Abstract
Let $ R$ be a ring, and $\alpha$ be an endomorphism of $ R$ . An additive mapping $ H : R \rightarrow R$ is called a left $ \alpha$-centralizer (resp. Jordan left $\alpha$-centralizer) if $ H ( xy ) = H ( x )\alpha(y)$ for all $ x; y \in R$ (resp. $ H ( x^ 2 ) = H ( x )\alpha ( x )$ for all $ x \in R$ ). The purpose of this paper is to prove two results concerning Jordan $\alpha$-centralizers and one result related to generalized Jordan $(\alpha ; \beta )$-derivations. The result which we refer state as follows: Let $ R$ be a 2-torsion-free semiprime ring, and $\alpha$ be an automorphism of $ R$ . If $ H : R \rightarrow R$ is an additive mapping such that $ H ( x^ 2 ) = H ( x )\alpha ( x )$ for every $ x \in $ R or $ H ( xyx ) = H ( x ) \alpha( yx )$ for all $ x; y \in R$ , then $ H$ is a left $\alpha$-centralizer on $ R$ . Secondly, this result is used to prove that every generalized Jordan $(\alpha ; \beta )$-derivation on a 2-torsion-free semiprime ring is a generalized $(\alpha ;\beta )$-derivation. Finally, some examples are given to demonstrate that the restrictions imposed on the hypothesis of the various theorems were not superfluous.
Highlights
It is well-known that Jordan derivations can be defined as d(x ◦ y) = d(x) ◦ y + x ◦ d(y), for all x, y ∈ R
In [17], it was shown that a Jordan left centralizer of a semiprime ring is a left centralizer, and each Jordan centralizer is a centralizer
Vukman [13] proved that every generalized Jordan derivation on a 2-torsion-free semiprime ring is a generalized derivation
Summary
Haetinger proved that in a 2-torsionfree semiprime ring R, for an endomorphism θ of R and for an additive mapping T :R → R such that T (xyx) = θ(x)T (y)θ(x) holds for all x, y ∈ R, T is a θ-centralizer of R. Molnar [12] proved that if R is a 2-torsion-free prime ring and T : R → R is an additive function such that T (xyx) = T (x)yx for all x, y ∈ R, T is a left (right) centralizer. Vukman [13] proved that every generalized Jordan derivation on a 2-torsion-free semiprime ring is a generalized derivation.
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