Generalized derivations with nilpotent, power-central, and invertible values in prime and semiprime rings

Let R be a ring, Q its symmetric Martindale quotient ring, C its extended centroid, I a nonzero ideal of R and F a generalized derivation with associated non-zero derivation d of R, and fixed integers. Let be a non-zero multilinear polynomial over C in t non-commuting variables, be any subset of R and . We prove the following results: If R is prime and for all , then is central valued on R.If R is prime and , for all , then is power central valued on R, unless .If R is semiprime and for all , then , for any and , that is there exists a central idempotent element such that , d vanishes identically on eQ and is central valued on .If R is semiprime and is zero or invertible in R, for all , then either R is a division ring or it is the ring of 2x2 matrices over a division ring, unless when , for any and .If R is prime and I is a non-zero right ideal of R such that and , for all , then is an identity on I.Let R be prime and I a non-zero right ideal of R such that and , for all . If there exists such that , then either is power central valued on R or is an identity on I, unless .(C)