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NOTES ON GENERALIZED DERIVATIONS ON LIE IDEALS IN PRIME RINGS

  • Dhara, Basudeb ;
  • Filippis, Vincenzo De
  • Published : 2009.05.31

Abstract

Let R be a prime ring, H a generalized derivation of R and L a noncommutative Lie ideal of R. Suppose that $u^sH(u)u^t$ = 0 for all u $\in$ L, where s $\geq$ 0, t $\geq$ 0 are fixed integers. Then H(x) = 0 for all x $\in$ R unless char R = 2 and R satisfies $S_4$, the standard identity in four variables.

Keywords

prime ring;derivation;generalized derivation;extended centroid;Utumi quotient ring

References

  1. K. I. Beidar, W. S. Martindale III, and A. V. Mikhalev, Rings with Generalized Identities, Monographs and Textbooks in Pure and Applied Mathematics, 196. Marcel Dekker, Inc., New York, 1996
  2. C.-M. Chang and Y.-C. Lin, Derivations on one-sided ideals of prime rings, Tamsui Oxf. J. Math. Sci. 17 (2001), no. 2, 139–145
  3. C. L. Chuang, GPIs having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc. 103 (1988), no. 3, 723–728
  4. B. Dhara and R. K. Sharma, Derivations with annihilator conditions in prime rings, Publ. Math. Debrecen 71 (2007), no. 1-2, 11–20
  5. T. S. Erickson, W. S. Martindale III, and J. M. Osborn, Prime nonassociative algebras, Pacific J. Math. 60 (1975), no. 1, 49–63
  6. C. Faith and Y. Utumi, On a new proof of Litoff's theorem, Acta Math. Acad. Sci. Hungar 14 (1963), 369–371 https://doi.org/10.1007/BF01895723
  7. B. Hvala, Generalized derivations in rings, Comm. Algebra 26 (1998), no. 4, 1147–1166 https://doi.org/10.1080/00927879808826190
  8. N. Jacobson, Structure of Rings, Amer. Math. Soc. Colloq. Pub., 37, Amer. Math. Soc., Providence, RI, 1964
  9. C. Lanski, Differential identities, Lie ideals, and Posner's theorems, Pacific J. Math. 134 (1988), no. 2, 275–297
  10. C. Lanski, An Engel condition with derivation, Proc. Amer. Math. Soc. 118 (1993), no. 3, 731–734
  11. T. K. Lee, Semiprime rings with differential identities, Bull. Inst. Math. Acad. Sinica 20 (1992), no. 1, 27–38
  12. 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
  13. I. N. Herstein, Topics in Ring Theory, Univ. of Chicago Press, Chicago, IL, 1969
  14. V. K. Kharchenko, Differential identities of prime rings, Algebra i Logika 17 (1978), no. 2, 220–238, 242–243 https://doi.org/10.1007/BF01670115
  15. T. K. Lee, Generalized derivations of left faithful rings, Comm. Algebra 27 (1999), no. 8, 4057–4073 https://doi.org/10.1080/00927879908826682

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