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 
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/
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