Kyungpook Mathematical Journal

  • 제56권2호
  • /
  • Pages.397-408
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  • 2016
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  • 1225-6951(pISSN)
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  • 0454-8124(eISSN)
경북대학교 자연과학대학 수학과 (Department of Mathematics, Kyungpook National University)
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Derivations with Power Values on Lie Ideals in Rings and Banach Algebras

  • 투고 : 2015.04.05
  • 심사 : 2016.02.02
  • 발행 : 2016.06.23
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초록

Let R be a 2-torsion free prime ring with center Z, U be the Utumi quotient ring, Q be the Martindale quotient ring of R, d be a derivation of R and L be a Lie ideal of R. If $d(uv)^n=d(u)^md(v)^l$ or $d(uv)^n=d(v)^ld(u)^m$ for all $u,v{\in}L$, where m, n, l are xed positive integers, then $L{\subseteq}Z$. We also examine the case when R is a semiprime ring. Finally, as an application we apply our result to the continuous derivations on non-commutative Banach algebras. This result simultaneously generalizes a number of results in the literature.

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참고문헌

  1. A. Ali, N. Rehman and A. Shakir, On Lie ideals with derivations as homomorphisms and anti-homomorphisms, Acta Math. Hungar., 101(1-2)(2003), 79-82. https://doi.org/10.1023/B:AMHU.0000003893.61349.98
  2. 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.
  3. H. E. Bell and L. C. Kappe, Rings in which derivations satisfy certain algebraic conditions, Acta Math. Hungar., 53(1989), 339-346. https://doi.org/10.1007/BF01953371
  4. J. Bergen and L. Carini, A note on derivations with power central values on a Lie ideal, Pacific J. Math., 132(2)(1988), 209-213. https://doi.org/10.2140/pjm.1988.132.209
  5. 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
  6. C. L. Chuang, Hypercentral derivations, J. Algebra, 166(1)(1994), 34-71. https://doi.org/10.1006/jabr.1994.1140
  7. V. De Filippis, Generalized derivations in prime rings and noncommutative Banach algebras, Bull. Korean Math. Soc., 45(2008), 621-629.
  8. T. S. Erickson, W. S. Martindale III and J. M. Osborn, Prime nonassociative algebras, Pacific J. Math., 60(1)(1975), 49-63. https://doi.org/10.2140/pjm.1975.60.49
  9. I. N. Herstein, Topics in Ring Theory, Univ. of Chicago Press, 1969.
  10. I. N. Herstein, Derivations of prime rings having power central values, Algebraist's Homage. Contemporary Mathematics. Vol. 13, Amer. Math. Soc., Providence, Rhode Island, 1982.
  11. N. Jacobson, Structure of Rings, Colloquium Publications Vol. XXXVII, Amer. Math. Soc., 190 Hope street, Provindence, R. I., 1956.
  12. 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
  13. V. K. Kharchenko, Differential identities of prime rings, Algebra Logic, 17(2)(1979), 155-168. https://doi.org/10.1007/BF01670115
  14. C. Lanski, An Engel condition with derivation, Proc. Amer. Math. Soc., 118(1993), 731-734. https://doi.org/10.1090/S0002-9939-1993-1132851-9
  15. T. K. Lee, Semiprime rings with differential identities, Bull. Inst. Math., Acad. Sin., 20(1)(1992), 27-38.
  16. P. H. Lee and T. L. Wong, Derivations cocentralizing Lie ideals, Bull. Inst. Math. Acad. Sin., 23(1)(1995), 1-5.
  17. W. S. Martindale III, Prime rings satisfying a generalized polynomial identity, J. Algebra, 12(4)(1969), 576-584. https://doi.org/10.1016/0021-8693(69)90029-5
  18. M. Mathieu and G. J. Murphy, Derivations mapping into the radical, Arch. Math., 57(1991), 469-474. https://doi.org/10.1007/BF01246745
  19. M. Mathieu and V. Runde, Derivations mapping into the radical II, Bull. Lond. Math. Soc., 24(1992), 485-487. https://doi.org/10.1112/blms/24.5.485
  20. K. H. Park, On derivations in noncommutative semiprime rings and Banach algebras, Bull. Korean Math. Soc., 42 (2005), 671-678. https://doi.org/10.4134/BKMS.2005.42.4.671
  21. E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc., 8(1958), 1093-1100.
  22. A. M. Sinclair, Continuous derivations on Banach algebras, Proc. Amer. Math. Soc., 20(1969), 166-170. https://doi.org/10.1090/S0002-9939-1969-0233207-X
  23. I. M. Singer and J. Wermer, Derivations on commutative normed algebras, Math. Ann., 129(1955), 260-264. https://doi.org/10.1007/BF01362370
  24. M. P. Thomas, The image of a derivation is contained in the radical, Ann. Math., 128(2)(1988), 435-460. https://doi.org/10.2307/1971432
  25. Y. Wang and H. You, Derivations as homomorphisms or anti-homomorphisms on Lie ideals, Acta Math. Sinica., 23(6)(2007), 1149-1152. https://doi.org/10.1007/s10114-005-0840-x