Glasnik Matematicki, Vol. 49, No. 2 (2014), 313-331.

ON SOME FUNCTIONAL EQUATIONS RELATED TO DERIVATIONS AND BICIRCULAR PROJECTIONS IN RINGS

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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