Remarks on Certain Identities with Derivations on Semiprime Rings

Let n be a fixed positive integer, let R be a (2n)! -torsion-free semiprime ring, let be an automorphism or an anti-automorphism of R, and let D (1) , D (2) : R -> R be derivations. We prove the following result: If (D (1) (2) (x) + D (2)(x)) (n) a similar to aEuro parts per thousand alpha(x) (n) = 0 holds for all x a R, then D (1) = D (2) = 0. The same is true if R is a 2-torsion free semiprime ring and F(x) A degrees beta(x) = 0 for all x a R, where F(x) = (D (1) (2) (x) + D (2)(x)) a similar to aEuro parts per thousand alpha(x), x aaEuro parts per thousand R, and beta is any automorphism or antiautomorphism on R.