On Ideals and Behavior of Quotient Rings via Generalized (α,β)-Derivations

Abstract

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.

1. Introduction

It is important to begin by reviewing some common knowledge regarding rings. The symbol $\mathcal{Z}\left(R\right)$ is using to denote the center of an associative ring R. For all $\mathsf{ı},\mathsf{ȷ}\in R$, the identities $[\mathsf{ı},\mathsf{ȷ}]=\mathsf{ı}\mathsf{ȷ}-\mathsf{ȷ}\mathsf{ı}$ and $\mathsf{ı}\circ \mathsf{ȷ}=\mathsf{ı}\mathsf{ȷ}+\mathsf{ȷ}\mathsf{ı}$ are known as Lie and Jordan products, respectively.

If $\mathsf{ı}R\mathsf{ȷ}=0$ for all $\mathsf{ı},\mathsf{ȷ}\in R$, where at least one of ı or ȷ must be zero, then R is referred to as a prime ring. If $\mathsf{ı}R\mathsf{ı}=0$ for all $\mathsf{ı}\in R$, where ı must be zero, then R is referred to as a semiprime ring. Let $\mathsf{ı}R\mathsf{ȷ}\subseteq P$ for all $\mathsf{ı},\mathsf{ȷ}\in R$. Then, a proper ideal P of R is considered prime when at least one of ı or ȷ is in P. Therefore, R is considered a prime ring if and only if $\left\{0\right\}$ is a prime ideal of R. An integral domain is a commutative domain with a multiplicative identity, while a ring without non-zero divisors is referred to as a domain. It is important to note that every integral domain is a prime ring, but the converse is not always true. Therefore, the concept of a prime ring is an extension of an integral domain.

The study of derivations on rings aims to analyze the behavior of a ring that satisfies certain algebraic identities. Here, we discuss some types of derivations related to our studies. An ordinary derivation, denoted as d, is an additive mapping from R to itself that satisfies the equation $d(\mathsf{ı}\mathsf{ȷ})=d(\mathsf{ı})\mathsf{ȷ}+\mathsf{ı}d(\mathsf{ȷ})$ for all $\mathsf{ı},\mathsf{ȷ}$ in R. This definition forms the basis for other types of derivations. The concept of an ordinary derivation has been extended to a generalized derivation, denoted as F, which is also an additive mapping from R to itself equipped with the equation $F(\mathsf{ı}\mathsf{ȷ})=F(\mathsf{ı})\mathsf{ȷ}+\mathsf{ı}d(\mathsf{ȷ})$ for all $\mathsf{ı},\mathsf{ȷ}$ in R. Let $\alpha$ and $\beta$ be automorphisms of R. An additive mapping d from R to itself satisfies the equation $d(\mathsf{ı}\mathsf{ȷ})=d(\mathsf{ı})\alpha (\mathsf{ȷ})+\beta (\mathsf{ı})d(\mathsf{ȷ})$ for all $\mathsf{ı},\mathsf{ȷ}$ in R, referred to as an $(\alpha ,\beta )$-derivation. For a fixed $\rho$, the map $d\rho :R⟶R$ defined by $d\rho (\mathsf{ı})={[\rho ,\mathsf{ı}]}_{\alpha ,\beta }$ for all $\mathsf{ı}\in R$ is an $(\alpha ,\beta )$-derivation, known as an $(\alpha ,\beta )$-inner derivation. An additive mapping F from R to itself satisfies the equation $F(\mathsf{ı}\mathsf{ȷ})=F(\mathsf{ı})\alpha (\mathsf{ȷ})+\beta (\mathsf{ı})d(\mathsf{ȷ})$ for each element $\mathsf{ı},\mathsf{ȷ}$ in R, referred to as the concept of a generalized $(\alpha ,\beta )$—derivation associated with d. The map $\mathsf{ı}⟶\rho \alpha (\mathsf{ı})+\beta (\mathsf{ı})\tau$, where $\rho ,\tau \in R$, is an example of a generalized $(\alpha ,\beta )$—derivation, known as a generalized $(\alpha ,\beta )$-inner derivation. Clearly, every generalized $(I{d}_R,I{d}_R)$-derivation of R is a generalized derivation of R, where $I{d}_R$ represents an identity map of R. If $d=0$, then $F(\mathsf{ı}\mathsf{ȷ})=F(\mathsf{ı})\alpha (\mathsf{ȷ})$ for all $\mathsf{ı},\mathsf{ȷ}$ in R, referred to as an $\alpha$-left multiplier mapping of R. Thus, a generalized $(\alpha ,\beta )$-derivation covers both the concepts of $(\alpha ,\beta )$-derivation and the $\alpha$-left multiplier mapping of R.

