Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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A NOTE ON GENERALIZED SKEW DERIVATIONS ON MULTILINEAR POLYNOMIALS

  • RAZA, MOHD ARIF (Department of Mathematics, Faculty of Science & Arts-Rabigh, King Abdulaziz University) ;
  • REHMAN, NADEEM UR (Department of Mathematics, Aligarh Muslim University) ;
  • GOTMARE, A.R. (GDM Arts, KRN Commerce and MD Science Colledge)
  • Received : 2020.05.14
  • Accepted : 2020.10.16
  • Published : 2021.01.30
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Abstract

Let R be a prime ring, Qr be the right Martindale quotient ring and C be the extended centroid of R. If �� be a nonzero generalized skew derivation of R and f(x1, x2, ⋯, xn) be a multilinear polynomial over C such that (��(f(x1, x2, ⋯, xn)) - f(x1, x2, ⋯, xn)) ∈ C for all x1, x2, ⋯, xn ∈ R, then either f(x1, x2, ⋯, xn) is central valued on R or R satisfies the standard identity s4(x1, x2, x3, x4).

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References

  1. M. Ashraf and N. Rehman, On commutativity of rings with derivations, Results Math. 42 (2002), 3-8. https://doi.org/10.1007/BF03323547
  2. N. Argac, V. De Filippis and H.G. Inceboz, Generalized derivations with power central values on multilinear polynomials on right ideals, Rend. Sem. Mat. Univ. Padova 120 (2008), 59-71. https://doi.org/10.4171/RSMUP/120-4
  3. K.I. Beidar, W.S. Martindale III and A.V. Mikhalev, Rings with Generalized Identities, Pure and Applied Mathematics, Marcel Dekker 196, New York, 1996.
  4. H.E. Bell and M.N. Daif, Remarks on derivations on semiprime rings, Int. J. Math. Math. Sci. 15 (1992), 205-206. https://doi.org/10.1155/S0161171292000255
  5. J. Bergen, Automorphisms with unipotent values, Rend. Circ. Math. Palermo Series II Tomo 31 (1982), 226-232. https://doi.org/10.1007/BF02844355
  6. C.M. Chang, Power central values of derivations on multilinear polynomials, Taiwanese J. Math 7 (2003), 329-338. https://doi.org/10.11650/twjm/1500575068
  7. J.C. Chang, On the identity h(x) = af(x) + g(x)b, Taiwanese J. Math 7 (2003), 103-113. https://doi.org/10.11650/twjm/1500407520
  8. M.C. Chou and C.K. Liu, An Engel condition with skew derivation, Monatash Math 158 (2009), 259-270. https://doi.org/10.1007/s00605-008-0043-5
  9. C.L. Chuang, GPIs having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc. 103 (1988), 723-728. https://doi.org/10.1090/S0002-9939-1988-0947646-4
  10. C.L. Chuang, Differential identities with automorphism and anti-automorphism-I, J. Algebra 149 (1992), 371-404. https://doi.org/10.1016/0021-8693(92)90023-F
  11. C.L. Chuang, Differential identities with automorphism and anti-automorphism-II, J. Algebra 160 (1993), 130-171. https://doi.org/10.1006/jabr.1993.1181
  12. C.L. Chuang and T.K. Lee, Identities with a single skew derivation, J. Algebra 288 (2005), 59-77. https://doi.org/10.1016/j.jalgebra.2003.12.032
  13. V. De Filippis and O.M. Di Vincenzo, Vanishing derivations and centralizers of generalized derivations on multilinear polynomials, Comm. Algebra 40 (2012), 1918-1932. https://doi.org/10.1080/00927872.2011.553859
  14. M. Hogan, A note on semiprime rings with deribvation, Int. J. Math. Math. Sci. 20 (1997), 413-415. https://doi.org/10.1155/S0161171297000562
  15. N. Jacobson, Structure of rings, Amer. Math. Soc. Colloq. Pub., Rhode Island, 1964.
  16. V.K. Kharchenko, Generalized identities with automorphisms, Algebra Logic 14 (1975), 132-148. https://doi.org/10.1007/bf01668425
  17. T.K. Lee, Derivations with invertiable values on a multilinear polynomial, Proc. Amer. Math. Soc. 119 (1993), 1077-1083. https://doi.org/10.1090/S0002-9939-1993-1156472-7
  18. U. Leron, Nil and power central polynomials in rings, Trans. Amer. Math. Soc. 202 (1975), 97-103. https://doi.org/10.2307/1997300
  19. W.S. Martindale III, Prime rings satisfying a generalized polynomial identity, J. Algebra 12 (1969), 576-584. https://doi.org/10.1016/0021-8693(69)90029-5
  20. M.A. Quadri, M.S., Khan and N. Rehman, Generalized derivations and commutativity of prime rings, Indian J. pure appl. Math. 34 (2003), 1393-1396.
  21. N. Rehman and M.A. Raza, On m-commuting mappings with skew derivations in prime rings, St. Petersburg Math. J. 27 (2016), 641-650. https://doi.org/10.1090/spmj/1411
  22. N. Rehman and M.A. Raza, On ideals with skew derivations of prime ring, Miskolc Math. Notes 15 (2014), 717-724. https://doi.org/10.18514/mmn.2014.1217
  23. L. Rowen, Polynomial identities in ring thory, Pure and Applied Math. 84, Academic Press, Inc., New York London, 1980.
  24. T.L. Wong, Derivation with power central values on multilinear polynomials, Algebra Colloq. 3 (1996), 369-378.