Advertisement

Generalized multiplicative derivations and commutativity of 3-prime near-rings

  • Mohammad AshrafEmail author
  • Mohammad Aslam Siddeeque
Article
  • 6 Downloads

Abstract

In the present paper, we investigate the commutativity of 3-prime near-rings satisfying certain conditions involving left generalized multiplicative derivations on semigroup ideals. Moreover, examples have been provided to justify the necessity of 3-primeness condition in the hypotheses of various results.

Keywords

3-Prime near-ring Semigroup ideal Multiplicative derivation Left generalized multiplicative derivation and commutativity 

Mathematics Subject Classification

16W25 16Y30 

Notes

Acknowledgements

This research is partially supported by UGC-BSR research start-up-Grant no. F.30-310/2016(BSR). The authors are also indebted to the referee for his/her valuable suggestions and comments.

References

  1. 1.
    Ashraf, M., Boua, A., Siddeeque, M.A.: Generalized multiplicative derivations in \(3\)-prime near-rings. Math. Slovaca 68(2), 331–338 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ashraf, M., Rehman, N.: On commutativity of rings with derivations. Results Math. 42(1–2), 3–8 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bell, H.E.: On prime near-rings with generalized derivations. Int. J. Math. Math. Sci. Article ID 490316, 5 (2008)Google Scholar
  4. 4.
    Bell, H.E.: On Derivations in Near-Rings II, pp. 191–197. Kluwer Academic Publishers, Dordrecht (1997)zbMATHGoogle Scholar
  5. 5.
    Bell, H.E., Boua, A., Oukhtite, L.: Semigroup ideals and commutativity in \(3\)-prime near-rings. Comm. Algebra 43, 1757–1770 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bell, H.E., Mason, G.: On derivations in near-rings, Near-rings and Near-fields (G. Betsch editor), North-Holland / American Elsevier, Amsterdam, vol. 137, pp. 31–35 (1987)Google Scholar
  7. 7.
    Boua, A., Kamal, A.A.M.: Some results on \(3\)-prime near-rings with derivations. Indian J. Pure Appl. Math. 47(4), 705–716 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Boua, A., Oukhtite, L.: On commutativity of prime near-rings with derivations. South East Asian Bull. Math. 37, 325–331 (2013)zbMATHGoogle Scholar
  9. 9.
    \({\ddot{G}}\)olbasi., \({\ddot{O}}.\): On generalized derivations of prime near-rings. Hacet. J. Math. Stat., 35(2), 173–180 (2006)Google Scholar
  10. 10.
    Havala, B.: Generalized derivations in rings. Comm. Algebra 26, 1147–1166 (1998)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kamal, A.A.M., Al-Shaalan, K.H.: Existence of derivations on near-rings. Math. Slovaca 63(3), 431–448 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Meldrum, J.D.P.: Near-rings and their links with groups, Res. Notes Math. \(134\), Pitman, Bostan, M.A., (1985) (Advanced Publishing Program) Google Scholar
  13. 13.
    Pilz, G.: Near-rings, 2nd edn, 23. North Holland /American Elsevier, Amsterdam (1983)Google Scholar
  14. 14.
    Wang, X.K.: Derivations in prime near-rings. Proc. Am. Math. Soc 121(2), 361–366 (1994)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsAligarh Muslim UniversityAligarhIndia

Personalised recommendations