Prime Rings with Generalized Derivations and Power Values on Lie Ideals

Abstract

AbstractSuppose \(\mathscr {R}\) is a prime ring that is non-commutative in structure and characteristic of \(\mathscr {R}\) is a positive integer apart from 2 and \(M=(-2)^{k-1}-1\), where k is any odd positive integers greater than one. Let the Utumi ring of quotients be denoted by \(\mathscr {Q}\), the extended centroid of \(\mathscr {R}\) by \(\mathscr {C}\). Consider \(\mathscr {L}\) to be Lie ideal of \(\mathscr {R}\) non-central in nature and \(\mathscr {T}\) be a non-zero generalized derivation of \(\mathscr {R}\). If \([\mathscr {T}(u^s), u^t]^m=[\mathscr {T}(u), u]\), for every \(u \in \mathscr {L}\), where m, s and t be the fixed positive integers such that \(m >1,~~s \ge 1\) and \(t \ge 1\), then one of the following situations prevails: (i): The standard identity \(s_4(x_1, \ldots , x_4)\) is satisfied by \(\mathscr {R}\) and there exists \(a \in \mathscr {Q}\) and \(\beta \in \mathscr {C}\) such that \(\mathscr {T}(x)= \beta x+ax+xa\), for every \(x \in \mathscr {R}\). (ii): there exists certain \(\theta \in \mathscr {C}\) such that \(\mathscr {T}(x)=\theta x\), for every \(x \in \mathscr {R}\). KeywordsPrime ringsGeneralized derivationMaximal right ring of quotientsGeneralized polynomial identity (GPI)Polynomial identity (PI)