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 $R$.