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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?

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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?

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