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Weak Jordan derivations of prime rings. (English) Zbl 1470.16043

Summary: Let \(R\) be a prime ring with extended centroid \(C\) and with maximal left ring of quotients \(Q_{ml}(R)\). An additive map \(\delta:R\to Q_{ml}(R)\) is called a weak Jordan derivation if \(\delta(x^2)-\delta(x)x-x\delta(x)\in C\) for all \(x\in R\). Applying the theory of functional identities and dealing with the low dimensional cases, we give a complete characterization of weak Jordan derivations of prime rings. Moreover, we generalize Brešar’s theorem concerning additive maps \(\delta:R\to RC+C\) satisfying \([\delta(x^2)-x\delta(x)-\delta(x)x,x]=0\) for all \(x\in R\).

MSC:

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