Remarks on Certain Identities with Derivations on Semiprime Rings - Baydar, N.; Fošner, A.; Strašek, R. (Ukrains’kyi Matematychnyi Zhurnal, 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 D_1 , D_2 : R → R be derivations. We prove the following result: If (D_1^2 (x) + D_2(x))^n ∘ α(x)^n = 0 holds for all x Є R, then D_1 = D_2 = 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) = (D_1^2 (x) + D_2(x)) ∘ α(x), x ∈ R, and β is any automorphism or antiautomorphism on R.