2021/12/08 by Erwin Kleinfeld, Kleinfeld, Erwin, Yoav Segev +1
Mathematics · #17A35 #17D05 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR) #Primary: 12E15 #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #Secondary: 11R52
paper · pdf · doi:10.48550/arxiv.2112.04250
openalex publication_date 2021/12/08 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28
Let R be a ring with \bf 1 which is not commutative. Assume that a non-zero commutator in R is not a zero divisor. Assume further that either R is alternative, but not associative, or R is associative and any commutator v∈ R satisfies: v2 is in the center of R. We prove that R has no zero divisors. Furthermore, if char(R)≠ 2, then the localization of R at its center is an octonion division algebra, if R is alternative and a quaternion division algebra, if R is associative. Our proof in both cases is essentially the same and it is elementary and rather self contained.