# $\mathscr{T}$-Commuting Generalized Derivations on Ideals and Semi-Prime Ideal-II

### Abstract

The study's primary purpose is to investigate the $\mathscr{A}/\mathscr{T}$ structure of a quotient ring, where $\mathscr{A}$ is an arbitrary ring and $\mathscr{T}$ is a semi-prime ideal of $\mathscr{A}$. In more details, we look at the differential identities in a semi-prime ideal of an arbitrary ring using $\mathscr{T}$-commuting generalized derivation. The article proves a number of statements. A characteristic representative of these assertions is, for example, the following Theorem 3: Let $\mathscr{A}$ be a ring with $\mathscr{T}$ a semi-prime ideal and $\mathscr{I}$ an ideal of $\mathscr{A}.$ If $(\lambda, \psi)$ is a non-zero generalized derivation of $\mathscr{A}$ and the derivation satisfies any one of the conditions:\

1)\ $\lambda([a, b])\pm[a, \psi(b)]\in \mathscr{T}$,\ 2) $\lambda(a\circ b)\pm a\circ \psi(b)\in \mathscr{T}$,

$\forall$ $a, b\in \mathscr{I},$ then $\psi$ is $\mathscr{T}$-commuting on $\mathscr{I}.$

Furthermore, examples are provided to demonstrate that the constraints placed on the hypothesis of the various theorems were not unnecessary.

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*Matematychni Studii*,

*57*(1), 98-110. https://doi.org/10.30970/ms.57.1.98-110

Copyright (c) 2022 N. U. Rehman, H. M. Alnoghashi

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