I am interested in the history of the proof of Artin's Theorem (on the diassociativity of alternative rings).

Question. When has Artin proved this theorem and where was it published for the first time?

This theorem easily implies the Artin–Zorn Theorem (saying that finite alternative rings are fields). Zorn published this result in 1930 and credited its discovery to Artin. So, this suggests that Artin should know his theorem before 1930. Also, I suspect that Ruth Moufang could also discover Artin's Theorem by studying the geometry of Moufang planes. So, what is the historical truth behind this theorem? Unfortunately, I cannot find the answer in available textbooks and papers (usually, textbooks just cite this theorem as Artin's Theorem, without any year or reference).

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    $\begingroup$ And who's buried in Grant's Tomb? $\endgroup$ Commented Jun 12 at 20:01
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    $\begingroup$ @SamHopkins Julia Grant, of course! $\endgroup$ Commented Jun 13 at 7:29

1 Answer 1


Artin published [1] on the related Wedderburn theorem, but Zorn does not cite a publication on the theorem he attributes to Artin in his 1930 paper [2]. Moufang [3] also cites Zorn in her 1935 publication on Artin's theorem as the only published source.

As discussed here, Zorn was Artin's Ph.D. student in Hamburg, Zorn's 1930 Ph.D thesis was on alternative algebras. The absence of a publication by Artin suggests Zorn credited Artin with the theorem because of a private communication between student and advisor.

[1] E. Artin, Zur Theorie der hyperkomplexen Zahlen, Abh. Math. Sem. Univ. Hamburg 5 (1927), 251–260.
[2] M. Zorn, Theorie der alternativen Ringe, Hamb. Abhandl. 8 (1930), 142-147.
[3] R. Moufang, Zur Struktur von Alternativkörpern, Mathem. Ann. 110 (1935), 416-430.

Follow up after comments on the relation between Artin's theorem and the Artin-Zorn theorem: The theorem which Zorn attributes to Artin, that in an alternative ring the subring generated by any two elements is associative, is not limited to finite rings, only the corollary, now known as the Artin-Zorn theorem is. I quote from Zorn's 1930 paper:

Herr Artin, der diese Arbeit veranlaßt hat, hat auch einen Teil der Ergebnisse, wie die Existenz des Einheitselements und die Reduzibilität in einfache Systeme unter der Annahme, das das halbeinfache System über einem Grundkörper der Charakteristik Null endlich ist, bewiesen, ferner den schönen Satz, daß ein aus zwei Elementen erzeugtes alternatives System assoziativ ist, mit der interessanten Folgerung: Ein nullteilerfreies alternatives System mit endlich vielen Elementen ist Galoisfeld. Darüber hinaus bin ich ihm für viele Hinweise und Ratschläge zu Dank verpflichtet.

Mr. Artin, who prompted this work, also proved part of the results, such as the existence of the identity element and the reducibility to simple systems under the assumption that the semisimple system over a ground field of characteristic zero is finite, as well as the beautiful theorem that an alternative system generated by two elements is associative, with the interesting corollary: A zero-divisor-free alternative system with finitely many elements is a Galois field. Furthermore, I am indebted to him for many suggestions and advice.
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    $\begingroup$ Thank you for the answer. What about Artin's Theorem on diassociativity of alternative rings? Was it also known to Artin (and his collaborators) before 1930 but not published? Or this theorem was discovered later? Because the diassociativity of alternating rings is a stronger result comparing the the Artin-Zorn Theorem (on finite alternative rings). $\endgroup$ Commented Jun 12 at 22:02
  • $\begingroup$ What Zorn calls "Artin's theorem" (the statement that in an alternative ring the subring generated by any two elements is associative) is not restricted to finite rings, is it? $\endgroup$ Commented Jun 13 at 8:36
  • $\begingroup$ Artin's Theorem holds for arbitrary alternative rings, not necessarily finite. $\endgroup$ Commented Jun 13 at 10:12
  • $\begingroup$ I added some text that hopefully resolves the distinction between Artin's theorem for arbitrary alternative rings and the Artin-Zorn theorem for finite rings. $\endgroup$ Commented Jun 13 at 15:10

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