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Saudi Arabia


Let R be a ring with involution ∗. An additive mapping T : R → R is called a left ∗-centralizer (resp. Jordan left ∗-centralizer) if T(xy) = T(x)y∗ (resp. T(x2) = T(x)x∗) holds for all x,y ∈ R, and a reverse left ∗-centralizer if T(xy) = T(y)x∗ holds for all x,y ∈ R. The purpose of this paper is to solve some functional equations involving Jordan left ∗-centralizers on some appropriate subsets of prime and semiprime rings with involution. In particular, we prove the following result: Let R be a 2-torsion free noncommutative semiprime ring with involution, I a ∗-ideal of R, and S,T : R → R be Jordan left ∗-centralizers satisfying the relation (S(x)◦T(x))S(x)−S(x)(S(x)◦ T(x)) = 0 for all x ∈ I. Then [S(x),T(x)] = 0 for all x ∈ I. Moreover, if R is a prime ring and S 6= 0 (T 6= 0), then there exists λ ∈C, the extended centroid of R, such that T = λ S (S = λ T). As an application, Jordan left ∗-centralizers of semiprime rings are characterized.

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