Generalized derivations with annihilator conditions in prime rings (Q5918739)

In this paper, the authors mainly consider an annihilating condition on power values of generalized derivations and obtain the following results. Theorem 1. Let \(R\) be a prime ring with symmetric Martindale quotient ring \(Q\) and extended centroid \(C\). Suppose that \(I\) is a nonzero ideal of \(R\), \(F\) is a generalized derivation of \(R\), \(m\) and \(n\) are two fixed positive integers, and \(a\in R\) is a nonzero element. If \(a((F(x\circ y))^m-(x\circ y)^n)=0\) for all \(x, y\in I\), then one of the following holds: \begin{itemize} \item [(a)] \(R\) is commutative. \item [(b)] \(m=n=1\) and there exists \(b\in Q\) such that \(F(x)=bx\) for all \(x\in R\) with \(ab=a\). \item [(c)] There exists \(b\in C\) such that \(F(x)=bx\) for all \(x\in R\) with \(b^m=1\) and \((x\circ y)^m=(x\circ y)^n\) for all \(x, y\in R\). \item [(d)] \(R\subseteq M_2(C)\), the ring of \(2\times 2\) matrices over \(C\); \(n=1\) and \(m\geq 2\) such that \(\alpha ^m=\alpha \) for all \(\alpha \in C\) and there exists \(b\in Q\) such that \(F(x)=bx\) for all \(x\in R\) with \(ab=a\). \item [(e)] \(R\subseteq M_2(C)\) and char\((R)=2\). \end{itemize} Theorem 2. Let \(R\) be a prime ring of char\((R)\not =2\), \(Q\) and \(C\) be its symmetric Martindale quotient ring and extended centroid, respectively. Suppose that \(I\subseteq R\) is a nonzero ideal, \(F\) is a generalized derivation of \(R\) with associated derivation \(d\), \(m\) and \(n\) are two fixed positive integers, and \(a\) is a nonzero element of \(R\). Assume that \(a((F(x\circ y))^m-(x\circ y)^n)\in Z(R)\) for all \(x, y\in I\). If there exist \(x_0, y_0\in I\) such that \(a((F(x_0\circ y_0))^m-(x_0\circ y_0) ^n)\not =0\), then either \begin{itemize} \item [(a)] there exists a field \(E\) such that \(R\subseteq M_2(E)\); or \item [(b)] \(a\in Z(R)\), \((x\circ y)^m-(x\circ y)^n\in Z(R)\) for any \(x, y\in R\) and there exist \(b\in Z(R)\) such that \(b^m=1\). \end{itemize}