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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 22nd 2024

    a stub entry, for the moment just to make the link work

    v1, current

    • CommentRowNumber2.
    • CommentAuthorJohn Baez
    • CommentTimeDec 22nd 2024

    I included a bunch of equivalent definitions of ’von Neumann regular ring’, and a proof of what’s essentially Lemma 5 from von Neumann’s original paper. This clarifies the funny-looking equation a=axaa = a x a.

    My lemma and proof are not correctly formatted. Maybe someone could do that? I always have to look up how to do it.

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorJ-B Vienney
    • CommentTimeDec 22nd 2024
    • (edited Dec 22nd 2024)

    I don’t know if it’s the best way but I just wrote \begin{lemma} \end{lemma} \begin{proof} \end{proof} and it works fine.

    • CommentRowNumber4.
    • CommentAuthorJohn Baez
    • CommentTimeDec 23rd 2024

    I showed all the first 5 definitions of von Neumann regular ring are equivalent.

    diff, v5, current

    • CommentRowNumber5.
    • CommentAuthorJohn Baez
    • CommentTimeDec 23rd 2024

    Added an “idea” section sketching some motivation from quantum foundations.

    diff, v5, current

  1. started examples section

    Anonymouse

    diff, v7, current

    • CommentRowNumber7.
    • CommentAuthorDavidRoberts
    • CommentTimeDec 23rd 2024

    More examples

    diff, v8, current

    • CommentRowNumber8.
    • CommentAuthorJohn Baez
    • CommentTimeDec 25th 2024

    Deleted my claim that von Neumann algebras are von Neumann regular rings. This now seems false to me, since I think L ()L^\infty(\mathbb{R}) has principal ideals that aren’t generated by idempotents (i.e. characteristic functions). For example, the ideal generated by a continuous ’bump function’.

    diff, v10, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeDec 25th 2024

    have replaced the broken link “[[Morita invariant]]” by “invariant under Morita equivalences

    diff, v11, current

    • CommentRowNumber10.
    • CommentAuthorJohn Baez
    • CommentTimeDec 27th 2024

    Added another characterization of von Neumann regularity: all finitely generated modules are projective.

    Btw, I believe type II 1\mathrm{II}_1 von Neumann algebras are von Neumann regular rings but I don’t know a reference where someone says this. This would be nice to include if it’s true.

    diff, v12, current

    • CommentRowNumber11.
    • CommentAuthorJ-B Vienney
    • CommentTimeDec 27th 2024

    Added reference book mentioned on Zulip

    diff, v13, current

    • CommentRowNumber12.
    • CommentAuthorJohn Baez
    • CommentTimeDec 30th 2024

    Corrected characterizations 8 and 9.

    diff, v16, current

    • CommentRowNumber13.
    • CommentAuthorJohn Baez
    • CommentTimeDec 30th 2024

    Temporarily reverted 8 and 9 now that I see the article refers to 9 as it was.

    diff, v16, current

    • CommentRowNumber14.
    • CommentAuthorJohn Baez
    • CommentTimeDec 30th 2024

    Changed

    One can characterize commutative von Neumann rings as certain subrings of fields.

    to

    One can characterize commutative von Neumann rings as certain subrings of products of fields.

    Von Neumann regular rings can have zero divisors, e.g. \mathbb{R} \oplus \mathbb{R}, so they are not all subrings of fields

    diff, v16, current

  2. Added reference

    Anonymouse

    diff, v23, current

  3. adding fact that commutative von Neumann regular rings are precisely the reduced zero-dimensional rings

    Anonymouse

    diff, v23, current

  4. Lombardi & Quitté also calls von Neumann regular rings “absolutely flat rings”

    Anonymouse

    diff, v23, current