Geodesic
Additive mappings satisfying algebraic identities in semiprime rings
Discussiones Mathematicae. General Algebra and Applications, Tome 43 (2023) no. 2, pp. 327-337

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Let R be a k-torsion free semiprime ring. Suppose that F, d : R→ R be two additive mappings which satisfy the algebraic identity F(x^2n)=F(x^n) α(x^n)+ β(x^n) d(x^n) for all x∈ R, where α and β are automorphisms on R. Then F is a generalized (α,β)-derivation with associated (α,β)-derivation d on R, where k∈{2,n,2n-1}. On the other hand, it is proved that f is a generalized Jordan left (α, β)-derivation associated with Jordan left (α, β)-derivation δ on R if they satisfy the algebraic identity f(x^2n)=α(x^n) f(x^n)+ β(x^n)δ(x^n) for all x∈ R together with some restrictions on R.
Keywords: semiprime rings, generalized $(\alpha, \beta)$-derivation, generalized left $(\alpha, \beta)$-derivation and additive mappings 
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     title = {Additive mappings satisfying algebraic identities in semiprime rings},
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Ansari, Abu Zaid. Additive mappings satisfying algebraic identities in semiprime rings. Discussiones Mathematicae. General Algebra and Applications, Tome 43 (2023) no. 2, pp. 327-337. http://geodesic.mathdoc.fr/item/DMGAA_2023_43_2_a10/