Abstract
Let M be a 2-torsion free -ring satisfies the condition
x y z=x y z for all x,y,zM and , . In section one ,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 aM, .Then T is a left centralizer or M
is commutative (res.,a right centralizer or M is commutative)
and so every Jordan centralizer on completely prime -ring M
is a centralizer .In section two ,we prove this problem but by
another way. In section three 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 a left centralizer(resp., is a
right centralizer) and so we prove that every Jordan centralizer
on -ring has a commutator non –zero divisor is a centralizer .
x y z=x y z for all x,y,zM and , . In section one ,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 aM, .Then T is a left centralizer or M
is commutative (res.,a right centralizer or M is commutative)
and so every Jordan centralizer on completely prime -ring M
is a centralizer .In section two ,we prove this problem but by
another way. In section three 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 a left centralizer(resp., is a
right centralizer) and so we prove that every Jordan centralizer
on -ring has a commutator non –zero divisor is a centralizer .