Based on these concepts, several researchers have published important articles on various types of rings, such as prime and semiprime rings, or appropriate subsets of them. Interested readers can find more details in the references [1,2,3,4].

In 1998, Creedon [5] generalized Posner’s first theorem (Theorem 1 in [6]). He proved that if $d_1$ and $d_2$ are derivations on a ring R such that the product $d_1{d}_2(\mathsf{ı})\in P$ for all $\mathsf{ı}\in R$, where the characteristic of $R/P$ is not two, then one of the derivations must belong to P, where P is a prime ideal of R. Almahdi et al. [7] discussed the commutativity of a factor ring $R/P$ without imposing any constraints on a ring R. They showed that either $R/P$ is a commutative integral domain or d maps R to a prime ideal P, where R is equipped with a derivation that satisfies the identity $\left[d\right(\mathsf{ı}),\mathsf{ı}],\mathsf{ȷ}]\in P$ for all $\mathsf{ı},\mathsf{ȷ}$ in R. It is evident that, by setting $P=\left\{0\right\}$, Posner’s second theorem can be directly derived. Afterwards, several researchers further investigated these findings. For instance, references [8,9,10,11] delved deeper into these studies. In 2022, Oukhtite et al. [12] examined the commutativity of a factor ring $R/P$ using endomorphisms instead of derivations. They hypothesized that R satisfies certain identities involving a prime ideal P by employing a partial range whose elements belong to a non-zero ideal I of R provided that $P⊊I$. A year later, Bouchannafa et al. [13] studied the commutativity of a factor ring $R/P$ where R is equipped with a generalized derivation. They conducted this study assuming that identities belong to the center of a factor ring $\mathcal{Z}(R/P)$, where P is a prime ideal of R, and the range of elements in these identities belongs to a non-zero ideal I of R such that $P⊊I$. In a recent study, Bera et al. [2] examined the effects of two generalized $(\alpha ,\beta )$-derivations on the behavior of prime and semiprime rings that satisfy certain algebraic identities involving a left ideal I of R, where $\alpha$ and $\beta$ are automorphisms of R. They further expanded their findings to scenarios where a left ideal encompasses the whole ring. In 2024, Alsowait et al. [14] explored the commutativity of a factor ring $R/P$, where R admits a generalized $(\alpha ,\beta )$-derivation associated with an $(\alpha ,\beta )$-derivation that satisfies certain identities involving a prime ideal P. They conducted their study assuming that $\alpha$ and $\beta$ are automorphisms of R, without imposing any restrictions on R being a prime or a semiprime ring.

Motivated by these findings, this article aims to further investigate the behavior of a factor ring $R/P$ by imposing two automorphisms, $\alpha$ and $\beta$, on any ring R that satisfies differential identities involving a prime ideal P. To achieve our goal, we utilize generalized $(\alpha ,\beta )$-derivations F and G associated with $(\alpha ,\beta )$-derivations d and g, respectively. The influence of the imposed identities on the behavior of $R/P$ is restricted in a partial range of elements that belong to a non-trivial ideal I of R provided that $P⊊\alpha \left(I\right)$ and $P⊊\beta \left(I\right)$. Furthermore, several important consequences and relevant special cases are discussed. Finally, an illustrative example has been included to emphasize the significance and importance of the various assumptions in our theorems.

2. Preliminaries

In this section, we list some useful facts and lemmas that facilitate the proofs of our findings. We begin with the following facts, which we apply freely throughout our discussion without explicitly mentioning them:

Fact 1.

