### Maja Fošner, Benjamin Marcen, Nejc Širovnik and Joso Vukman

Faculty of Logistics, University of Maribor, Mariborska cesta 7, 3000 Celje, Slovenia
e-mail: maja.fosner@fl.uni-mb.si

Faculty of Logistics, University of Maribor, Mariborska cesta 7, 3000 Celje, Slovenia
e-mail: benjamin.marcen@fl.uni-mb.si

Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, 2000 Maribor, Slovenia
e-mail: nejc.sirovnik@uni-mb.si

Faculty of Mathematics, Natural Sciences and Information Technologies , University of Primorska, Glagoljaška 8, 6000 Koper
and
Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia
e-mail: joso.vukman@uni-mb.si, joso.vukman@gmail.com

Abstract.   In this paper we prove the following result. Let n≥ 1 be some fixed integer and let R be a prime ring with 2n < char(R) ≠ 2. Suppose there exist additive mappings S,T:R → R satisfying the relations

S(x2n)=S(x)x2n-1+xT(x)x2n-2+x2S(x)x2n-3+ ⋯ +x2n-1T(x),

T(x2n)=T(x)x2n-1+xS(x)x2n-2+ x2T(x)x2n-3+ ⋯ +x2n-1S(x)

for all x R. In this case S and T are of the form 2S(x)=D(x)+ζ (x), 2T(x)=D(x)-ζ (x) for all x R, where D:R → R is a derivation and ζ is an additive mapping, which maps R into its extended centroid. Besides, ζ (x2n)=0 for all x R. Functional equations related to bicircular projections are also investigated.

2010 Mathematics Subject Classification.   16R60, 16W10, 39B05.

Key words and phrases.   Derivation, functional identity, bicircular projection.

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DOI: 10.3336/gm.49.2.06

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