Jordan $\varphi$-centralizers on Semiprime and Involution Rings

Abstract

The intention of the current investigation is to demonstrate that if an additive mapping $\mathcal{H}:R\to R$ fulfills any one of the following identities:
\begin{enumerate}
\item [$(i)$] $3\mathcal{H}(r^{3p})=\mathcal{H}(r^p)\varphi(r^{2p})+\varphi(r^p) \mathcal{H}(r^{p})\varphi(r^p)+ \varphi(r^{2p})\mathcal{H}(r^p)$
\item [$(ii)$] $2\mathcal{H}(r^{2p})=\mathcal{H}(r^p)\varphi(r^{p})+\varphi(r^p)\mathcal{H}(r^{p})$
\item [$(ii)$] $\mathcal{H}(r^{3p})=\varphi(r^p) \mathcal{H}(r^{p}) \varphi(r^p)$ for all $r\in R$,
\end{enumerate}
then $\mathcal{H}$ is a $\varphi$-centralizer on $R$, where $R$ is any suitable, torsion-free semiprime ring and $p$ is a fixed integer greater than or equal to 1. As a result of the primary theorems, involution $I_v$ related observations are also provided.  We will also consider   criticism and discussion alongside the proofs of theorems. Suitable  examples given in favor of justification.