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
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.
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.
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. July 2024;36(36):215-224. doi:10.24330/ieja.1470687
Chicago
Kleinfeld, Erwin, and Yoav Segev. “A Uniform Characterization of the Octonions and the Quaternions Using Commutators”. International Electronic Journal of Algebra 36, no. 36 (July 2024): 215-24. https://doi.org/10.24330/ieja.1470687.
EndNote
Kleinfeld E, Segev Y (July 1, 2024) A uniform characterization of the Octonions and the Quaternions using commutators. International Electronic Journal of Algebra 36 36 215–224.
IEEE
E. Kleinfeld and Y. Segev, “A uniform characterization of the Octonions and the Quaternions using commutators”, IEJA, vol. 36, no. 36, pp. 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 (July 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 and Yoav Segev. “A Uniform Characterization of the Octonions and the Quaternions Using Commutators”. International Electronic Journal of Algebra, vol. 36, no. 36, 2024, pp. 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.