On -Centralizers of Prime and Semiprime Rings

Abstract

The purpose of this paper is to prove the following result : Let R be a 2-torsion free ring and T : RR an additive mapping such that 2T(x2) = T(x)(x) + (x)T(x) holds for all x  R. . In this case T is left and right -centralizer , if one of the following statements hold (i) R semiprime ring has a commutator which is not a zero divisor . (ii) R is a non commutative prime ring . (iii) R is a commutative semiprime ring , where  be surjective endomorphism of