On 2-Skew-Commuting Mappings of Prime Rings

Breşsar proved in [13] that if R is a prime ring of characteristic not 2, C is an extended centroid of R and an additive mapping f : R → R satisfies the functional identity f (x)x sup 2 /sup ? x sup 2 /sup f (x) = 0 ∀x ∈ R, then there exits λ ∈C and an additive mapping ξ : R → C such that f (x)= λx+ξ(x) ∀x ∈ R. The purpose of this paper is to prove that if R is a prime ring with characteristic of not 2 and 3 and if an additive mapping f : R → R satisfies functional identity f (x)x sup 2 /sup + x sup 2 /sup f (x) = 0 ∀x ∈ I, then f (x) = 0 ∀x ∈ I. We will conclude this paper by introducing a conjecture about the solution of an important functional identity.