Some Identities Involving Generalized Left (T,F)-Derivations in Semiprime Rings

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Some Identities Involving Generalized Left (T,F)-Derivations in Semiprime Rings

Sachhidananda Mondal

Author Affiliations

¹Shyamsundarpur High School, P. O.- Hanschara, Dist: Purba Medinipur, 721625, INDIA

Res. J. Mathematical & Statistical Sci., Volume 3, Issue (5), Pages 5-10, May,12 (2015)

Abstract

Let us suppose that R be a semiprime ring with two epimorphisms ?, f and I a nonzero ideal of R. By generalized left (?,f)derivation of R we mean an additive mapping F : R ? R such that F(ab) = ?(a)F(b) + f(b)d(a) holds for all a, b ? R, where d : R ? R is a left(?,f)derivation of R. The goal of the present paper is to study the following identities: (i) F([a, b])= ?([a, b]), (ii) F(a?b)= ?(a?b), (iii) F([a, b])= ?(a?b), (iv) F(a?b) = ?([a, b]), (v) F(ab)±?(ab)?Z(R), (vi) F(ab) ± ?(ba) ? Z(R), for all a, b ? I. Mathematics Subject Classification 2010. 16W25,16W80, 16N60.

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[ref1] E.C. Posner, Derivations in prime rings, Proc. Amer. Math.Soc., 8, 1093-1100 (1957)
Google Scholar

[ref2] M. Bre ar and J. Vukman, On left derivations and relatedmappings, Proc. Amer. Math. Soc., 110(1), 7-16 (1990)
Google Scholar

[ref3] M. Ashraf and S. Ali, On generalized Jordan left derivationsin rings, Bull. Korean Math. Soc., 45(2), 253-261 (2008)
Google Scholar

[ref4] X. W. Xu, H. Y. Zhang, A note on generalized left (θ, )-derivations in prime rings, Hacettepe J. Math. and Stat.,40(4), 523–529 (2011)
Google Scholar

[ref5] M. Ashraf, S. Ali, N. Rehman and M.R. Mozumder, Ongeneralized left (α,β)-derivations in rings, Bul. Iranian Math.Soc., 38(4), 893-905 (2012)
Google Scholar

[ref6] M.N. Daif and H. E. Bell, Remarks on derivations onsemiprime rings, internat., J. Math. & Math. Sci., 15(1), 205-206, (1992)

[ref7] I.N. Herstein, Rings with involution, Univ. of Chicago Press, Chicago (1976)
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[ref8] B. Dhara, S. Kar and S. Mondal, Commutativity theorems onprime and semiprime rings with generalized (σ,τ )-derivations, Bol. Soc. Paran. Mat, 32(1), 109-122 (2014)
Google Scholar