On commutativity of semiprime rings with generalized derivations. (Q842007)

A generalized derivation on the ring \(R\) is an additive map \(f\colon R\to R\) for which there exists a derivation on \(R\) (which the author assumes to be nonzero) such that \(f(xy)=f(x)y+xd(y)\) for all \(x,y\in R\). This paper establishes that certain conditions involving one or two generalized derivations force a semiprime ring to have a nonzero central ideal, and hence force a prime ring to be commutative. A sample of such conditions: (a) \(R\) has generalized derivations \(f\) and \(g\) such that \(f(x)x=xg(x)\) for all \(x\in R\); (b) \(R\) has a generalized derivation \(f\) such that \(f(xy-yx)=xy-yx\) for all \(x,y\in R\). The results generalize results previously known for derivations.