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GENERALIZED DERIVATIONS IN PRIME RINGS AND NONCOMMUTATIVE BANACH ALGEBRAS

  • Published : 2008.11.30

Abstract

Let R be a prime ring of characteristic different from 2, C the extended centroid of R, and $\delta$ a generalized derivations of R. If [[$\delta(x)$, x], $\delta(x)$] = 0 for all $x\;{\in}\;R$ then either R is commutative or $\delta(x)\;=\;ax$ for all $x\;{\in}\;R$ and some $a\;{\in}\;C$. We also obtain some related result in case R is a Banach algebra and $\delta$ is either continuous or spectrally bounded.

References

  1. K. I. Beidar, Rings with generalized identities, Moscow Univ. Math. Bull. 33 (1978), no. 4, 53-58
  2. K. I. Beidar, W. S. Martindale III, and V. Mikhalev, Rings with Generalized Identities, Monographs and Textbooks in Pure and Applied Mathematics, 196. Marcel Dekker, Inc., New York, 1996
  3. M. Bresar and M. Mathieu, Derivations mapping into the radical. III, J. Funct. Anal. 133 (1995), no. 1, 21-29 https://doi.org/10.1006/jfan.1995.1116
  4. C. L. Chuang, GPIs having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc. 103 (1988), no. 3, 723-728
  5. T. S. Erickson, W. S. Martindale III, and J. M. Osborn, Prime nonassociative algebras, Pacific J. Math. 60 (1975), no. 1, 49-63 https://doi.org/10.2140/pjm.1975.60.49
  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, PI-algebras, Lecture Notes in Mathematics, Vol. 441. Springer-Verlag, Berlin-New York, 1975
  9. N. Jacobson, Structure of Rings, American Mathematical Society Colloquium Publications, Vol. 37. Revised edition American Mathematical Society, Providence, R.I. 1964
  10. B. E. Johnson and A. M. Sinclair, Continuity of derivations and a problem of Kaplansky, Amer. J. Math. 90 (1968), 1067-1073 https://doi.org/10.2307/2373290
  11. V. K. Kharchenko, Differential identities of prime rings, Algebra and Logic 17 (1978), no. 2, 155-168 https://doi.org/10.1007/BF01670115
  12. B.-D. Kim, Derivations of semiprime rings and noncommutative Banach algebras, Commun. Korean Math. Soc. 17 (2002), no. 4, 607-618 https://doi.org/10.4134/CKMS.2002.17.4.607
  13. T. K. Lee, Generalized derivations of left faithful rings, Comm. Algebra 27 (1999), no. 8, 4057-4073 https://doi.org/10.1080/00927879908826682
  14. T. K. Lee, Semiprime rings with differential identities, Bull. Inst. Math. Acad. Sinica 20 (1992), no. 1, 27-38
  15. 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
  16. M. Mathieu and G. J. Murphy, Derivations mapping into the radical, Arch. Math. (Basel) 57 (1991), no. 5, 469-474 https://doi.org/10.1007/BF01246745
  17. M. Mathieu and V. Runde, Derivations mapping into the radical. II, Bull. London Math. Soc. 24 (1992), no. 5, 485-487 https://doi.org/10.1112/blms/24.5.485
  18. K.-H. Park, On derivations in noncommutative semiprime rings and Banach algebras, Bull. Korean Math. Soc. 42 (2005), no. 4, 671-678 https://doi.org/10.4134/BKMS.2005.42.4.671
  19. E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093-1100
  20. A. M. Sinclair, Continuous derivations on Banach algebras, Proc. Amer. Math. Soc. 20 (1969), 166-170
  21. I. M. Singer and J. Werner, Derivations on commutative normed algebras, Math. Ann. 129 (1955), 260-264 https://doi.org/10.1007/BF01362370
  22. M. P. Thomas, The image of a derivation is contained in the radical, Ann. of Math. (2) 128 (1988), no. 3, 435-460 https://doi.org/10.2307/1971432
  23. J. Vukman, A result concerning derivations in noncommutative Banach algebras, Glas. Mat. Ser. III 26(46) (1991), no. 1-2, 83-88

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