Abstract
Let R be a prime ring of characteristic different from 2 and k >= 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)) circle ([r(1), r(2)](k)) = 0 (2) G([r(1), r(2)](k)) circle G([r(3), r(4)](k)) = [r(1), r(2)](k) circle [r(3), r(4)](k) (3) G([r(1), r(2)])circle(k) G([r(3), r(4)]) = 0 (4) G([r(1), r(2)])circle(k) G([r(3), r(4)]) = [r(1), r(2)]circle(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.