Since the terminology "normal", "normalized", etc has different meanings in mathematics (some geometric in flavor, like when referring to perpendicularity) and I just read in Eisenbud's book on Commutative Algebra that what I knew as the integral closure of an $R$-algebra $S$, where $S$ is a commutative ring, is also called the normalization of $R$ in $S$, then I cannot help but wonder where does this terminology come from?

In the introductory section of chapter 4 in Eisenbud's book, he gives some examples that ilustrate how the normalization of a ring behaves "better" than the ring itself. Some are related to algebraic geometry and algebraic varieties, and others to algebraic number theory and algebraic number fields.

So my question is the following.

What is the origin and what motivated the terminology of normalization in commutative algebra?

Thank you very much.

  • 5
    $\begingroup$ Dear Adrian, There is a theorem that you might know, called "Noether normalization", which states that any finite type algebra over a field can be written as a finite extension of a polynomial algebra. I think (but can't support it with appropriate evidence) that normalization in the name of this theorem refers to the idea of putting an object into some kind of normal form. Since this result sits in the same general circle of ideas as integral closures and such, perhapsthere is a connection between the uses of the word normalization here and in the theory of integral closure. Regards, $\endgroup$ – Matt E Sep 24 '11 at 21:49

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