Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

On *-bimultipliers, Generalized *-biderivations and Related Mappings

  • Received : 2010.07.22
  • Accepted : 2011.04.06
  • Published : 2011.09.23
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper we dene the notions of left *-bimultiplier, *-bimultiplier and generalized *-biderivation, and to prove that if a semiprime *-ring admits a left *-bimultiplier M, then M maps R ${\times}$ R into Z(R). In Section 3, we discuss the applications of theory of *-bimultipliers. Further, it was shown that if a semiprime *-ring R admits a symmetric generalized *-biderivation G : R ${\times}$ R ${\rightarrow}$ R with an associated nonzero symmetric *-biderivation R ${\times}$ R ${\rightarrow}$ R, then G maps R ${\times}$ R into Z(R). As an application, we establish corresponding results in the setting of $C^*$-algebra.

Keywords

References

  1. Shakir Ali, On generalized $\ast$-derivations in $\ast$-rings, Plastin Journal of Mathematics, (submitted for publications).
  2. P. Ara and M. Mathieu, Local Multipliers of $C^{\ast}$-Algebras, Springer Monograph in Mathematics, Springer-Verlag, London, 2003.
  3. M. Ashraf and Shakir Ali, On $({\alpha}, {beta})^{\ast}$-derivations in $H^{\ast}$-algebras, Advances in Algebra, 2(1)(2009), 23-31.
  4. H. E. Bell and 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
  5. M. Bresar and J. Vukman, On some additive mappings in rings with involution, Aequationes Math., 38(1989), 178-185. https://doi.org/10.1007/BF01840003
  6. M. Bresar and J. Vukman, On left derivations and related mappings, Proc. Amer. Math. Soc., 110(1990), 7-16. https://doi.org/10.1090/S0002-9939-1990-1028284-3
  7. M. Bresar, W. S. Martindale III and C. R. Miers, Centralizing maps in prime rings with involution, J. Algebra, 161(1993), 432-357. https://doi.org/10.1006/jabr.1993.1223
  8. I. N. Herstein, Rings with involution, The Univ. of Chicago Press, Chicago 1976.
  9. B. Hvala, Generalized derivations in rings, Comm. Algebra, 26(4)(1998), 1147-1166. https://doi.org/10.1080/00927879808826190
  10. G. Maksa, A remark on symmetric biadditive functions having nonnegative diagonalization, Glasnik Math., 15(35)(1980), 279-282.
  11. G. Maksa, On the trace of symmetric biderivations, C. R. Math. Rep. Acad. Sci. Canada, 9(1987), 303-307.
  12. N. M. Muthana, Rings endowed with biderivations and other biadditive mappings, Ph. D. Dissertation, Girls College of Education Jeddeh, Saudi Arbia, 2005.
  13. N. M. Muthana, Left centralizer traces, generalized derivation, left bimultiplier, The Aligarh Bull. Maths., 26(2)(2007), 33-45.
  14. 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
  15. B. Zalar, On centralizers of semiprime rings, Comment. Math. Univ. Carol., 32(1991), 609-614.

Cited by

  1. On Jordan ∗-mappings in rings with involution vol.24, pp.1, 2016, https://doi.org/10.1016/j.joems.2014.12.006
  2. GENERALIZED (α, β)*-DERIVATIONS AND RELATED MAPPINGS IN SEMIPRIME *-RINGS vol.05, pp.02, 2012, https://doi.org/10.1142/S1793557112500155