摘要: | Let R be a prime ring with extended centroid C, λ a nonzero left ideal of R and f (X1, . . . , Xt ) a nonzero multilinear polynomial over C. Suppose that d and δ are derivations of R such that d( f (x1, . . . , xt )) f (x1, . . . , xt ) − f (x1, . . . , xt )δ( f (x1, . . . , xt )) ∈ C for all x1, . . . , xt ∈ λ. Then either d = 0 and λδ(λ) = 0 or λC = RCe for some idempotent e in the socle of RC and one of the following holds: (1) f (X1, . . . , Xt ) is central-valued on eRCe; (2) λ(d + δ)(λ) = 0 and f (X1, . . . , Xt )2 is central-valued on eRCe; (3) char R = 2 and eRCe satisfies st4(X1, X2, X3, X4), the standard polynomial identity of degree 4. |