The purpose of this paper is to study the commutativity of a prime ring ℛ with anti-automorphism ψ and generalized derivation ( ℱ , ξ ) satisfying certain algebraic identities. Furthermore, we describe the structure of any additive map which is ψ-centralizing on a 2-torsion free prime ring with anti-automorphism ψ. We also provide examples to show that various conditions imposed on the hypotheses of our results are essential.

Let A be any arbitrary associative ring, P a semiprime ideal, and J a nonzero ideal of A . In this study, using multiplicative (generalized)-derivations, we explore the behavior of semiprime ideals that satisfy certain algebraic identities. Moreover, examples are provided to demonstrate that the restrictions imposed on the hypotheses of the various theorems are necessary.

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.

Let A be a unital *-algebra containing a non-trivial projection. We prove that if a map ?: A ? A such that ?(S ? T ? U) = ?(S) ? T ? U + S ? ?(T) ? U + S ? T ? ?(U) for all S, T, U ? A, then ? is an additive *-derivation.