Let $\alpha :R⟶R$ be an epimorphism of a ring R. If I is a non-zero ideal of R, then so is $\alpha \left(I\right)$.

Fact 2.

Let α and β be epimorphisms of R. If $F:R⟶R$ is a generalized $(\alpha ,\beta )$-derivation associated with an $(\alpha ,\beta )$-derivation d, then $F\pm \alpha$ is also.

Remark 1.

Let I be a non-zero ideal of any ring R, and let P be a prime ideal of R such that $P⊊\alpha \left(I\right)$ (resp. $P⊊\beta \left(I\right)$), where α and β are epimorphisms of R. For any $\mathsf{ı},\mathsf{ȷ}\in R$, if $\mathsf{ı}\alpha \left(I\right)\mathsf{ȷ}\subseteq P$ (resp. $\mathsf{ı}\beta \left(I\right)\mathsf{ȷ}\subseteq P$), then either $\mathsf{ı}\in P$ or $\mathsf{ȷ}\in P$.

Proof. 

Assume that $\mathsf{ı}\alpha \left(I\right)\mathsf{ȷ}\subseteq P$ for some $\mathsf{ı},\mathsf{ȷ}\in R$. That is, $\mathsf{ı}\alpha \left(RIR\right)\mathsf{ȷ}\subseteq P$ for some $\mathsf{ı},\mathsf{ȷ}\in R$. Hence, $\mathsf{ı}\alpha \left(R\right)\alpha \left(I\right)\alpha \left(R\right)\mathsf{ȷ}\subseteq P$ for some $\mathsf{ı},\mathsf{ȷ}\in R$. Since $\alpha$ and $\beta$ are epimorphisms of R, we obtain $\mathsf{ı}R\alpha \left(I\right)R\mathsf{ȷ}\subseteq P$ for some $\mathsf{ı},\mathsf{ȷ}\in R$. By the primeness of P, we obtain $\mathsf{ı}\in P$, $\mathsf{ȷ}\in P$, or $\alpha \left(I\right)\subseteq P.$ In the last case, we obtain a contradiction.

By using similar arguments, we can prove this in the case $\mathsf{ı}\beta \left(I\right)\mathsf{ȷ}\subseteq P$. □

Lemma 1.

Let I be a non-zero right (resp. left) ideal of a ring R, and let P be a prime ideal of R such that $P⊊\beta \left(I\right)$ (resp. $P⊊\alpha \left(I\right)$). Suppose α and β are automorphisms of R. If d is an $(\alpha ,\beta )$-derivation of R such that $d\left(I\right)$ is contained in P, then $d\left(R\right)$ is also contained in P.

Proof. 

Assume that I is a non-zero right ideal of R. From the initial assumption, for all $\mathsf{ı}\in I$, $\mathsf{\ell }\in R$, we have $d(\mathsf{ı}\mathsf{\ell })=d(\mathsf{ı})\alpha (\mathsf{\ell })+\beta (\mathsf{ı})d(\mathsf{\ell })\in P$. Since $d\left(I\right)\subseteq P$, then the previous relation reduces to $\beta (\mathsf{ı})d(\mathsf{\ell })\in P$. By replacing ℓ with $\mathsf{\ell }\mathsf{ȷ}$ in the last relation and applying it, we obtain $\beta (\mathsf{ı})\beta (\mathsf{\ell })d(\mathsf{ȷ})\in P$ for all $\mathsf{ı}\in I$, $\mathsf{\ell },\mathsf{ȷ}\in R$. For any $\tau \in R$, replacing ℓ by ${\beta }^{-1}\left(\tau \right)$ in the last relation, we obtain $\beta (\mathsf{ı})Rd(\mathsf{ȷ})\subseteq P$ for all $\mathsf{ı}\in I$, $\mathsf{ȷ}\in R$. Since $\beta \left(I\right)$ is a right ideal of R and $P⊊\beta \left(I\right)$, there exists ${\mathsf{ı}}_{\circ }\in I$ such that $\beta \left({\mathsf{ı}}_{\circ }\right)\notin P$. Since P is a prime ideal, we obtain $d(\mathsf{ȷ})\in P$ for all $\mathsf{ȷ}\in R$, and thus we obtain the desired result.

