Symmetric bi-centralizers on semiprime rings

Abstract EN

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(x x, y) = x F( , y) x fulfilled for all x,y, 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