On Jordan*- Centralizers On Gamma Rings With Involution

Abstract

    Let M be a 2-torsion free-ring with involution satisfies the condition xyz=xyz for all x,y,zM and ,.an additive mapping *: M→Mis called Involution if and only if (ab)*=b* a*and (a*)*=a . In section one of this paper ,we prove if M be a completely prime-ring and T:M→M an additive mapping such that T(aa)=T(a)a* (resp., T(aa)=a* T(a ))holds for all aM,.Then T is an anti- left *centralizer or M is commutative (res.,an anti- right* centralizer  or M is commutative) and so every Jordan* centralizer on completely prime-ring M is an anti- *centralizer or M is commutative. In section two we prove that every Jordan* left centralizer (resp., every Jordan* right centralizer) on-ring has a commutator right non-zero divisor(resp., on-ring has a commutator left non-zero divisor)is an anti- left *centralizer(resp., is an anti- right *centralizer) and so we prove that every Jordan* centralizer on -ring has a commutator non –zero divisor is an anti-* centralizer .