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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]