<b>Jordan \alpha-centralizers in rings and some applications</b> - doi: 10.5269/bspm.v26i1-2.7405

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.