LIE IDEALS AND DERIVATIONS OF $\sigma$-PRIME RINGS

  • Published : 2010.02.28

Abstract

Let R be a 2-torsion free $\sigma$-prime ring with an involution $\sigma$, U a nonzero square closed $\sigma$-Lie ideal, Z(R) the center of Rand d a derivation of R. In this paper, it is proved that d = 0 or $U\;{\subseteq}\;Z(R)$ if one of the following conditions holds: (1) $d(xy)\;-\;xy\;{\in}\;Z(R)$ or $d(xy)\;-\;yx\;{\in}Z(R)$ for all x, $y\;{\in}\;U$. (2) $d(x)\;{\circ}\;d(y)\;=\;0$ or $d(x)\;{\circ}\;d(y)\;=\;x\;{\circ}\;y$ for all x, $y\;{\in}\;U$ and d commutes with $\sigma$.

Keywords

References

  1. L. Oukhtite & S. Salhi: Centralizing automorphisms and Jordan left derivations of $\sigma$-prime rings. Advances in Algebra 1 (2008), 19-26.
  2. L. Oukhtite & S. Salhi: Lie ideals and derivations of $\sigma$-prime rings. Int. J. Algebra 1 (2007), 25-30. https://doi.org/10.12988/ija.2007.07003
  3. L. Oukhtite & S. Salhi: On commutativity of $\sigma$-prime rings. Glasnik Mathematicki. 41 (2006), 57-64. https://doi.org/10.3336/gm.41.1.05
  4. L. Oukhtite & S. Salhi: On derivations in $\sigma$-prime rings. Int. J. Algebra 1 (2007), 241-246. https://doi.org/10.12988/ija.2007.07025
  5. L. Oukhtite & S. Salhi: Derivations and commutativity of $\sigma$-prime rings. Int. J. Contemp. 1 (2006), 439-448.
  6. L. Oukhtite & S. Salhi: $\sigma$-Lie ideals with derivations as homomorphisms and antihomomorphisms. Int. J. Algebra 1 (2007), 235-239,
  7. L. Oukhtite & S. Salhi: On generalized derivations of $\sigma$-prime rings. Afr. Diaspora J. Math. 5 (2007), 19-23.
  8. L. Oukhtite, S. Salhi & L. Taoufiq: On generalized derivations and commutativity in $\sigma$-prime rings. Int. J. Algebra 1 (2007), 227-230. https://doi.org/10.12988/ija.2007.07022
  9. L. Oukhtite, S. Salhi & L. Taoufiq: Jordan generalized derivations on $\sigma$-prime rings. Int. J. Algebra 1 (2007), 231-234. https://doi.org/10.12988/ija.2007.07023
  10. M. Ashraf & N. Rehman: On derivations and commutativity in prime rings. East- West J. Math. 3 (2001), 87-91.
  11. M. Ashraf & N. Rehman: On commutativity of rings with derivations. Result. Math. 42 (2002), 3-8. https://doi.org/10.1007/BF03323547
  12. S. Huang: On commutativity of $\sigma$-prime rings. Journal of Algebra and Discrete Structure 6 (2008), 89-93.