# Noether normalization over $\mathbb{Z}$

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: http://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.

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.