Derivations cocentralizing multilinear polynomials on left ideals

Let R be a prime ring with extended centroid C, lambda a nonzero left ideal of R and f (X-1,..., X-t) a nonzero multilinear polynomial over C. Suppose that d and delta are derivations of R such that d(f(x(1),..., x(t))) f(x(1),..., x(t)) - f(x(1),..., x(t))delta(f(x(1),..., x(t))) is an element of C for all x(1),..., x(t) is an element of lambda Then either d = 0 and lambda delta(lambda) = 0 or lambda C = RCe for some idempotent e in the socle of RC and one of the following holds: (1) f (X-1,..., X-t) is central-valued on eRCe; (2) lambda(d + delta)(lambda) = 0 and f (X-1, ..., X-t)(2) is central-valued on eRCe; (3) char R = 2 and eRCe satisfies st (4)(X-1, X-2, X-3, X-4), the standard polynomial identity of degree 4.