Generalized skew derivations acting as Jordan homomorphism (Q6630776)

Let \(f:R\longrightarrow R\) an additive map defined on an associative ring \(R\). It is called \textit{Jordan homomorphism} on \(T\subseteq R\) if \(f(x^2)=f(x)^2\) for all \(x\in T\).\N\NMany papers have studied homomorphisms or anti-homomorphisms in prime and semiprime rings satisfying particular conditions, or some additive maps that act as homomorphisms or anti-homomorphisms, to describe the behaviour of the involved maps.\N\NHere the authors consider \(H_1\) and \(H_2\), two generalized skew derivations defined on \(R\), a prime ring with characteristic different from 2, in the case in which it is satisfied the following condition: \[H_1\biggl(H_2(X^2)\biggr)=H_2(X)^2\] for all \(X\in f(R)\), where \(f(R)=\{f(x_1,\ldots,x_n):x_i\in R\}\) is the set of all evaluations of a non-central multilinear polynomial over \(C\) (the extended centroid of \(R\)).\N\NUnder the previous hypothesis, they prove that one of the following holds:\N\begin{itemize}\N\item[1.] \(H_2=0\);\N\item[2.] \(f(R)^2\subseteq Z(R)\) there exist \(\lambda\in C\), \(a,b\in Q_r\) and an invertible element \(q\in Q_r\) such that \(H_1(x)=ax+qxq^{-1}b\), \(H_2(x)=\lambda x\), for all \(x\in R\), with \(a+b=\lambda\).\N\item[3.] \(f(R)^2\subseteq Z(R)\) and there exist \(\lambda\in C\), \(a,b\in Q_r\) and an invertible element \(q\in Q_r\) such that \(H_1(x)=ax+qxq^{-1}b\), \(H_2(x)=\lambda qxq^{-1}\), for all \(x\in R\), with \(a+b=\lambda\).\N\end{itemize}\Nwhere \(Q_r\) is the Martindale quotient ring.