## Generalized Jordan derivations on semiprime rings and its applications in range inclusion problems.(English)Zbl 1246.16033

The paper considers various generalizations of Jordan derivations. For instance, a generalized Jordan triple derivation of a ring $$R$$ is defined as an additive map $$d\colon R\to R$$ satisfying $$d(xyx)=d(x)yx+xf(y)x+xyf(x)$$, where $$f$$ is a Jordan triple derivation. It is shown that on $$2$$-torsion free semiprime rings such maps are necessarily generalized derivations. Some related extensions of higher derivations are also studied, and applications to Banach algebras are given.

### MSC:

 16W25 Derivations, actions of Lie algebras 16N60 Prime and semiprime associative rings 47B47 Commutators, derivations, elementary operators, etc.
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### References:

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