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On certain identity related to Herstein theorem on Jordan derivations. (English) Zbl 1345.16044

In this paper the authors study a 6-torsion-free prime ring \(R\) and suppose \(D\colon R\to R\) be an additive mapping satisfying the relation \(2D(x^4)=D(x^3)x+x^3D(x)+D(x)x^3+xD(x^3)\) for all \(x\in R\) to show that \(D\) is a derivation. This result is related to a classical result of Herstein, which states that any Jordan derivation on a 2-torsion-free prime ring is a derivation.
The title of this paper need to become more special where the authors, not determine their work on ring, algebra…etc. The results of J. Vukman [Glas. Mat., III. Ser. 46, No. 1, 43-48 (2011; Zbl 1217.47073)] represent the motivation of this paper. The authors depend on Theorem 4 which is focussed on 8-free Lie subring of \(R\) to prove Theorem 3 which studies 6-torsion free prime ring without mention what is the relation between Lie subring and prime ring. In fact, the authors use a good technique to present their results here, they depend on the commutators in large area of work. It is new addition of the field of derivations and Ring Theory. Finally, they close their paper by a good list of references which is mixed between old and some update.

MSC:

16W25 Derivations, actions of Lie algebras
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
16N60 Prime and semiprime associative rings
16R60 Functional identities (associative rings and algebras)
39B05 General theory of functional equations and inequalities

Citations:

Zbl 1217.47073
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References:

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