ON CENTRALIZERS ON SOME GAMMA RING

Abstract

Let M be a 2-torsion free  -ring satisfies the conditionx y z=x y z for all x,y,zM and  ,   . In section one ,weprove if M be a completely prime -ring and T:M→M anadditive mapping such that T(a a)=T(a) a (resp., T(a a)=aT(a ))holds for all aM,  .Then T is a left centralizer or Mis commutative (res.,a right centralizer or M is commutative)and so every Jordan centralizer on completely prime  -ring Mis a centralizer .In section two ,we prove this problem but byanother way. In section three we prove that every Jordan leftcentralizer(resp., every Jordan right centralizer) on -ring hasa commutator right non-zero divisor(resp., on -ring has acommutator left non-zero divisor)is a left centralizer(resp., is aright centralizer) and so we prove that every Jordan centralizeron  -ring has a commutator non –zero divisor is a centralizer .