Posner’s Second Theorem and an Annihilator Condition with Generalized Derivations

Let R be a prime ring of characteristic different from 2, with extended centroid C, U its two-sided Utumi quotient ring, delta not equal 0 a non-zero generalized derivation of R, f(x1, ..,xn) a non-central multilinear polynomial over C in n non-commuting variables, a element of R such that a[delta(f(r1, .., rn)), f(r1, .., rn)] = 0, for any r1, .., rn element of R. Then one of the following holds: 1. a = 0; 2. there exists lambda element of C such that delta(x) = lambda x, for all x element of R; 3. there exist q element of U and lambda element of C such that delta(x) = (q +lambda )x+xq, for all x element of R, and f(x1, .., xn)2 is central valued on R.