Keywords : Prime


Rajaa C .Shaheen

Journal of Al-Qadisiyah for Computer Science and Mathematics, 2011, Volume 3, Issue 1, Pages 1-6

Let M be a 2-torsion free -ring with involution satisfies the condition x y z=x y z for all x,y,z M and , .an additive mapping *: M→Mis called Involution if and only if (a b)*=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(a a)=T(a) a* (resp., T(a a)=a* T(a ))holds for all a M, .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 .