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
Article

Abstract

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 \({n \not \equiv 0 (\mod {4})}\) .

Keywords

Quaternion ring Modular integers Structure 

Mathematics Subject Classification

11R52 16-99 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Gross B.H., Lucianovic M.W.: On cubic rings and quaternion rings. J. Number Theory 129(6), 1468–1478 (2009)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Hahn, A.J.: Quadratic algebras, Clifford algebras, and arithmetic Witt groups. Universitext. Springer, New York, (1994)Google Scholar
  3. 3.
    Kanzaki T.: On non-commutative quadratic extensions of a commutative ring. Osaka J. Math 10, 597–605 (1973)MathSciNetMATHGoogle Scholar
  4. 4.
    Knus, M.A.: Quadratic and Hermitian forms over rings, volume 294 of Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences). Springer, Berlin (With a foreword by I. Bertuccioni) (1991)Google Scholar
  5. 5.
    Miguel C.J., Serôdio R.: On the structure of quaternion rings over \({\mathbb{Z}_{p}}\). Int. J. Algebra 5(25-28), 1313–1325 (2011)MathSciNetMATHGoogle Scholar
  6. 6.
    O’Meara, T.O.: Introduction to quadratic forms. Classics in Mathematics. Springer, Berlin (Reprint of the 1973 edition) (2000)Google Scholar
  7. 7.
    Özdemir M.: The roots of a split quaternion. Appl. Math. Lett. 22(2), 258–263 (2009)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Özen M., Güzeltepe M.: Cyclic codes over some finite quaternion integer rings. J. Franklin Inst. 348(7), 1312–1317 (2011)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Pierce, R.S.: Associative algebras, volume 88 of Graduate Texts in Mathematics. Studies in the History of Modern Science, vol. 9. Springer, New York (1982)Google Scholar
  10. 10.
    Rosen, K.H.: Elementary number theory and its applications, 4th edn. Addison-Wesley, Reading (2000)Google Scholar
  11. 11.
    Shah T., Rasool S.S.: On codes over quaternion integers. Appl. Algebra Eng. Commin. Comput. 24(6), 477–496 (2013)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Tuganbaev, A.A.: Quaternion algebras over commutative rings. (Russian). Mat. Zametki. 53(2), 126–131 (1993). Translation in Math. Notes 53(1–2), 204–207 (1993)Google Scholar
  13. 13.
    Voight J.: Characterizing quaternion rings over an arbitrary base. J. Reine Angew. Math. 657, 113–134 (2011)MathSciNetMATHGoogle Scholar
  14. 14.
    Voight, J.: Identifying the matrix ring: algorithms for quaternion algebras and quadratic forms. In: Alladi, K., Bhargava, M., Savitt, D., Tiep, P.H. (eds.) Quadratic and Higher Degree Forms, volume 31 of Developments in Mathematics, pp. 255–298. Springer, New York (2013)Google Scholar

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

Personalised recommendations