Let R be a prime ring with nontrivial idempotents. We present a characterization of a tri-additive map f : R 3 → R such that f(x, y, z) = 0 for all x, y, z ∈ R with xy = yz = 0. As an application, we show that, in a prime ring with nontrivial idempotents, any local generalized (α, β)-derivation (or a generalized Jordan triple (α, β)-derivation) is a generalized (α, β)-derivation.
This is a preview of subscription content, log in via an institution to check access.
References
K. I. Beidar, W. S. Martindale, III, and A.V. Mikhalev, Rings with generalized identities,” Monographs and Textbooks in Pure and Appl. Math., Marcel Dekker, New York (1996), 196.
M. Bresar, “Characterizing homomorphisms, derivations, and multipliers in rings with idempotents,” Proc. Roy. Soc. Edinburgh, Sec. A, 137, 9–21 (2007).
M. Brešar and J. Vukman, “Jordan (Θ, φ)-derivations,” Glas. Mat. Ser. III, 26(46), 13–17 (1991).
J.-C. Chang, “On the identity h(x) = af(x) + g(x)b,” Taiwanese J. Math., 7, 103–113 (2003).
M. A. Chebotar, W.-F. Ke, and P.-H. Lee, “Maps characterized by action on zero products,” Pacific J. Math., 216, 217–228 (2004).
C.-L. Chuang and T.-K. Lee, “Derivations modulo elementary operators,” J. Algebra, 338, 56–70 (2011).
I. N. Herstein, Topics in Ring Theory, University of Chicago Press, Chicago (1969).
R. V. Kadison, “Local derivations,” J. Algebra, 130, 494–509 (1990).
D. R. Larson and A. R. Sourour, “Local derivations and local automorphisms of B(X),” Proc. Symp. Pure Math., 51, 187–194 (1990).
T.-K. Lee, “Semiprime rings with differential identities,” Bull. Inst. Math. Acad. Sinica (N.S.), 20, No. 1, 27–38 (1992).
T.-K. Lee, “Generalized derivations of left faithful rings,” Comm. Algebra, 27, No. 8, 4057–4073 (1999).
T.-K. Lee and K.-S. Liu, “Generalized skew derivations with algebraic values of bounded degree,” Houston J. Math. (to appear).
C.-K. Liu and W.-K. Shiue, “Generalized Jordan triple (θ, ϕ)-derivations on semiprime rings,” Taiwanese J. Math., 11, 1397–1406 (2007).
Y. Wang, “Local generalized derivations in prime rings with idempotents,” Algebra Colloq., 17, 295–300 (2010).
Author information
Authors and Affiliations
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 69, No. 6, pp. 848–853, June, 2017.
Rights and permissions
About this article
Cite this article
Mozumder, M.R., Jamal, M.R. Tri-Additive Maps and Local Generalized (α, β)-Derivations. Ukr Math J 69, 986–992 (2017). https://doi.org/10.1007/s11253-017-1408-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-017-1408-5