention what the general version of Noether Normalization is that Hochster proves in his notes:

Let $D$ be a domain, and $R$ a finitely generated $D$ algebra. There exists a nonzero $f \in D$, and a finite injective ring map $D_f[X_1,\dots,X_n] \hookrightarrow R_f$. Here the $X_i$ are indeterminates.

Note how the above version implies Noether Normalization over a field. Although, if you know some basic scheme theory, I feel like Qing Liu's answer involving constructible sets is equally enlightening.

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    @Rankeya - If $D_f = 0$ then $R_f=0$ too, so that's okay. Perhaps the issue is the notion of algebraic independence over $D$?2015-11-07