A UNIFORM CHARACTERIZATION OF THE OCTONIONS AND THE QUATERNIONS USING COMMUTATORS

Erwin Kleinfeld, Yoav Segev

Research output: Contribution to journalArticlepeer-review

Abstract

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 ∈ R satisfies: v2 is in the center of R. We show (using commutators) that if R contains no divisors of zero and char(R) ≠ 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.

Original languageEnglish
Pages (from-to)215-224
Number of pages10
JournalInternational Electronic Journal of Algebra
Volume36
Issue number36
DOIs
StatePublished - 12 Jul 2024

Keywords

  • Octonion algebra
  • Quaternion algebra
  • commutator
  • division algebra
  • zero divisor

ASJC Scopus subject areas

  • Algebra and Number Theory

Fingerprint

Dive into the research topics of 'A UNIFORM CHARACTERIZATION OF THE OCTONIONS AND THE QUATERNIONS USING COMMUTATORS'. Together they form a unique fingerprint.

Cite this