On certain functional equation in prime rings

Abstract

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