On generalized homoderivations of prime rings

Let $\mathscr{A}$ be a ring with its center $\mathscr{Z}(\mathscr{A}).$ An additive mapping $\xi\colon \mathscr{A}\to \mathscr{A}$ is called a homoderivation on $\mathscr{A}$ if $\forall\ a,b\in \mathscr{A}\colon\quad \xi(ab)=\xi(a)\xi(b)+\xi(a)b+a\xi(b).$ An additive map $\psi\colon \mathscr{A}\to \mathscr{A}$ is called a generalized homoderivation with associated homoderivation $\xi$ on $\mathscr{A}$ if $\forall\ a,b\in \mathscr{A}\colon\quad\psi(ab)=\psi(a)\psi(b)+\psi(a)b+a\xi(b).$ This study examines whether a prime ring $\mathscr{A}$ with a generalized homoderivation $\psi$ that fulfils specific algebraic identities is commutative. Precisely, we discuss the following identities: $\psi(a)\psi(b)+ab\in \mathscr{Z}(\mathscr{A}),\quad\psi(a)\psi(b)-ab\in \mathscr{Z}(\mathscr{A}),\quad\psi(a)\psi(b)+ab\in \mathscr{Z}(\mathscr{A}),$ $\psi(a)\psi(b)-ab\in \mathscr{Z}(\mathscr{A}),\quad\psi(ab)+ab\in \mathscr{Z}(\mathscr{A}),\quad\psi(ab)-ab\in \mathscr{Z}(\mathscr{A}),$ $\psi(ab)+ba\in \mathscr{Z}(\mathscr{A}),\quad\psi(ab)-ba\in \mathscr{Z}(\mathscr{A})\quad (\forall\ a, b\in \mathscr{A}).$ Furthermore, examples are given to prove that the restrictions imposed on the hypothesis of the various theorems were not superfluous.