Multilinear Polynomials and Cocentralizing Conditions In Prime Rings

Let R be a noncommutative prime ring of characteristic different from 2, let Z(R) be its center, let U be the Utumi quotient ring of R, let C be the extended centroid of R, and let f(x(1), ..., x(n)) be a noncentral multilinear polynomial over C in n noncommuting variables. Denote by f(R) the set of all evaluations of f(x(1), ..., x(n)) on R. If F and G are generalized derivations of R such that [[F(x), x], [G(y), y]] is an element of Z(R) for any x, y is an element of f(R), then one of the following holds: (1) there exists alpha is an element of C such that F(x) = alpha x for all x is an element of R; (2) there exists beta is an element of C such that G(x) = beta x for all x is an element of R; (3) f(x(1), ..., x(n))(2) is central valued on R and either there exist a is an element of U and alpha is an element of C such that F(x) = ax + xa + alpha x for all x is an element of R or there exist c is an element of U and beta is an element of C such that G(x) = cx + xc + beta x for all x is an element of R; (4) R satisfies the standard identity s(4)(x(1), ..., x(4)) and either there exist a is an element of U and alpha is an element of C such that F(x) = ax + xa + alpha x for all x is an element of R or there exist c is an element of U and beta is an element of C such that G(x) = cx + xc + beta x for all x is an element of R.