Let n be a fixed positive integer, let R be a (2n)! -torsion-free semiprime ring, let \( \alpha \) be an automorphism or an anti-automorphism of R, and let D1, D2 : R → R be derivations. We prove the following result: If (D12(x) + D2(x))n ∘ α(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) ° β(x) = 0 for all x ∈ R, where F(x) = (D12(x) + D2(x)) ∘ α(x), x ∈ R, and β is any automorphism or antiautomorphism on R.
Positive Integer Additive Mapping Prime Ring Associative Ring Semiprime Ring
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