Certain additive mappings on semiprime rings and their characterization

Abstract

The objective of this article is to show that an additive mapping $\mathcal{H}:\mathcal{A}\to \mathcal{A}$ is a $\phi$-centralizer on $\mathcal{A}$ if it satisfies one of the following identities:
\begin{enumerate}
\item [$(i)$] $\mathcal{H}(a_1^{p}a_2^{p}+a_2^{p}a_1^{p})=\mathcal{H}(a_1^p)\phi(a_2^{p})+\phi(a_2^p)\mathcal{H}(a_1^{p})$
\item [$(ii)$] $2\mathcal{H}(a_1^{p}a_2^{p})=\mathcal{H}(a_1^p)\phi(a_2^{p})+\phi(a_2^p)\mathcal{H}(a_1^{p})$
\end{enumerate}
for all $a_1,a_2\in \mathcal{A}$, where $p\geq 1$ is a fixed integer, $\phi$ is a surjective endomorphism on a $p!$-torsion free semiprime ring $\mathcal{A}$. Some extensions of these results are also presented in the setting of ring with involution $``\star"$. Furthermore, we also give the verity of examples that illustrate and enrich the subject matter.