Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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SKEW n-DERIVATIONS ON SEMIPRIME RINGS

  • Xu, Xiaowei (College of Mathematics Jilin University) ;
  • Liu, Yang (College of Mathematics Jilin University) ;
  • Zhang, Wei (College of Mathematics Jilin University)
  • Received : 2012.03.25
  • Published : 2013.11.30
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Abstract

For a ring R with an automorphism ${\sigma}$, an n-additive mapping ${\Delta}:R{\times}R{\times}{\cdots}{\times}R{\rightarrow}R$ is called a skew n-derivation with respect to ${\sigma}$ if it is always a ${\sigma}$-derivation of R for each argument. Namely, if n - 1 of the arguments are fixed, then ${\Delta}$ is a ${\sigma}$-derivation on the remaining argument. In this short note, from Bre$\check{s}$ar Theorems, we prove that a skew n-derivation ($n{\geq}3$) on a semiprime ring R must map into the center of R.

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References

  1. A. Ali and D. Kumar, Ideals and symmetric $({\sigma}, {\sigma})$-biderivations on prime rings, Aligarh Bull. Math. 25 (2006), no. 2, 9-18.
  2. N. Argac, On prime and semiprime rings with derivations, Algebra Colloq. 13 (2006), no. 3, 371-380. https://doi.org/10.1142/S1005386706000320
  3. N. Argac and M. S. Yenigul, Lie ideals and symmetric bi-derivations of prime rings, Symmetries in Science, VI (Bregenz, 1992), 41-45, Plenum, New York, 1993.
  4. N. Argac and M. S. Yenigul, Lie ideals and symmetric bi-derivations of prime rings, Pure Appl. Math. Sci. 44 (1996), no. 1-2, 17-21.
  5. M. Ashraf, On symmetric bi-derivations in rings, Rend. Istit. Mat. Univ. Trieste 31 (1999), no. 1-2, 25-36.
  6. M. Ashraf and N. Rehman, On symmetric $({\sigma}, {\sigma})$-biderivations, Aligarh Bull. Math. 17 (1997/98), 9-16.
  7. K. I. Beidar, W. S. Martindale 3rd, and A. V. Mikhalev, Rings with Generalized Identities, Marcel Dekker, Inc., New York, 1996.
  8. M. Bresar, On certain pairs of functions of semiprime rings, Proc. Amer. Math. Soc. 120 (1994), no. 3, 709-713. https://doi.org/10.1090/S0002-9939-1994-1181158-3
  9. M. Bresar, Functional identities of degree two, J. Algebra 172 (1995), no. 3, 690-720. https://doi.org/10.1006/jabr.1995.1066
  10. M. Bresar, On generalized biderivations and related maps, J. Algebra 172 (1995), no. 3, 764-786. https://doi.org/10.1006/jabr.1995.1069
  11. S. Ceran and M. Asci, On traces of symmetric bi-$({\sigma}, {\tau})$ derivations on prime rings, Algebras Groups Geom. 26 (2009), 203-214.
  12. Q. Deng, Symmetric biderivations and commutativity of prime rings, J. Math. Res. Exposition 16 (1996), no. 3, 427-430.
  13. Q. Deng, On a conjecture of Vukman, Internat. J. Math. Math. Sci. 20 (1997), no. 2, 263-266. https://doi.org/10.1155/S0161171297000355
  14. A. Fosner, Prime and semiprime rings with symmetric skew 3-derivations, submitted to Aequationes Mathematicae.
  15. M. Hongan and H. Komatsu, On the module of differentials of a noncommutative algebra and symmetric biderivations of a semiprime algebra, Comm. Algebra 28 (2000), no. 2, 669-692. https://doi.org/10.1080/00927870008826851
  16. S. L. Huang and S. T. Fu, Remarks on generalized derivations and symmetric bideriva- tions of prime rings, J. Math. Study 40 (2007), no. 4, 360-364.
  17. Y. S. Jung and K. H. Park, On prime and semiprime rings with permuting 3-derivations, Bull. Korean Math. Soc. 44 (2007), no. 4, 789-794. https://doi.org/10.4134/BKMS.2007.44.4.789
  18. P. Kannappan, Jordan derivation and functional equations, Glas. Mat. Ser. III 29(49) (1994), no. 2, 305-310.
  19. M. A. Khan, Remarks on symmetric biderivations of rings, Southeast Asian Bull. Math. 27 (2003), no. 4, 631-640.
  20. V. K. Kharchenko, Skew derivations of semiprime rings, Siberian Math. J. 32 (1991), no. 6, 1045-1051.
  21. G. Maksa, A remark on symmetric biadditive functions having nonnegative diagonalization, Glas. Mat. Ser. III 15(35) (1980), no. 2, 279-282.
  22. N. M. Muthana, Orthogonality of traces and derivations in semiprime rings, Aligarh Bull. Math. 26 (2007), no. 1, 49-60.
  23. K. H. Park, On 4-permuting 4-derivations in prime and semiprime rings, J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 14 (2007), no. 4, 271-278.
  24. K. H. Park, On prime and semiprime rings with symmetric n-derivations, J. Chungcheong Math. Soc. 22 (2009), 451-458.
  25. E. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093-1100. https://doi.org/10.1090/S0002-9939-1957-0095863-0
  26. J. Vukman, Symmetric bi-derivations on prime and semi-prime rings, Aequationes Math. 38 (1989), no. 2-3, 245-254. https://doi.org/10.1007/BF01840009
  27. J. Vukman, Two results concerning symmetric bi-derivations on prime rings, Aequationes Math. 40 (1990), no. 2-3, 181-189. https://doi.org/10.1007/BF02112294
  28. Y. Wang, Symmetric bi-derivation of prime rings, J. Math. Res. Exposition 22 (2002), no. 3, 503-504.
  29. M. S. Yenigul and N. Argac, Ideals and symmetric bi-derivations of prime and semi-prime rings, Math. J. Okayama Univ. 35 (1993), 189-192.
  30. J. M. Zhan, T-symmetric bi-derivations of prime rings, Qufu Shifan Daxue Xuebao Ziran Kexue Ban 29 (2003), no. 3, 16-18.

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