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

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.