Generalized Derivations with Power Central Values on Multilinear Polynomials on Right Ideals

Let K be a commutative ring with unity, R a prime K-algebra, with extended centroid C and right Utumi quotient ring U, g a non-zero generalized derivation of R. Suppose that f(x(1), . . . . ,x(n)) is a multilinear polynomial over K, I is a non-zero right ideal of R and m >= 1, a fixed integer. We prove the following results: If g(f(r(1), . . . . , r(n)))(m) = 0, for all r(1), . . . . , r(n) is an element of I, then one of the following holds: (1) [f(r(1), . . . . . , X-n), Xn+1] Xn+2 is an identity for I; (2) g(x) = ax for all X is an element of R, where a is an element of U such that aI = 0; (3) g(x) = ax + [q, x] for all X is an element of R, where a, q is an element of U such that aI = 0 and [q, I] I = 0. If there exist a(1), . . . , a(n) is an element of I such that g(f(a(1), . . . . , a(n)))(m) not equal 0 and g(f (r(1), . . . , r(n)))(m) is an element of Z(R), for all r(1), . . . . , r(n) is an element of I, then one of the following holds: (1) f(x(1), . . . . , x(n))x(n+1) is an identity for I; (2) f(x(1), . . . . , x(n)) is central valued on R; (3) g(x) = ax for a is an element of C and f(x(1), . . . . , x(n)) is power central valued on R; (4) R satisfies s(4), the standard identity in four variables.