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.
Digital Object Identifier (DOI)
Alahamdi, Adel; Alhazmi, Husain; Ali, Shakir; and Nadim Khan, Abdul
"A characterization of Jordan left ∗-centralizers in rings with involution,"
Applied Mathematics & Information Sciences: Vol. 11
, Article 47.
Available at: https://dc.naturalspublishing.com/amis/vol11/iss2/47