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nLab: Normed Ring

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    • CommentRowNumber1.
    • CommentAuthorCorbennick
    • CommentTimeFeb 29th 2024
    In the page on normed rings the axiom |-x|=|x| is missing.
    It is imposed in the definition of a normed group, but not here.
    This has consequences. A Banach-ring is defined to be a complete normed ring.
    For it to be called complete, it needs a topology.
    One usually uses d(x,y)=|x-y| as a metric to build one.
    But the axioms, as they stand, do not imply that it is symmetric.
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 1st 2024
    • (edited Mar 1st 2024)

    I see that you added the condition to the entry normed ring.

    Just noting that the previous form of the definition was following the reference given, Def. 1.2.1 in Berkovich 2009. (?)

    Maybe worth commenting on and referencing different definitions?

    • CommentRowNumber3.
    • CommentAuthorCorbennick
    • CommentTimeMar 1st 2024
    Right. I did so. However I am unfamiliar to the habits around here, I just added it to the reference. Maybe that's not the appropriate style. Feel free to change it.
    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMar 1st 2024

    Thanks. The comment is fine, but best would be to add a reference which suports your claim. (Not that I am doubting it, but it would still be good to add a reference.)

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeMar 1st 2024

    It looks like Remark 2.2 is also false unless this axiom is added, since a normed group includes the axiom ρ(g −1)=ρ(g)\rho(g^{-1}) = \rho(g).

    I wonder if Berkovich just made a typo?

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMar 2nd 2024

    added the original reference:

    The extra axiom in not stated there, either.

    In Bosch, Güntzer & Remmert 1984 the axiom is also not stated, but they prove that it follows from their axioms (Prop. 3).

    • CommentRowNumber7.
    • CommentAuthorʇɐ
    • CommentTimeMar 2nd 2024

    Bosch–Güntzer–Remmert is about non-archimedean norms though (prop. 3 uses the ultrametric inequality), and Gelfand’s first axiom for normed rings is that they’re Banach spaces (so the paper is actually about Banach algebras)

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMar 3rd 2024

    This looks like it gives the definition you are after:

    • Shigeo Ozaki, Sadao Kashiwagi, Teruo Tsuboi, Note on Normed Rings, Science Reports of the Tokyo Bunrika Daigaku, Section A 4 98/103 (1953) 277-282 [jstor:43700402]

    (first page)

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeMar 3rd 2024

    After the definition (here), I have replaced the pointer to Berkovich 2009, def. 1.2.1 and the claim that Berkovich is missing an axiom (instead the issue seems to be that Berkovich discusses the non-Archimedean case) by pointer to Ozaki, Kashiwagi & Tsuboi 1953.

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