On certain functional equation in prime rings

Abstract

The purpose of this paper is to prove the following result. Let RR be prime ring of characteristic different from two and three, and let F:R→RF:R\to R be an additive mapping satisfying the relation F(x3)=F(x2)x−xF(x)x+xF(x2)F\left({x}^{3})=F\left({x}^{2})x-xF\left(x)x+xF\left({x}^{2}) for all x∈Rx\in R. In this case, FF is of the form 4F(x)=D(x)+qx+xq4F\left(x)=D\left(x)+qx+xq for all x∈Rx\in R, where D:R→RD:R\to R is a derivation, and qq is some fixed element from the symmetric Martindale ring of quotients of RR.

Keywords

• prime ring
• derivation
• jordan derivation
• two-sided centralizer
• functional equation
• 16r60
• 16w25
• 39b05