LetH be a real or complex Hilbert space with dim(H) > 1, B(H) be algebra of all bounded linear operators on H and A(H) ? B(H) be a standard operator algebra on H. If D : A(H) ? B(H) is a linear mapping satisfying D(An+1) = Pn i=0 AiD(A)(A*)n?i for all A ? A(H), then D is a Jordan *-derivation on A(H). Later, we discuss some algebraic identities on semiprime rings.

In the present note we anaugrate the idea of symmetric generalized bi-semiderivation on rings and prove some classical commutativity results for generalized bi-semiderivation. Moreover, our main objective is to extend the main theorem in \cite{VJ} for biderivation to the case of symmetric generalized bi-semiderivation on prime ring.

The goal of this study is to bring out the following conclusion. Let $ R $ be a noncommutative prime ring with $ 2(m+n)! $ torsion freeness and let $ m $ and $ n $ be fixed, non-negative integers and $ d, g $ be Jordan derivations on $ R $. If $ x^{m+n}d(x)+x^mg(x)x^n\in Z(R) $ or $ d(x)x^{m+n}+x^mg(x)x^n\in Z(R) $ or $ x^{n}d(x)x^{m}+x^mg(x)x^n\in Z(R) $ then $ d = g = 0 $ follows for every $ x\in R $.

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.