A Note on Skew-commuting Automorphisms in Prime Rings

ur Rehman, Nadeem;Bano, Tarannum

  • Received : 2014.02.15
  • Accepted : 2014.04.22
  • Published : 2015.03.23


Let R be a prime ring with center Z, I a nonzero ideal of R, and ${\sigma}$ a non-trivial automorphism of R such that $\{(x{\circ}y)^{\sigma}-(x{\circ}y)\}^n{\in}Z$ for all $x,y{\in}I$. If either char(R) > n or char (R) = 0, then R satisfies $s_4$, the standard identity in 4 variables.


Prime ring;Ideal;Automorphism


  1. N. Argac, H. G. Inceboz, Derivation of prime ring and semiprime rings, J. Korean Math. Soc., 46(5)(2009), 997-1005.
  2. M. Ashraf, N. Rehman, On commutativity of rings with derivations, Results Math., 42(1-2)(2002), 3-8.
  3. K. I. Beidar, W. S. Martindale III, A. V. Mikhalev, Rings with Generalized Identities, New York-Basel-Hong Kong: Marcel Dekker, Inc., (1996), 51-95.
  4. H. E. Bell, W. S. Martindale, Centralizing mappings of semiprime rings, Can. Math. Bull., 30(1987), 92-101.
  5. M. Bresar, centralizing mappings and derivations in prime rings, J. Algebra, 156(1993), 385-394.
  6. M. Bresar, On skew commuting mappings of rings, Bull. Aust. Math. Soc., 47(1993), 291-296.
  7. M. Bresar, Functional identities: a survey, Contemporary Math., 259(2000), 93-109.
  8. C. L. Chuang, GPIs having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc., 103(3)(1988), 723-728,.
  9. C. L. Chuang, Differential identities with automorphisms and antiautomorphisms II, J. Algebra, 160(1993), 130-171.
  10. N. Divinsky, On commuting automorphisms of rings, Trans. R. Soc. Can. Sect. III, 49(1955), 19-22.
  11. J. S. Erickson, W. S. Martindale, J. M. Osborn, Prime nonassociative algebras, Pacific J. Math., 60(1975), 49-63.
  12. N. Jacobson, PI-algebras: An Introduction, Lecture Notes in Mathematics, Vol. 441. Berlin HeidelbergNew York: Springer Verlag, (1975).
  13. C. Lanski, Left ideals and derivations in semiprime ring, J. Algebra, 277(2004), 658-667.
  14. J. Luh, A note on commuting automorphisms of rings, Amer. Math. Mont., 77(1970), 61-62.
  15. W. S. Martindale, Prime rings satisfying a generalized polynomial identity, J. Algebra, 12(1969), 576-584.
  16. J. H. Mayne, Centralizing automorphisms of prime ring, Canad. Math. Bull., 19(1976), 113-115.
  17. E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc., 8(1957), 1093-1100.
  18. Y. Wang, Power-centralizing automorphisms of Lie ideals in prime ring, Comm. Algebra, 34(2006), 609-615.

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