Geodesic
Commutativity with Derivations of Semiprime Rings
Discussiones Mathematicae. General Algebra and Applications, Tome 40 (2020) no. 2, pp. 165-175.

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Let R be a 2-torsion free semiprime ring with the centre Z(R), U be a non-zero ideal and d : R → R be a derivation mapping. Suppose that R admits(1) a derivation d satisfying one of the following conditions:(i) [d(x), d(y)] - [x, y] ∈ Z(R) for all x, y ∈ U,(ii) [d^2(x), d^2(y)] - [x, y] ∈ Z(R) for all x, y in U,(iii) [d(x)^2 , d(y)^2] - [x, y] ∈ Z(R) for all x, y ∈ U,(iv) [d(x^2), d(y^2)] - [x, y] ∈ Z(R) for all x, y ∈ U,(v) [d(x), d(y)] - [x^2, y^2] ∈ Z(R) for all x, y ∈ U.(2) a non-zero derivation d satisfying one of the following conditions:(i) d ( [ d(x), d(y) ] ) - [x, y] ∈ Z(R) for all x, y ∈ U,(ii) d ( [ d(x), d(y) ] ) + [x, y] ∈ Z(R) for all x, y ∈ U.Then R contains a non-zero central ideal.
Keywords: semiprime rings, derivations, torsion free rings, central ideal 
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Atteya, Mehsin Jabel. Commutativity with Derivations of Semiprime Rings. Discussiones Mathematicae. General Algebra and Applications, Tome 40 (2020) no. 2, pp. 165-175. http://geodesic.mathdoc.fr/item/DMGAA_2020_40_2_a2/

[1] M.A. Chaudhry and Allah-Bakhsh Thaheem, On a pair of (α, β) -derivation of semiprime rings, Aequation Math. 69 (2005) 224–230. doi:10.1007/s00010-004-2763-5

[2] H.E. Bell and W.S. Martindal III, Centralizing mappings of semiprime rings, Canad. Math. Bull. 30 10 (1987) 92–101. doi:10.4153/CMB-1987-014-x

[3] M. Brešar, On the distance of the compositions of two derivations to the generalized derivations, Glasgow Math. J. 33 (1991) 80–93.

[4] M.N. Daif and H.E. Bell, Remarks on derivations on semiprime rings, Int. J. Math. and Math. Sci. 15 (1992) 205–206. doi:10.1155/S0161171292000255

[5] M.N. Daif, Commutativity results for semiprime rings with derivations, Int. J. Math. and Math. Sci. 21 (1998) 471–474. doi:10.1155/S0161171298000660

[6] V. De Filippis, Automorphisms and derivations in prime rings, Rcndiconti di Mathematica, Serie VII, 19 (1999) 393–404.

[7] I.N. Herstein, Topics in Ring Theory (University of Chicago Press, Chicago, 1969).

[8] C. Lanski, An Engel condition with derivation for left ideals, Proc. Amer. Math. Soc. 125 (1997) 339–345. doi:10.1090/S0002-9939-97-03673-3

[9] A.H. Majeed and Mehsin Jabel, Some results of prime and semiprime rings with derivations, Um-Salama Sci. J. 3 (2005) 508–516.

[10] Mehsin Jabel and A.H. Majeed, On derivations of semiprime rings, Antarctica J. Math. 4 (2007) 131–138.

[11] M.J. Atteya, On generalized derivations of semiprime rings, Int. J. Alg. 12 (2010) 591–598.

[12] A. Nakajima, On categorical properties of generalized derivations, Sci. Math. 2 (1999) 345–352.

[13] L. Oukhtite and S. Salhi, On derivations in σ-prime rings, Int. J. Alg. 1 (2007) 241–246. doi:10.12988/ija.2007.07025

[14] B. Zalar, On centralizers of semiprime rings, Comment. Math. Univ. Carolinae 32 (1991) 609–614.

[15] I.N. Herstein, Rings with Involution (University of Chicago Press, Chicago, Illinois, 1976).

[16] M. Brešar, Centralizing mappings and derivations in prime rings, J. Alg. 156 (1993) 385–394.

[17] X. Xu, Y. Liu and W. Zhang, Skew N-derivations on semiprime rings, Bull. Korean Math. Soc. 50 (2013) 1863–1871. doi:10.4134/BKMS.2013.50.6.1863

[18] P.M. Cohn, Further Algebra and Applications, 1st edition (Springer-Verlag, London, Berlin, Heidelberg, 2003). doi:10.1007/978-1-4471-0039-3

[19] I.N. Herstein, A commutativity theorem, J. Alg. 38 (1976) 112–118. doi:10.1016/0021-8693(76)90248-9

[20] E.C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957) 1093–1100. doi:10.2307/2032686