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Let R be a prime ring of characteristic different from 2, C its extended centroid, d a nonzero derivation of R, f (x1, . . . , xn) a multilinear polynomial over C, ρ a nonzero right ideal of R andm > 1 a fixed integer such that ([d(f(r1, . . . , rn)), f(r1, . . . , rn)])m = [d(f(r1, . . . , rn)), f(r1, . . . , rn)] for all r1, . . . , rn ∈ ρ. Then either [f(x1,..., xn), xn+1]xn+2 is an identity for ρ or d(ρ)ρ = 0. [ABSTRACT FROM AUTHOR]

Let R be a prime ring, I a nonzero ideal of R, and a ε R. Suppose that σ is a nontrivial automorphism of R such that af{(σ(x 0 y))n - (x 0 y)m} = 0 or a{(σ([x; y]))n - ([x; y])m} = 0 for all x,y ε I, where n and m are fixed positive integers. We prove that if char(R) > n + 1 or char(R) = 0, then either a = 0 or R is commutative. [ABSTRACT FROM AUTHOR]