A Characterization of the Quaternions Using Commutators

Erwin Kleinfeld, Yoav Segev

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Abstract

Let R be an associative ring with 1, which is not commutative. Assume that any non-zero commutator v ∈ R satisfies: v2 is in the centre of R, and v is not a zero divisor. We prove that R has no zero divisors, and that if char(R) ≠ 2, then the localisation of R at its centre is a quaternion division algebra. Our proof is elementary and self contained.

Original languageEnglish
Pages (from-to)1-4
JournalThe Mathematical Proceedings of the Royal Irish Academy
Volume122A
Issue number1
DOIs
StatePublished - 21 Jul 2021

Keywords

  • math.RA
  • Primary: 12E15

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