By using similar arguments, we can prove this in the case of a non-zero left ideal. □

Remark 2.

By setting $\alpha =\beta =I{d}_R$ in the previous lemma, we can directly obtain Lemma 2.2 in [11].

Lemma 2

(Lemma 1.2 in [10]). Let R be a ring. If P is a prime ideal of a given ring, then $R/P$ is a commutative integral domain if $[\mathsf{ı},\mathsf{ȷ}]\in P$ for every $\mathsf{ı},\mathsf{ȷ}\in R$.

Now we introduce a similar lemma for a subset I of R, where I is a non-zero ideal of R and $\alpha$ and $\beta$ are automorphisms on R. This is a generalization of Lemma 2.3 in [11] and is as follows:

Lemma 3.

Consider two ideals, P and $I\ne 0$, of a ring R, where P is a prime ideal provided that $P⊊\alpha \left(I\right)$ and $P⊊\beta \left(I\right)$. Then, $R/P$ is a commutative integral domain if $\left[\alpha \right(\mathsf{ı}),\beta (\mathsf{ȷ}\left)\right]\in P$ holds for every $\mathsf{ı},\mathsf{ȷ}\in I$, where α and β are epimorphisms of R.

Proof. 

We have the following hypothesis:

$$\left[\alpha \right(\mathsf{ı}),\beta (\mathsf{ȷ}\left)\right]\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I.$$

(1)

By replacing ı with $\mathsf{\ell }\mathsf{ı}$ in (1), we obtain

$$\alpha (\mathsf{\ell })\left[\alpha \right(\mathsf{ı}),\beta (\mathsf{ȷ}\left)\right]+\left[\alpha \right(\mathsf{\ell }),\beta (\mathsf{ȷ}\left)\right]\alpha (\mathsf{ı})\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I,\mathsf{\ell }\in R.$$

(2)

Left multiplying Equation (1) by $\alpha (\mathsf{\ell })$ and comparing it with Equation (2), we find

$$\left[\alpha \right(\mathsf{\ell }),\beta (\mathsf{ȷ}\left)\right]\alpha (\mathsf{ı})\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I,\mathsf{\ell }\in R.$$

(3)

Replacing ȷ with $\tau \mathsf{ȷ}$ in Equation (3), we obtain

$$\beta \left(\tau \right)\left[\alpha \right(\mathsf{\ell }),\beta (\mathsf{ȷ}\left)\right]\alpha (\mathsf{ı})+\left[\alpha \right(\mathsf{\ell }),\beta (\tau \left)\right]\beta (\mathsf{ȷ})\alpha (\mathsf{ı})\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I,\mathsf{\ell },\tau \in R.$$

(4)

Left multiplying Equation (3) by $\beta \left(\tau \right)$ and comparing it with Equation (4), we find

$$\left[\alpha \right(\mathsf{\ell }),\beta (\tau \left)\right]\beta (\mathsf{ȷ})\alpha (\mathsf{ı})\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I,\mathsf{\ell },\tau \in R.$$

(5)

Putting $\mathsf{ȷ}=s\mathsf{ȷ}r$ in Equation (5), where $s,r\in R$, we obtain

$$\left[\alpha \right(\mathsf{\ell }),\beta (\tau \left)\right]\beta \left(s\right)\beta (\mathsf{ȷ})\beta \left(r\right)\alpha (\mathsf{ı})\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I,\mathsf{\ell },\tau ,s,r\in R.$$

As $\alpha ,\beta$ are epimorphisms of R, we obtain

$$[\mathsf{\ell },\tau ]R\beta (\mathsf{ȷ})R\alpha (\mathsf{ı})\subseteq P\mathrm{for}\mathrm{all}\mathsf{ı}\in I,\mathsf{\ell },\tau \in R.$$

(6)

