Ashraf, Mohammad;Siddeeque, Mohammad Aslam

  • Received : 2012.08.14
  • Published : 2013.10.31


In this paper, we introduce the notion of permuting $n$-derivations in near-ring N and investigate commutativity of addition and multiplication of N. Further, under certain constrants on a $n!$-torsion free prime near-ring N, it is shown that a permuting $n$-additive mapping D on N is zero if the trace $d$ of D is zero. Finally, some more related results are also obtained.


left near-rings;zero symmetric;derivations;permuting n-derivations


  1. K. I. Beidar, Y. Fong, and X. K. Wang, Posner and Herstein theorems for derivations of 3-prime near-rings, Comm. Algebra 24 (1996), no. 5, 1581-1589.
  2. H. E. Bell, On derivations in near-rings. II, Nearrings, nearfields and K-loops (Hamburg, 1995), 191-197, Math. Appl., 426, Kluwer Acad. Publ., Dordrecht, 1997.
  3. H. E. Bell and G. Mason, On derivations in near-rings, Near-rings and near-fields (Tubingen, 1985), 31-35, North-Holland Math. Stud., 137, North-Holland, Amsterdam, 1987.
  4. G. Maksa, A remark on symmetric biadditive functions having nonnegative diagonalization, Glas. Mat. Ser. III 15(35) (1980), no. 2, 279-282.
  5. G. Maksa, On the trace of symmetric bi-derivations, C. R. Math. Rep. Acad. Sci. Canada 9 (1987), no. 6, 303-307.
  6. M. A. Ozturk, Permuting tri-derivations in prime and semi prime rings, East. Asian. Math. J. 15 (1999), no. 2, 177-190.
  7. M. A. Ozturk and Y. B. Jun, On trace of symmetric bi-derivations in near-rings, Int. J. Pure Appl. Math. 17 (2004), no. 1, 95-102.
  8. K. H. Park, On prime and semi prime rings with symmetric n-derivations, J. Chungcheong Math. Soc. 22 (2009), no. 3, 451-458.
  9. K. H. Park and Y. S. Jung, On permuting 3-derivations and commutativity in prime near-rings, Commun. Korean Math. Soc. 25 (2010), no. 1, 1-9.
  10. G. Pilz, Near-rings, 2nd ed., 23, North Holland/American Elsevier, Amsterdam, 1983.

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