DOI QR코드

DOI QR Code

NOTES ON (σ, τ)-DERIVATIONS OF LIE IDEALS IN PRIME RINGS

  • Golbasi, Oznur (Department of Mathematics Faculty of Science Cumhuriyet University) ;
  • Oguz, Seda (Department of Secondary School Science and Mathematics Education Faculty of Education Cumhuriyet University)
  • Received : 2011.01.10
  • Published : 2012.07.31

Abstract

Let R be a prime ring with center Z and characteristic different from two, U a nonzero Lie ideal of R such that $u^2{\in}U$ for all $u{\in}U$ and $d$ be a nonzero (${\sigma}$, ${\tau}$)-derivation of R. We prove the following results: (i) If $[d(u),u]_{{\sigma},{\tau}}$ = 0 or $[d(u),u]_{{\sigma},{\tau}}{\in}C_{{\sigma},{\tau}}$ for all $u{\in}U$, then $U{\subseteq}Z$. (ii) If $a{\in}R$ and $[d(u),a]_{{\sigma},{\tau}}$ = 0 for all $u{\in}U$, then $U{\subseteq}Z$ or $a{\in}Z$. (iii) If $d([u,v])={\pm}[u,v]_{{\sigma},{\tau}}$ for all $u{\in}U$, then $U{\subseteq}Z$.

Acknowledgement

Supported by : Cumhuriyet University

References

  1. N. Argac, On prime and semiprime rings with derivations, Algebra Colloq. 13 (2006), no. 3, 371-380. https://doi.org/10.1142/S1005386706000320
  2. M. Ashraf and N. Rehman, On (${\sigma}$, ${\tau}$)-derivations in prime rings, Arch. Math. (Brno) 38 (2002), no. 4, 259-264.
  3. R. Awtar, Lie and Jordan structure in prime rings with derivations, Proc. Amer. Math. Soc. 41 (1973), 67-74. https://doi.org/10.1090/S0002-9939-1973-0318233-5
  4. N. Aydin, A note on (${\sigma}$, ${\tau}$)-derivations in prime rings, Indian J. Pure Appl. Math. 39 (2008), no. 4, 347-352.
  5. N. Aydin and K. Kaya, Some generalizations in prime rings with (${\sigma}$, ${\tau}$)-derivations, Doga Mat. 16 (1992), no. 3, 169-176.
  6. H. E. Bell and W. S. Martindale III, Centralizing mappings of semiprime rings, Canad. Math. Bull. 30 (1987), no. 1, 92-101. https://doi.org/10.4153/CMB-1987-014-x
  7. I. Bergen, I. N. Herstein, and J. W. Kerr, Lie ideals and derivation of prime rings, J. Algebra 71 (1981), no. 1, 259-267. https://doi.org/10.1016/0021-8693(81)90120-4
  8. M. N. Daif and H. E. Bell, Remarks on derivations on semiprime rings, Internat. J. Math. Math. Sci. 15 (1992), no. 1, 205-206. https://doi.org/10.1155/S0161171292000255
  9. E. Guven, Lie ideals in prime rings with (${\sigma}$, ${\tau}$)-derivation, Int. J. Algebra 3 (2009), no. 17-20, 857-862.
  10. I. N. Herstein, A note on derivations, Canad. Math. Bull. 21 (1978), no. 3, 369-370. https://doi.org/10.4153/CMB-1978-065-x
  11. H. Kandamar and K. Kaya, Lie ideals and (${\sigma}$, ${\tau}$)-derivations in prime rings, Hacettepe Bull. Natural Sci. and Engeneering 21 (1992), 29-33.
  12. P. H. Lee and T. K. Lee, Lie ideals of prime rings with derivations, Bull. Inst. Math. Acad. Sinica 11 (1983), no. 1, 75-79.
  13. J. Mayne, Centralizing automorphisms of prime rings, Canad. Math. Bull. 19 (1976), no. 1, 113-115. https://doi.org/10.4153/CMB-1976-017-1
  14. J. Mayne, Centralizing automorphisms of Lie ideals in prime rings, Canad. Math. Bull. 35 (1992), no. 4, 510-514. https://doi.org/10.4153/CMB-1992-067-0
  15. E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093-1100. https://doi.org/10.1090/S0002-9939-1957-0095863-0
  16. M. F. Smiley, Remarks on the commutativity of rings, Proc. Amer. Soc. 10 (1959), 466-470. https://doi.org/10.1090/S0002-9939-1959-0105427-X