JORDAN HIGHER CENTRALIZERS ON SEMIPRIME RINGS AND RELATED MAPPINGS

Abstract

Abstract. We prove that every Jordan higher left (right) central-izer on a 2-torsion free semiprime ring is a higher left (right) cen-tralizer which is to generalize the result of Zalar [18]. 1. Introduction and PreliminariesThroughout this note, Rwill represent an associative ring. For a;b2R, let ab babe denoted by [a;b] and let N be the set of all positiveintegers.A derivation (resp. Jordan derivation) is an additive mapping  :R!Rsatisfying (ab) = a(b) + (a)bfor all a;b2R(resp. (a 2 ) =a(a) + (a)afor all a2R). A left (resp. right) centralizer of Risan additive mapping ’: R!Rwhich satis es ’(ab) = ’(a)b(resp.’(ab) = a’(b)) for all a;b2R:If a;x2R, then L x (a) = xais a leftcentralizer and R x (a) = axis a right centralizer.It is obvious that every derivation is a Jordan derivation. But theconverse is in general not true. Herstein [10] proved that the converseis true on 2-torsion free prime rings and latter on, Bresar [5] extendedthis result to 2-torsion free semiprime rings. Bresar, Vukman ([6], [15]),Deng [8] and Ashraf et al. [1] studied Jordan left derivations and leftderivations on prime rings and semiprime rings, which are in a closeconnection with so-called commuting mappings. In [18], Zalar provedthat every Jordan left (right) centralizer on a 2-torsion free semiprimering is a left (right) centralizer.