Let R be a prime ring with characteristic different from 2 and U be a Lie ideal of R. In the paper, we initiate the study of generalized Jordan left derivations on Lie ideals of R and prove that every generalized Jordan left derivation on U is a generalized left derivation on U. Further, it is shown that generalized Jordan left biderivation associated with the left biderivation on U is either U ⊆ Z(R) or a right bicentralizer on U.
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References
S. Ali, “On generalized left derivations in rings and Banach algebras,” Aequat. Math., 81, 209–226 (2011).
M. Ashraf and S. Ali, “On generalized Jordan left derivations in rings,” Bull. Korean Math. Soc., 45, No. 2, 253–261 (2008).
M. Ashraf and N. Rehman, “On Lie ideals and Jordan left derivations of prime rings,” Arch. Math. (Brno), 36, 201–206 (2000).
M. Ashraf, N. Rehman, and S. Ali, “On Jordan left derivations of Lie ideals in prime rings,” South Asian Bull. Math., 25, 379–382 (2001).
J. Bergen, I. N. Herstein, and J. W. Kerr, “Lie ideals and derivations of prime rings,” J. Algebra, 71, 259–267 (1981).
K. I. Beidar, W. S. Martinedale III, and A.V. Mikhaleu, “Ring with generalized identities,” Monographs and Textbooks in Pure and Appl. Math., Markel Dekker, New York (1995).
M. Bresar, “On generalized bi-derivation and related maps,” J. Algebra, 172, 764–786 (1995).
M. Bresar and J. Vukman, “On left derivations and related mappings,” Proc. Amer. Math. Soc., 110, 7–16 (1990).
Q. Deng, “On Jordan left derivations,” Math. J. Okayama Univ., 34, 145–147 (1992).
I. N. Herstein, “On the Lie structure of associative rings,” J. Algebra, 14, No. 4, 561–571 (1970).
I. N. Herstein, Topics in Ring Theory, Univ. Chicago Press, Chicago (1969).
K. W. Jun and B. D. Kim, “A note on Jordan left derivations,” Bull. Korean Math. Soc., 33, No. 2, 221–228 (1996).
W. S. Martindale III, “Prime ring satisfying a generalized polynomial identity,” J. Algebra, 12, 576–584 (1969).
N. M. Muthana, “Left centralizer traces, generalized biderivations, left bi-multipliers, and generalized Jordan biderivations,” Aligarh Bull. Math., 26, No. 2, 33–45 (2007).
N. Rehman and M. Hongan, “Generalized Jordan derivations on Lie ideals associated with Hochschild 2-cocycles of rings,” Rend. Circ. Mat. Palermo, 60(2), No. 3, 437–444 (2011).
J. Vukman, “Jordan left derivations on semiprime rings,” Math. J. Okayama Univ., 39, 1–6 (1997).
S. M. A. Zaidi, M. Ashraf, and S. Ali, “On Jordan ideals and left (θ, θ) -derivation in prime rings,” Int. J. Math. Math. Sci., 37, 1957–1965 (2004).
B. Zalar, “On centralizers of semiprime rings,” Comment. Math. Univ. Carol., 32, No. 4, 609–614 (1991).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 8, pp. 1118–1125, August, 2013.
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Rehman, N., Ansari, A.Z. On Lie Ideals and Generalized Jordan Left Derivations of Prime Rings. Ukr Math J 65, 1247–1256 (2014). https://doi.org/10.1007/s11253-014-0855-5
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DOI: https://doi.org/10.1007/s11253-014-0855-5