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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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