Generalized Jordan derivations on prime rings and standard operator algebras. (Q1428961)

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.