Contents

Journal of Combinatorial Mathematics and Combinatorial Computing

  • Research article

On the theorems of Frobenius, Zorn and Hopf’s commutativity

Alassane Diouf1, Elhassan Idnarour2, Mbayang Amar1, Abdellatif Rochdi2
1Département de Mathématiques et Informatique, Faculté des Sciences et Techniques, Université Cheikh Anta Diop, 5005 Dakar (Sénégal)
2Département de Mathématiques et Informatique, Faculté des Sciences Ben M’Sik, Université Hassan II, 7955 Casablanca (Morocco)
  • Received: 01/02/2025
  • Accepted: 23/02/2025
  • Published Online: 17/03/2025

Abstract

Let \(A\) be a real algebra. It is called locally complex algebra if every non-zero element generates a subalgebra isomorphic to either \(\mathbb{R}\) or \(\mathbb{C}.\) It is said to satisfy the  uniqueness of the square root except the sign if the equation \(x^2=y^2\) implies \(y=\pm x.\) We show the following:
1. Every locally complex algebra is a quadratic algebra.
2. Every alternative locally complex algebra is isomorphic to either \(\mathbb{R},\) \(\mathbb{C},\) \(\mathbb{H}\) or \(\mathbb{O}.\)
3. Every commutative locally complex algebra without divisors of zero is isomorphic to \(\mathbb{R}\) or \(\mathbb{C}.\)
4. Every finite-dimensional algebra satisfying the uniqueness of the square root except the sign has dimension \(\leq 2\) and contains non-zero idempotents.

Keywords: quadratic (alternative, flexible, locally complex) algebra, divisors of zero, Cayley-Dickson process, quaternions, octonions, sedenions, nearly absolute-valued algebra

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Citation
Alassane Diouf, Elhassan Idnarour, Mbayang Amar, Abdellatif Rochdi. On the theorems of Frobenius, Zorn and Hopf’s commutativity[J], Journal of Combinatorial Mathematics and Combinatorial Computing, Volume 124. 219-227. https://doi.org/10.61091/jcmcc124-14.