The purpose of this article is to enhance the previous studies regarding the behavior of a quotient ring $\Im /\wp$, where ℘ is a prime ideal in a ring ℑ. In particular, we are going to explore more general scenarios whenever a ring ℑ admits a generalized $(\alpha ,\beta )$-derivation associated with an $(\alpha ,\beta )$-derivation ∂ that satisfies certain criteria involving ℘, where $\alpha$ and $\beta$ are automorphisms on ℑ. Moreover, we provide some examples to demonstrate the importance of the assumptions made in our results.

The current article focuses on studying the behavior of a ring $\Re /\Pi$ when ℜ admits generalized derivations $\Psi$ and $\Omega$ with associated derivations $\varphi$ and $\delta$, respectively. These derivations satisfy specific differential identities involving $\Pi$, where $\Pi$ is a prime ideal of an arbitrary ring ℜ, not necessarily prime or semiprime. Furthermore, we explore some consequences of our findings. To emphasize the necessity of the primeness of $\Pi$ in the hypotheses of our various theorems, we provide a list of examples.

The purpose of this paper is to examine the behavior of a factor ring $R/P$ with a partial range I of a ring R, where P is a prime ideal of R and I is a non-zero ideal of R such that $P⊊I$. In order to accomplish this objective, we will utilize specific algebraic identities that involve P and are related to generalized $(\alpha ,\beta )$-derivations F and G, associated with $(\alpha ,\beta )$-derivations d and g, respectively, where $\alpha$ and $\beta$ are automorphisms of R. Additionally, we will list some important related consequences. Finally, we will provide an illustrative example to emphasize the importance of the hypotheses imposed in our theorems.

In this article, we will investigate the commutativity of the factor ring $\Re/P$, where $P$ is a prime ideal of any ring $\Re$. This investigation will be carried out using generalized $P$-derivations $\mho$ and $\amalg$ associated with $P$-derivations $\chi$ and $\propto$, respectively, that satisfy specific functional identities linking $\Re$ to $P$. Moreover, we will discuss some related results. Finally, to reinforce the importance of our assumption regarding the primeness of $P$, we will provide some examples.

Let $\tau$ and $\sigma :X⟶X$ be automorphisms of an arbitrary associative ring X, and let L be a prime ideal of X. The main objective of this article is to combine the notions of generalized L-derivations and $(\tau ,\sigma )$-L-derivations by introducing and analyzing a novel additive mapping $\Pi :X\to X$ called a generalized $(\tau ,\sigma )$-L-derivation associated with a $(\tau ,\sigma )$-L-derivation $\pi$. Later, we will examine the algebraic properties of a factor ring $X/L$ under the influence of certain algebraic expressions containing this generalized $(\tau ,\sigma )$-L-derivation and lying in a prime ideal L. Through our main findings, we establish certain results under different conditions. It also provides various illustrative examples to show that our primeness hypotheses in various theorems are not exaggerated.

In this article, for a prime ideal $\rho$ of an arbitrary ring ℜ, we study the commutativity of the quotient ring $\Re /\rho$, whenever ℜ admits a generalized reverse derivation $\vartheta$ associated with a reverse derivation ∂ that satisfies certain identities in $\rho$. Additionally, we show that, for some cases, the range of the generalized reverse derivation $\vartheta$ lies in the prime ideal $\rho$. Moreover, we explore several consequences and special cases. Throughout, we provide examples to demonstrate that various restrictions in the assumptions of our results are essential.