# On generalized homoderivations of prime rings

### Abstract

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.

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*Matematychni Studii*,

*60*(1), 12-27. https://doi.org/10.30970/ms.60.1.12-27

Copyright (c) 2023 N. Rehman, E. K. Sogutcu, H. M. Alnoghashi

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Matematychni Studii is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) license.