Generalized derivations on power values of Lie ideals in prime and semiprime rings. (Q2260309)

Summary: Let \(R\) be a 2-torsion free ring and let \(L\) be a noncentral Lie ideal of \(R\), and let \(F\colon R\to R\) and \(G\colon R\to R\) be two generalized derivations of \(R\). We analyse the structure of \(R\) in the following cases: (a) \(R\) is prime and \(F(u^m)=G(u^n)\) for all \(u\in L\) and fixed positive integers \(m\neq n\); (b) \(R\) is prime and \(F((u^pv^q)^m)=G((v^ru^s)^n)\) for all \(u,v\in L\) and fixed integers \(m,n,p,q,r,s\geq 1\); (c) \(R\) is semiprime and \(F((uv)^n)=G((vu)^n)\) for all \(u,v\in [R,R]\) and fixed integer \(n\geq 1\); and (d) \(R\) is semiprime and \(F((uv)^n)=G((vu)^n)\) for all \(u,v\in R\) and fixed integer \(n\geq 1\).