Abstract
The purpose of this paper is to prove the following result: Let R be a 2-torsion free ring and T: R?R an additive mapping such that T is left (right) Jordan ?-centralizers on R. Then T is a left (right) ?-centralizer of R, if one of the following conditions hold (i) R is a semiprime ring has a commutator which is not a zero divisor . (ii) R is a non commutative prime ring . (iii) R is a commutative semiprime ring, where ? be surjective endomorphism of R . It is also proved that if T(x?y)=T(x)??(y)=?(x)?T(y) for all x, y ? R and ?-centralizers of R coincide under same condition and ?(Z(R)) = Z(R) .
Highlights
Throughout this paper, R will represent an associative ring with the center Z
If R is a ring with involution *, every additive mapping E: R R which satisfies E(x2) = E(x)x* + xE(x) for all x R is called Jordan *derivation
BreŠar and Zalar obtained a representation of Jordan *-derivations in terms of left and right centralizers on the algebra of compact operators on a Hilbert space
Summary
Throughout this paper, R will represent an associative ring with the center Z. A left (right) -centralizer of R is an additive mapping T: R R which satisfies T(xy) = T(x) (y) (T(xy) = (x)T(y)) for all x, y R. A centralizer of R is an additive mapping which is both left and right centralizer.
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