On identities with composition of generalized derivations

Let R be a prime ring with extended centroid C, Q maximal right ring of quotients of R, RC central closure of R such that dime (RC) > 4, f (X1, ..., Xn) a multilinear polynomial over C that is not central-valued on R, and f (R) the set of all evaluations of the multilinear polynomial f (X1, ..., Xn) in R. Suppose that G is a nonzero generalized derivation of R such that G2(u)u ? C for all u ? f (R). Then one of the following conditions holds: (i) there exists a ? Q such that a2 = 0 and either G(x) = ax for all x ? R or G(x) = xa for all x ? R; (ii) there exists a ? Q such that 0 ? a2 e C and either G(x) = ax for all x ? R or G(x) = x a for all x ? R and f (X1, ..., Xn)2 is central-valued on R; (iii) char(R) = 2 and one of the following holds: (a) there exist a, b e Q such that G(x) = ax + xb for all x ? R and a2 = b2 e C; (b) there exist a, b e Q such that G(x) = ax + xb for all x ? R, a2, b2 e C and f (X1, ..., Xn)2 is central-valued on R; (c) there exist a ? Q and an X-outer derivation d of R such that G(x) = ax + d(x) for all x ? R, d2 = 0 and a2 + d(a) = 0; (d) there exist a ? Q and an X-outer derivation d of R such that G(x) = ax + d(x) for all x ? R, d2 = 0, a2 + d(a) e C and f (X1, ..., Xn)2 is central-valued on R. Moreover, we characterize the form of nonzero generalized derivations G of R satisfying G2 (x) = Ax for all x ? R, where A e C. © Canadian Mathematical Society 2017.