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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]

Let N be a prime near-ring and σ, τ be automorphisms of N. Suppose that F is a nonzero left generalized (σ, σ) -derivation of N associated with a nonzero (σ, σ)-derivation d. In the present paper, our objective is to prove that N is a commutative ring if one of the following conditions holds: (i) F([x, y]) = 0, (ii) F(x o y) = 0, (iii) F([x,y]) = σ([x, y]), (iv) F([x,y]) = σ(-xy + yx) for all x,y ∈ N. Further, suppose that F is a left generalized (σ, τ)-derivation of N associated with a nonzero (σ, τ)-derivation d. If F acts as a homomorphism or an anti-homomorphism on N, then F = τ. Finally, an example is given to demonstrate that the hypothesis of primeness in the above results is essential. [ABSTRACT FROM AUTHOR]