On skew-commuting generalized skew derivations in prime and semiprime rings (Q6580163)

Let \(R\) be a noncommutative prime algebra of characteristic different from \(2\), with right Martindale quotient ring \(Q_r\) and extended centroid \(C\). Let \(m,n\geq 1\) be fixed integers and \(F\) be a nonzero generalized skew derivation of \(R\). In this paper, the authors study the set \(S=\{ F(x^m)x^n+x^nF(x^m)\mid x\in R\}\) and prove that its left annihilator in \(R\) is identically zero. Using the above result, they investigate the identity \(F(x)x^n+x^nF(x)=0\) for semiprime algebras. More precisely, Suppose that \(R\) is a noncommutative semiprime algbera of characteristic different from \(2\), with right Martindale quotient ring \(Q_r\) and extended centroid \(C\). Let \(n\geq 1\) be a fixed integer and \(F\) a nonzero generalized skew derivation of \(R\). If \(F(x)x^n+x^nF(x)=0\) holds true for all \(x\in R\), then \(R\) contains a nonzero central ideal. The two main results obtained in this article under review should be useful and helpful for all non-commutative algebraists and functional analysts.