Advances in Applied Clifford Algebras

, Volume 25, Issue 4, pp 875–887 | Cite as

On the Structure of Quaternion Rings Over \({\mathbb{Z}/n\mathbb{Z}}\)

  • José María Grau
  • Celino Miguel
  • Antonio M. Oller-Marcén


In this paper we investigate the structure of \({\left(\frac{a,b}{\mathbb{Z}/n \mathbb{Z}}\right)}\), the quaternion rings over \({\mathbb{Z}/n \mathbb{Z}}\). It is proved that these rings are isomorphic to \({\left(\frac{-1,-1}{\mathbb{Z}/n \mathbb{Z}}\right)}\) if \({ a \equiv b \equiv -1 (\mod {4})}\) or to \({\left(\frac{1,1}{\mathbb{Z}/n \mathbb{Z}}\right)}\) otherwise. We also prove that the ring \({\left(\frac{a,b}{\mathbb{Z}/n \mathbb{Z}}\right)}\) is isomorphic to \({\mathbb{M}_2(\mathbb{Z}/n \mathbb{Z})}\) if and only if n is odd and that all quaternion algebras defined over \({\mathbb{Z}/n \mathbb{Z}}\) are isomorphic if and only if \({n \not \equiv 0 (\mod {4})}\) .


Quaternion ring Modular integers Structure 

Mathematics Subject Classification

11R52 16-99 


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Copyright information

© Springer Basel 2015

Authors and Affiliations

  • José María Grau
    • 1
  • Celino Miguel
    • 2
  • Antonio M. Oller-Marcén
    • 3
  1. 1.Departamento de MatemáticasUniversidad de OviedoOviedoSpain
  2. 2.Department of Mathematics, Instituto de TelecomunicaçõesBeira Interior UniversityCovilhãPortugal
  3. 3.Centro Universitario de la Defensa de ZaragozaSaragossaSpain

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