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In the Stacks Project, Tag 032F, we find:

Definition. Let $R$ be a domain with field of fractions $K$.

  1. We say $R$ is N-1 if the integral closure of $R$ in $K$ is a finite $R$-module.
  2. We say $R$ is N-2 or Japanese if for any finite extension $L/K$ of fields the integral closure of $R$ in $L$ is finite over $R$.

Why do they call these rings 'N-1' and 'N-2'? What is the reason behind this terminology? (Other sources denote them the same way.)

(I guess the 'Japanese' terminology for the latter is to pay tribute to the Japanese mathematicians that studied these kinds of rings properties, such as Masayoshi Nagata.)

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    $\begingroup$ The term “Japanese ring” (resp. “universally Japanese ring”) for N-1 and N-2 was introduced by Grothendieck (im honor of Nagata but also Akizuki at least). My guess is that N stands for “Nagata” and the dash is a hyphen not a minus sign. You should mention that your quote is from the Stacks project. $\endgroup$
    – Gro-Tsen
    Commented Aug 29, 2023 at 11:16
  • $\begingroup$ @Gro-Tsen, re, in fact, since, as you point out, this is directly from Stacks Tag 032F, we can see there that, as in the quote, the original had not only a hyphen in place of a minus, but an italic N in place of a math-mode $N$. Since it's probably safe to assume that de Jong would have written $N - 1$ if that's what was meant, I think there is strong evidence that at least de Jong interprets it as you do. $\endgroup$
    – LSpice
    Commented Aug 29, 2023 at 13:33
  • $\begingroup$ It would be clearer if subscripts were used instead of hyphens. $\endgroup$
    – Ben McKay
    Commented Aug 29, 2023 at 15:11
  • $\begingroup$ Although the hypothesis that N comes from Nagata sounds like a good hypothesis, it's still a hypothesis. To verify it, I guess we would need to see the account of some mathematician or an explicit discussion on this regard on the literature (or some other historiographical proof). For example, some of us may have heard in the past the story that the '$\mathcal{O}$' notation for the structure sheaf of a ringed space was chosen 'in honor of Oka.' But apparently this explanation is not the true story. $\endgroup$
    – Elías Guisado Villalgordo
    Commented Aug 29, 2023 at 16:01
  • $\begingroup$ N = Nihon (romanisation of 日本, Japan in Japanese) perhaps ? $\endgroup$
    – Dat Minh Ha
    Commented Jan 16 at 19:58

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