On Functional Equation Related to a Class of Generalized Inner Derivations in Prime Rings

The main purpose of this paper is to prove the following result. Let R be a prime ring of char(R) = 0 or char(R) > 4, and let T : R -> R be an additive mapping satisfying the relation T (x(4)) = T (x)x(3) - xT(x(2))x + x(3)T (x) for all x is an element of R. In this case, T is of the form 4T (x) = qx + xq for all x is an element of R, where q is some fixed element from the symmetric Martindale ring of quotients of R. Some functional equations related to bicircular projections are also investigated.