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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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/

[1] S. Ali, On generalized left derivations in rings and Banach algebras, Aequat. Math. 81 (2011) 209–226. https://doi.org/10.1007/s00010-011-0070-5

[2] S. Ali and C. Haetinger, Jordan $\alpha$-centralizer in rings and some applications, Bol. Soc. Paran. Mat. 26 (2008) (1–2) 71–80. https://doi.org/10.5269/bspm.v26i1-2.7405

[3] A.Z. Ansari, On identities with additive mappings in rings, Iranian J. Math. Sci. Inform. 15 (1) (2020) 125–133. http://ijmsi.ir/article-1-1051-en.html

[4] A.Z. Ansari and F. Shujat, Additive mappings covering generalized $(\alpha_1, \alpha_2)$-derivations in semiprime rings, Gulf J. Math. 11 (2) (2021) 19–26. https://gjom.org/index.php/gjom/article/view/495

[5] M. Ashraf and S. Ali, On generalized Jordan left derivations in rings, Bull. Korean Math. Soc. 45 (2) (2008) 253–261. https://doi.org/10.4134/BKMS.2008.45.2.253

[6] J.M. Cusack, Jordan derivations in rings, Proc. Amer. Math. Soc. 53 (2) (1975) 321–324. https://doi.org/10.1090/S0002-9939-1975-0399182-5

[7] F. Shujat and A.Z. Ansari, Additive mappings satisfying certain identities of semiprime rings, Preprint in Bulletin of Korean Mathematical Society.

[8] I.N. Herstein, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957) 1104–1110. https://doi.org/10.1090/S0002-9939-1957-0095864-2

[9] C. Lanski, Generalized derivations and n-th power maps in rings, Comm. Algebra 35 (11) (2007) 3660–3672. https://doi.org/10.1080/00927870701511426

[10] E.C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. (1957) 1093–1100. https://doi.org/10.1090/S0002-9939-1957-0095863-0

[11] J. Vukman, On left Jordan derivations on rings and Banach algebras, Aequationes Math. 75 (2008) 260–266. https://doi.org/10.1007/s00010-007-2872-z

[12] S.M.A. Zaidi, M. Ashraf and S. Ali, On Jordan ideals and left $(\theta,\theta)$-derivation in prime rings, Int. J. Math. Math. Sci. 37 (2004) 1957–1965. https://doi.org/10.1155/S0161171204309075

[13] B. Zalar, On centralizers of semiprime rings, Comment. Math. Univ. Carol. 32 (1991) 609–614. http://eudml.org/doc/247321

[14] J. Zhu and C. Xiong, Generalized derivations on rings and mappings of P-preserving kernel into range on Von Neumann algebras, Acta Math. Sinica 41 (1998) 795–800. https://doi.org/10.12386/A19980133