On generalized derivations in prime ring with skew-commutativity conditions

Let \(R\) be a prime ring of characteristic different from \(2\) and \(k\ge 1\) fixed positive integer. If \(R\) admits a generalized derivation \(G\) associated with a deviation \(d\), here we study the following cases: (1) \(G([r_1,r_2]_k)\circ ([r_1,r_2]_k)=0\) (2) \(G([r_1,r_2]_k)\circ G([r_3,r_4]_k)=[r_1,r_2]_k\circ [r_3,r_4]_k\) (3) \(G([r_1,r_2])\circ _k G([r_3,r_4])=0\) (4) \(G([r_1,r_2])\circ _k G([r_3,r_4])=[r_1,r_2]\circ _k [r_3,r_4]\). We obtain a description of the structure of \(R\) and information on the form of \(G\) in terms of the commutativity of \(R\) and the multiplication by a specific element from the extended centroid of \(R\).