Results on Generalized Derivations in Prime Rings with Involution

Let \(\mathscr {A}\) be a prime ring with the center \(\mathscr {Z}(\mathscr {A})\) and a second-kind involution \(*\) in which generalized derivations fulfil specific algebraic identities. Assume that \(\psi \) and \(\phi \) are generalized derivations associated with \(\xi \) and \(\mu \) on \(\mathscr {A},\) respectively. In this article, we examine the following situation: (i) \(\psi (x)x^*-x\phi (x^*)\in \mathscr {Z}(\mathscr {A}),\) (ii) \(\psi (x)x^*-x^*\phi (x)\in \mathscr {Z}(\mathscr {A}),\) (iii) \(\psi (x)x-x\phi (x^*)\in \mathscr {Z}(\mathscr {A}),\) (iv) \(\psi (x)x-\) \(x^*\phi (x)\in \mathscr {Z}(\mathscr {A}),\) (v) \(\psi (x)x-x^*\phi (x^*)\in \mathscr {Z}(\mathscr {A}),\) \(\forall \) \(x\in \mathscr {A}.\)