Posner's Second theorem and an annihilator condition with generalized derivations

Abstract

Let R be a prime ring of characteristic different from 2, with extended centroid C, U its two-sided Utumi quotient ring, $deltaneq 0$ a non-zero generalized derivation of R, f $(x_1 , .., x_n )$ a non-central multilinear polynomial over C in n non-commuting variables, $a epsilon R$ such that $a[delta(f (r_1 , .., r_n )]$, $f (r_1 , .., r_n )] = 0$, for any $r_1 , .., r_n epsilon R$. Then one of the following holds: 1. a = 0; 2. there exists $lambda epsilon C$ such that $delta(x) = lambda x$, for all $x epsilon R$; 3. there exist $q epsilon U$ and $lambda epsilon C$ such that $delta(x) = (q + lambda)x + xq$, for all $x epsilon R$, and $f (x_1, .., x_n )^2$ is central valued on R.