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Generalized Jordan derivations on prime rings and standard operator algebras. (English) Zbl 1058.16031
Let $$R$$ be a ring. An additive map $$\delta\colon R\to R$$ is called a generalized derivation if there exists a derivation $$\tau\colon R\to R$$ such that $$\delta(xy)=\delta(x)y+x\tau(y)$$ for all $$x,y\in R$$. The authors define a generalized Jordan derivation as an additive map $$\delta\colon R\to R$$ such that there exists a Jordan derivation $$\tau\colon R\to R$$ satisfying $$\delta(x^2)=\delta(x)x+x\tau(x)$$ for all $$x\in R$$. In a similar fashion they define a generalized Jordan triple derivation, extending the known concept of a Jordan triple derivation. Using a standard approach they extend known results on Jordan (triple) derivations on prime rings and standard operator algebras to the generalized ones.

##### MSC:
 16W10 Rings with involution; Lie, Jordan and other nonassociative structures 16N60 Prime and semiprime associative rings 16W20 Automorphisms and endomorphisms 47B47 Commutators, derivations, elementary operators, etc. 47L10 Algebras of operators on Banach spaces and other topological linear spaces
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