Annihilators of power values of generalized skew derivations on Lie ideals in prime rings (Q6639844)

A ring \(R\) is defined as \textit{prime} if for any \(a, b \in R\), the condition \(aRb = \{0\}\) implies either \(a = 0\) or \(b = 0\). The \textit{center} of \(R\), denoted \(Z(R)\), comprises elements that commute with every element of \(R\). A Lie ideal \(L\) of \(R\) satisfies \([l, r] \in L\) for all \(l \in L\) and \(r \in R\), where \([x, y] = xy - yx\) denotes the commutator.\N\NA \textit{skew derivation} \(d\) of \(R\) is an additive map satisfying \(d(xy) = d(x)y + \alpha(x)d(y)\) for all \(x, y \in R\), where \(\alpha\) is an automorphism of \(R\). Generalized skew derivations extend this concept: an additive map \(F\) satisfies \(F(xy) = F(x)y + \alpha(x)d(y)\) for all \(x, y \in R\), where \(d\) is a skew derivation.\N\NThe article explores the structural properties of prime rings, particularly focusing on the behavior of generalized skew derivations on their Lie ideals. The main theorem (Theorem 1.1) provides a significant result that relates the centrality of Lie ideals to specific annihilator conditions involving generalized skew derivations. This study builds upon a rich background of algebraic research, including the works of Posner, Herstein, and others, and addresses open questions about annihilator conditions and polynomial identities in prime rings. The main result of this paper is described in Theorem 1.1 as follows.\N\NTheorem 1.1: Let \(R\) be a prime ring of characteristic different from 2, \(n \geq 1\) a fixed integer, \(C\) the extended centroid of \(R\), \(F\) a generalized skew derivation of \(R\), and \(L\) a Lie ideal of \(R\). If there exists \(0 \neq a \in R\) such that \(a(F(xy) - yx)^n = 0\) for all \(x, y \in L\), then \(L\) is central, unless \(R\) satisfies the standard polynomial identity \(s_4(x_1, \ldots, x_4)\).\N\NThis theorem identifies the necessary and sufficient conditions under which a Lie ideal \(L\) of a prime ring \(R\) becomes central, thereby enriching the understanding of the interaction between generalized skew derivations and the ring structure.\N\NThe proof is divided into two cases based on whether the skew derivation \(d\) is \textit{inner} or \textit{outer}:\N\begin{itemize}\N\item Inner Case: Here, \(d(x) = cx + \alpha(x)c\) for some \(c \in R\). The proof motivates Proposition 3.3, which relates the annihilator condition to the centrality of \(L\) using the structure of Martindale quotient rings.\N\item Outer Case: The analysis involves polynomial identities and annihilator conditions, showing that non-centrality of \(L\) leads to a contradiction unless \(R\) satisfies \(s_4\).\N\end{itemize}\N\NThe result of this research also motivates some interesting research topics for further investigation:\N\N\begin{itemize}\N\item[1.] Theorem 1.1 provides sufficient conditions for centrality but does not fully characterize all cases where \(L\) might fail to be central. Further exploration is an interesting topic.\N\item[2.] The results are specific to prime rings. Extending these findings to semiprime or more general classes of rings would be challenging.\N\item[3.] The restriction to characteristic different from 2 is essential in the proof. Investigating whether similar results hold for rings of positive characteristic (e.g., \(p \geq 3\)) could uncover new insights.\N\end{itemize}