Remarks on certain identities with derivations on semiprime rings - Fosner, A.; Baydar, N.; Strasek, R. (Український математичний журнал, 2014)
Let n be a fixed positive integer, let R be a (2n)! -torsion-free semiprime ring, let α be an automorphism or an anti-automorphism of R, and let D₁,D₂:R→R be derivations. We prove the following result: If (D²₁(x) + D₂(x))ⁿ ∘ α(x)ⁿ = 0 holds for all xЄR, then D₁=D₂=0. The same is true if R is a 2-torsion free semiprime ring and F(x) ° β(x) = 0 for all x ∈ R, where F(x)=(D²₁(x) + D₂(x)) ∘ α(x), x ∈ R, and β is any automorphism or antiautomorphism on R.