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, (Formula presented.) and (Formula presented.) fixed integers. Let (Formula presented.) be a non-zero multilinear polynomial over C in t non-commuting variables, (Formula presented.) be any subset of R and (Formula presented.). We prove the following results: If R is prime and (Formula presented.) for all (Formula presented.), then (Formula presented.) is central valued on R. If R is prime and (Formula presented.), for all (Formula presented.), then (Formula presented.) is power central valued on R, unless (Formula presented.). If R is semiprime and (Formula presented.) for all (Formula presented.), then (Formula presented.), for any (Formula presented.) and (Formula presented.), that is there exists a central idempotent element (Formula presented.) such that (Formula presented.), d vanishes identically on eQ and (Formula presented.) is central valued on (Formula presented.). If R is semiprime and (Formula presented.) is zero or invertible in R, for all (Formula presented.), then either R is a division ring or it is the ring of 2 × 2 matrices over a division ring, unless when (Formula presented.), for any (Formula presented.) and (Formula presented.). If R is prime and I is a non-zero right ideal of R such that (Formula presented.) and (Formula presented.), for all (Formula presented.), then (Formula presented.) is an identity on I. Let R be prime and I a non-zero right ideal of R such that (Formula presented.) and (Formula presented.), for all (Formula presented.). If there exists (Formula presented.) such that (Formula presented.), then either (Formula presented.) is power central valued on R or (Formula presented.) is an identity on I, unless (Formula presented.). © 2018, © 2018 Taylor & Francis Group, LLC.