Generalized derivations and left ideals in prime and semiprime rings. (Q643170)

Summary: Let \(R\) be an associative ring, \(\lambda\) a nonzero left ideal of \(R\), \(d\colon R\to R\) a derivation and \(G\colon R\to R\) a generalized derivation. In this paper, we study the following situations in prime and semiprime rings: (1) \(G(x\circ y)=a(xy\pm yx)\); (2) \(G[x,y]=a(xy\pm yx)\); (3) \(d(x)\circ d(y)=a(xy\pm yx)\); for all \(x,y\in\lambda\) and \(a\in\{0,1,-1\}\).