Generalized Derivations on Power Values of Lie Ideals in Prime and Semiprime Rings

Let R be a 2-torsion free ring and let L be a noncentral Lie ideal of R, and let F:R→R and G:R→R be two generalized derivations of R. We will analyse the structure of R in the following cases: (a) R is prime and F(um)=G(un) for all u∈L and fixed positive integers m≠n; (b) R is prime and F((upvq)m)=G((vrus)n) for all u,v∈L and fixed integers m,n,p,q,r,s≥1; (c) R is semiprime and F((uv)n)=G((vu)n) for all u,v∈[R,R] and fixed integer n≥1; and (d) R is semiprime and F((uv)n)=G((vu)n) for all u,v∈R and fixed integer n≥1.