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 D1,D2: R › R be derivations. We prove the following result: If (D2 1(x) + D2(x)) ? ?(x)n = 0 holds for all x ? R, then D1= D2= 0. The same is true if R is a 2-torsion free semiprime ring and F(x) ? ß(x) = 0 for all x ? R, where F(x) = (D1 2(x) + D2(x)) ? ?(x), x ? R, and ß is any automorphism or antiautomorphism on R. © 2015 Springer Science+Business Media New York