Commutativity of $$*$$-Prime Rings with Generalized Derivations on $$*$$-Jordan Ideals

Abstract

AbstractThe purpose of this paper is to find commutativity conditions in \(*\)-prime rings with generalized derivations, where \('*'\) is involution of the second kind. More specifically, it is shown that if a 2-torsion free \(*\)-prime ring \(\mathscr {R}\) with involution of the second kind satisfies any of the following assertions: (i) \([\mathscr {F}(\alpha ),\alpha ^{*}]\in Z(\mathscr {R}),\) (ii) \(\mathscr {F}(\alpha )\circ (\alpha ^{*}) \in Z(\mathscr {R}),\) (iii) \(\mathscr {F}(\alpha )\circ \mathfrak {d}(\alpha ^{*})\pm \alpha \circ \alpha ^{*} \in Z(\mathscr {R})\) and (iv) \([\mathscr {F}(\alpha ),\mathfrak {d}(\alpha ^{*})]\pm \alpha \circ \alpha ^{*}\in Z(\mathscr {R}),\) where \(\mathscr {F}\) is a generalized derivation associated with a derivation \( \mathfrak {d}\) such that \( \mathfrak {d}\) is commuting with \(*\) and \(\alpha \) varies over a nonzero \(*\)-Jordan ideal of \(\mathscr {R}\), then \(\mathscr {R}\) is commutative.Keywords\(*\)-prime ringsGeneralized derivationsInvolution