Araştırma Makalesi
BibTex RIS Kaynak Göster
Yıl 2024, Cilt: 36 Sayı: 36, 215 - 224, 12.07.2024
https://doi.org/10.24330/ieja.1470687

Öz

Kaynakça

  • R. H. Bruck and E. Kleinfeld, The structure of alternative division rings, Proc. Amer. Math. Soc., 2 (1951), 878-890.
  • E. Kleinfeld, Simple alternative rings, Ann. of Math., 58(3) (1953), 544-547.
  • E. Kleinfeld, A Characterization of the Cayley Numbers, Math. Assoc. America Studies in Mathematics, Vol. 2, pp. 126–143, Prentice-Hall, Englewood Cliffs, N. J., 1963.
  • E. Kleinfeld and Y. Segev, A short characterization of the octonions, Comm. Algebra, 49(12) (2021), 5347-5353.
  • E. Kleinfeld and Y. Segev, A characterisation of the quaternions using commutators, Math. Proc. R. Ir. Acad., 122A(1) (2022), 1-4.
  • R. D. Schafer, An Introduction to Nonassociative Algebras, Pure and Applied Mathematics, Vol. 22, Academic Press, New York-London, 1966.

A uniform characterization of the Octonions and the Quaternions using commutators

Yıl 2024, Cilt: 36 Sayı: 36, 215 - 224, 12.07.2024
https://doi.org/10.24330/ieja.1470687

Öz

Let $R$ be a ring which is not commutative. Assume that either $R$ is alternative, but not associative, or $R$ is associative and any commutator $v\in R$ satisfies: $v^2$ is in the center of $R.$ We show (using commutators) that if $R$ contains no divisors of zero and $\text{char}(R)\ne 2,$ then $R//C,$ the localization of $R$ at its center $C,$ is the octonions in the first case and the quaternions, in latter case. Our proof in both cases is essentially the same and it is elementary and rather self contained. We also give a short (uniform) proof that if a non-zero commutator in $R$ is not a zero divisor (with mild additional hypothesis when $R$ is alternative, but not associative (e.g.~that $(R,+)$ contains no $3$-torsion), then $R$ contains no divisors of zero.

Kaynakça

  • R. H. Bruck and E. Kleinfeld, The structure of alternative division rings, Proc. Amer. Math. Soc., 2 (1951), 878-890.
  • E. Kleinfeld, Simple alternative rings, Ann. of Math., 58(3) (1953), 544-547.
  • E. Kleinfeld, A Characterization of the Cayley Numbers, Math. Assoc. America Studies in Mathematics, Vol. 2, pp. 126–143, Prentice-Hall, Englewood Cliffs, N. J., 1963.
  • E. Kleinfeld and Y. Segev, A short characterization of the octonions, Comm. Algebra, 49(12) (2021), 5347-5353.
  • E. Kleinfeld and Y. Segev, A characterisation of the quaternions using commutators, Math. Proc. R. Ir. Acad., 122A(1) (2022), 1-4.
  • R. D. Schafer, An Introduction to Nonassociative Algebras, Pure and Applied Mathematics, Vol. 22, Academic Press, New York-London, 1966.
Toplam 6 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Cebir ve Sayı Teorisi
Bölüm Makaleler
Yazarlar

Erwin Kleinfeld Bu kişi benim

Yoav Segev Bu kişi benim

Erken Görünüm Tarihi 19 Nisan 2024
Yayımlanma Tarihi 12 Temmuz 2024
Gönderilme Tarihi 1 Şubat 2024
Kabul Tarihi 8 Nisan 2024
Yayımlandığı Sayı Yıl 2024 Cilt: 36 Sayı: 36

Kaynak Göster

APA Kleinfeld, E., & Segev, Y. (2024). A uniform characterization of the Octonions and the Quaternions using commutators. International Electronic Journal of Algebra, 36(36), 215-224. https://doi.org/10.24330/ieja.1470687
AMA Kleinfeld E, Segev Y. A uniform characterization of the Octonions and the Quaternions using commutators. IEJA. Temmuz 2024;36(36):215-224. doi:10.24330/ieja.1470687
Chicago Kleinfeld, Erwin, ve Yoav Segev. “A Uniform Characterization of the Octonions and the Quaternions Using Commutators”. International Electronic Journal of Algebra 36, sy. 36 (Temmuz 2024): 215-24. https://doi.org/10.24330/ieja.1470687.
EndNote Kleinfeld E, Segev Y (01 Temmuz 2024) A uniform characterization of the Octonions and the Quaternions using commutators. International Electronic Journal of Algebra 36 36 215–224.
IEEE E. Kleinfeld ve Y. Segev, “A uniform characterization of the Octonions and the Quaternions using commutators”, IEJA, c. 36, sy. 36, ss. 215–224, 2024, doi: 10.24330/ieja.1470687.
ISNAD Kleinfeld, Erwin - Segev, Yoav. “A Uniform Characterization of the Octonions and the Quaternions Using Commutators”. International Electronic Journal of Algebra 36/36 (Temmuz 2024), 215-224. https://doi.org/10.24330/ieja.1470687.
JAMA Kleinfeld E, Segev Y. A uniform characterization of the Octonions and the Quaternions using commutators. IEJA. 2024;36:215–224.
MLA Kleinfeld, Erwin ve Yoav Segev. “A Uniform Characterization of the Octonions and the Quaternions Using Commutators”. International Electronic Journal of Algebra, c. 36, sy. 36, 2024, ss. 215-24, doi:10.24330/ieja.1470687.
Vancouver Kleinfeld E, Segev Y. A uniform characterization of the Octonions and the Quaternions using commutators. IEJA. 2024;36(36):215-24.