On Θ-centralizers of Semiprime Rings (ii)

Centralizers on prime and semiprime rings

The purpose of this paper is to investigate identities satisfied by centralizers on prime and semiprime rings. We prove the following result: Let R be a noncommutative prime ring of characteristic different from two and let S and T be left centralizers on R. Suppose that [S(x), T (x)]S(x) + S(x)[S(x), T (x)] = 0 is fulfilled for all x ∈ R. If S 6= 0 (T 6= 0) then there exists λ from the extende...

Centralizers on semiprime rings

The main result: Let R be a 2-torsion free semiprime ring and let T : R → R be an additive mapping. Suppose that T (xyx) = xT (y)x holds for all x, y ∈ R. In this case T is a centralizer.

An identity related to centralizers in semiprime rings

The purpose of this paper is to prove the following result: Let R be a 2torsion free semiprime ring and let T : R → R be an additive mapping, such that 2T (x) = T (x)x + xT (x) holds for all x ∈ R. In this case T is left and right centralizer.

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 su...

On (θ, θ)-Derivations in Semiprime Rings