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Let R be a prime ring of characteristic not 2, U a nonzero square closed Lie ideal of R and F;G;H be the generalized derivations with associated derivations d,δh of R respectively. In the present paper, we study the situations if one the follwoing holds (1) F(u)G(v)±H(uv)ϵ Z(R), (2) F(u)F (v)±H(vu) ϵ Z(R), for all u,v ϵ U, then U ⊆Z(R). [ABSTRACT FROM AUTHOR]

Let R be a prime ring and set [x,y]1 = [x,y] = D xy - yx for all x,y ∊ R and inductively [x,y]k = [[x,y]k-1,y] for k > 1. We apply the theory of generalized polynomial identities with automorphism and skew derivations to obtain the following result: Let R be a prime ring and I a nonzero ideal of R. Suppose that (δ,φ) is a skew derivation of R such that δ([x,y])= [x,] = n for all x;y ∊ I , then R is commutative. [ABSTRACT FROM AUTHOR]