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: