A Note on Jordan Left ∗-Centralizers in Rings with Involution

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 be a ∗-closed ideal of R, and let 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 16 Abdul Nadim Khan, Mohammad Shadab Khan and Shakir Ali (T 6= 0), then there exists λ ∈ C such that T = λS (S = λT ). Mathematics Subject Classification: 16N60, 16W10