Symmetric Bi-Centralizers on Semiprime Rings

Let R be a 2-torsion free semiprime ring and F: R×R→ R be a symmetric Bi-additive mapping. The purpose of this paper is to prove the following results: (1) If F(x2, y)= F(x, y)x fulfilled for all x,y ∈R, then F is a symmetric left Bi-centralizer. (2) If F(xωx, y) = x F(ω, y) x fulfilled for all x,y, ω∈R, then F is a symmetric Bi-centralizer (3) Let R be a 2-torsion free semiprime ring with an identity element and F:R×R→ R be a symmetric Bi-additive mapping such that F(x3, y) = x F(x, y)x fulfilled for all x,y ∈R, then F is a symmetric Bi-centralizer.