What is the normalization in the Noether normalization lemma?

Question

As I understand it, the normalization of a domain $A$ is its integral closure in its field of fractions.

Why do we call the Noether normalization lemma by that name? What ring is being normalized?

This is, to some extend, explained in this question. – Dietrich Burde Mar 28 '17 at 20:32 — Dietrich Burde, Mar 28 '17 at 20:32

score 1 · Accepted Answer · edited Apr 13 '17 at 12:58

Geometrically Noether Normalization means that for any irreducible projective variety $X$ there is a finite map $\phi:X→ℙ^n$. In a sense, this helps to obtain a normalization of a variety, which is an attempt to make things nice by assuming integrally closeness. Here are some links about this:

https://mathoverflow.net/questions/81420/noether-normalization-vs-normalization-of-varieties

noether normalization theorem geometric meaning

What is the meaning of normalization of varieties in complex geometry?

What is the normalization in the Noether normalization lemma?

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