DOI QR코드

DOI QR Code

On Semiprime Rings with Generalized Derivations

Khan, Mohd Rais;Hasnain, Mohammad Mueenul

  • Received : 2011.11.29
  • Accepted : 2012.04.03
  • Published : 2013.12.23

Abstract

In this paper, we investigate the commutativity of a semiprime ring R admitting a generalized derivation F with associated derivation D satisfying any one of the properties: (i) $F(x){\circ}D(y)=[x,y]$, (ii) $D(x){\circ}F(y)=F[x,y]$, (iii) $D(x){\circ}F(y)=xy$, (iv) $F(x{\circ}y)=[F(x) y]+[D(y),x]$, and (v) $F[x,y]=F(x){\circ}y-D(y){\circ}x$ for all x, y in some appropriate subsets of R.

Keywords

Commutators;Derivation;Ideals;Semiprime-ring

References

  1. E. Albas, N. Argac, Generalized derivations of prime rings, Algebra Colloq., 11(2)(2004), 399-410.
  2. N. Argac, On prime and semiprime rings with derivations, Algebra Colloq., 13(3)(2006), 371-380. https://doi.org/10.1142/S1005386706000320
  3. M. Ashraf, A. Ali and R. Rani, On generalized derivations of prime rings, Southeast Asian Bull. Math., 29(2005), 669-675.
  4. M. Ashraf, N. Rehman, On commutativity of rings with derivations, Results Math., 42(1-2)(2002), 3-8. https://doi.org/10.1007/BF03323547
  5. M. Ashraf and N. Rehman, On derivations and commutativity in prime rings, East- West J. Math., 3(1)(2001), 87-91.
  6. H. E. Bell, W. S. Martindale III, Centralizing mappings of semiprime rings, Canad. Math. Bull., 30(1987), 92-101. https://doi.org/10.4153/CMB-1987-014-x
  7. H. E. Bell, Some commutativity results involving derivations, Trends in Theory of Rings and Modules, S. T. Rizvi and S. M. A. Zaidi (Eds), Anamaya publisher, New Delhi, India (2005).
  8. M. Bresar, On distance of the composition of two derivations to the generalized derivations, Glasgo Math. J., 33(1991), 89-93. https://doi.org/10.1017/S0017089500008077
  9. M. N. Daif and H. E. Bell, Remarks on derivations on semiprime rings, Internat. J. Math. & Math. Sci., 15(1)(1992), 205-206. https://doi.org/10.1155/S0161171292000255
  10. J. H. Mayne, Centralizing mappings of prime rings, Canad. Math. Bull., 27(1984), 122-126. https://doi.org/10.4153/CMB-1984-018-2
  11. E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc., 8(1957), 1093-1100.
  12. N. Rehman, On commutativity of rings with generalized derivations, Math. J. Okayama Univ., 44(2002), 43-49.
  13. M. Ashraf, N. Rehman and M. Rahman, On generalized derivations and commutativ- ity of rings, Int. J. Math., Game Theory and Algebra, 18(1)(2008), 19-24.
  14. B. Hvala, Generalized derivations in rings, Comm. Algebra, 26(1998), 1147-1166. https://doi.org/10.1080/00927879808826190