Power values of generalized skew derivations with annihilator conditions on Lie ideals (Q826564)

Let \(R\) be a ring and let \(M\) be a simple right \(R\)-module. Now let \(m\) be a non-zero member of \(M\) such that \(mR=M\). Hence, there exists a non-zero element \(n\) of \(M\) such that there is an element of \(R\) that induces and endomorphism of \(M\) transforming \(m\) to \(n\). This condition motivates the natural question which asked whether this can be generalized to arbitrary finite-tuples of element. Furthermore, what are the necessary and sufficient conditions on the tuples \((x_1, x_2,\dots, x_n)\) and \((y_1, y_2,\dots, y_n)\) separately, so there exists an element of \(R\) with the property that \(x_ir=y_i\) for all \(i \in \{1,2,\dots,n\}\) where \(r \in R\). A well-known theorem which is called the Jacobson density theorem gives a positive answer on tuples affirmatively which provided that the \(x_i\) are linearly independent over \(D\), where \(D\) is a division ring. In this paper, the authors use the extended Jacobson density theorem to characterize the forms of generalized skew derivations \(\delta\) and \(\varrho\) of a prime ring \(R\) with maximal right ring of questions \(Q\), extended centroid \(C\), and non-commutative Lie ideal \(L\) in case of the following equation \(p(\delta(u)^{l_1}\varrho{u}^{l_2}\delta{u}^{l_3}\varrho{u}^{l_4}\dots\varrho(u)^{l_k})^n=0,\) for all \(u \in L,\) where \(0\neq p \in R, l_1, l_2,\dots,l_k\) are fixed nonnegative integers with \(l_1 \neq 0\) and \(n\) is a positive fixed integer.