Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지
DOI QR코드

DOI QR Code

NILPOTENT VALUES OF GENERALIZED DERIVATION IN PRIME AND SEMIPRIME RINGS

  • Received : 2020.10.03
  • Accepted : 2020.12.27
  • Published : 2021.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

Let I be a nonzero ideal of a prime ring R and m, n are positive integers. If R admits a generalized derivation F satisfying F(x)nF(y)m - xnym = 0 for any y, x ∈ I, then F(x) = ax for any x ∈ R and am+n = 1, where a ∈ C, the extended centroid of R.

Keywords

References

  1. M. Ashraf, A. Ali, S. Ali, Some commutativity theorems for rings with generalized derivations, Southeast Asian Bull. Math. 31 (2007), 415-421.
  2. M. Daif, H.E. Bell, Remarks on derivations on semiprime rings, Int. J. Math. Math. Sci. 15 (1992), 205-206. https://doi.org/10.1155/S0161171292000255
  3. C.L. Chuang, GPIs having coefficents in Utumi quotient rings, Proc. Amer. Math. Soc. 103 (1988), 723-728. https://doi.org/10.1090/S0002-9939-1988-0947646-4
  4. V.D. Filippis, On a subset with nilpotent values in a prime ring with derivation, Boll. U.M.I. 8 (2002), 833-838.
  5. I.N. Herstein, Center-like elements in prime rings, Journal of Algebra 60 (1979), 567-574. https://doi.org/10.1016/0021-8693(79)90102-9
  6. M. Hongan, A note on semiprime rings with derivation, Int. J. Math. Math. Sci. 20 (1997), 413-415. https://doi.org/10.1155/S0161171297000562
  7. N. Jacobson, Structure of rings, Amer. Math. Soc. Colloq. Pub., RI, 1964.
  8. N. Jacobson, PI-Algebras, Springer Verlag, Lecture Notes in Math., New York, 1975.
  9. T.K. Lee, Generalized derivations of left faithful rings, Commun. Algebra 27 (1999), 4057-4073. https://doi.org/10.1080/00927879908826682
  10. T.K. Lee, Semiprime rings with differential identities, Bull. Inst. Math. Acad. Sin. 8 (1992), 27-38.
  11. K.I. Beidar, W.S. Martindale III, A.V. Mikhalev, Rings with generalized identities, Marcel Dekker Inc., New York, 1996.
  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. T. Erickson, W.S. Martindale III and J.M. Osborn, Prime nonassociative algebras, Pacific Journal of Mathematics 15 (1975), 49-63
  14. M. Ashraf, N. Rehman, On derivation and commutativity in prime rings, East-West J. Math. 3 (2001), 87-91.
  15. M.A. Quadri, M. Shadab Khan, N. Rehman, Generalized derivations and commutativity of prime rings, Indian J. Pure Appl. Math. 34 (2003), 1393-1396.