Since $P⊊\alpha \left(I\right)$ and $P⊊\beta \left(I\right)$, and from the primeness of P, we obtain $[\mathsf{\ell },\tau ]\in P$ for all $\mathsf{\ell },\tau \in R$. By utilizing Lemma 2, we can conclude that $R/P$ is a commutative integral domain. □

In Lemma 2.1 in [2], it was shown that a non-zero $(\alpha ,\beta )$-derivation $d\left(R\right)\subseteq \mathcal{Z}\left(R\right)$ if it satisfies $\left[d\right(\mathsf{ı}),\beta (\mathsf{ı}\left)\right]=0$ for all ı in a semiprime ring R. Additionally, in Lemma 2.1 in [13], without assuming that R is semiprime, it was concluded that either $R/P$ is a commutative integral domain or $d(\mathsf{ı})\subseteq P$ when R admits a derivation d that satisfies $\left[d\right(\mathsf{ı}),\mathsf{ı}]\in P$ for all $\mathsf{ı}\in I$, where P is a prime ideal and I is a non-zero ideal of R such that $P⊊I$. In the following lemma, we derive an improved version of the previous results in the context of a generalized $(\alpha ,\beta )$-derivation F instead of d, without additional restrictions on R as follows:

Lemma 4.

Consider two ideals, P and $I\ne 0$, of a ring R, where P is a prime ideal and $\alpha ,\beta$ are automorphisms provided that $P⊊\alpha \left(I\right)$ and $P⊊\beta \left(I\right)$. If R is equipped with a generalized $(\alpha ,\beta )$-derivation F associated with d such that $\left[\alpha \right(\mathsf{ı}),F(\mathsf{ı}\left)\right]\in P$ for all $\mathsf{ı}\in I$, then $R/P$ is a commutative integral domain or d maps R to the prime ideal P.

Proof. 

The given hypothesis states the following:

$$\left[\alpha \right(\mathsf{ı}),F(\mathsf{ı}\left)\right]\in P\mathrm{for}\mathrm{all}\mathsf{ı}\in I.$$

(7)

Linearizing the last equation, we obtain

$$\left[\alpha \right(\mathsf{ȷ}),F(\mathsf{ı}\left)\right]+\left[\alpha \right(\mathsf{ı}),F(\mathsf{ȷ}\left)\right]\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I.$$

(8)

Setting $\mathsf{ȷ}=\mathsf{ȷ}\mathsf{ı}$ in Equation (8), and then using Equation (7), we find

$$\begin{array}{c}\hfill \left[\alpha \right(\mathsf{ȷ}),F(\mathsf{ı}\left)\right]\alpha (\mathsf{ı})+\left[\alpha \right(\mathsf{ı}),F(\mathsf{ȷ}\left)\right]\alpha (\mathsf{ı})+\beta (\mathsf{ȷ})\left[\alpha \right(\mathsf{ı}),d(\mathsf{ı}\left)\right]+\left[\alpha \right(\mathsf{ı}),\beta (\mathsf{ȷ}\left)\right]d(\mathsf{ı})\in P.\end{array}$$

Right multiplying Equation (8) by $\alpha (\mathsf{ı})$ and comparing it with the last equation, we deduce

$$\beta (\mathsf{ȷ})\left[\alpha \right(\mathsf{ı}),d(\mathsf{ı}\left)\right]+\left[\alpha \right(\mathsf{ı}),\beta (\mathsf{ȷ}\left)\right]d(\mathsf{ı})\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I.$$

Taking $\mathsf{ȷ}=\mathsf{\ell }\mathsf{ȷ}$ in the last equation and utilizing it, we obtain

$$\left[\alpha \right(\mathsf{ı}),\beta (\mathsf{\ell }\left)\right]\beta (\mathsf{ȷ})d(\mathsf{ı})\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ},\mathsf{\ell }\in I.$$

Putting $\mathsf{ȷ}=s\mathsf{ȷ}r$ in the last relation, where $s,r\in R$, we obtain

$$\left[\alpha \right(\mathsf{ı}),\beta (\mathsf{\ell }\left)\right]\beta \left(s\right)\beta (\mathsf{ȷ})\beta \left(r\right)d(\mathsf{ı})\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ},\mathsf{\ell }\in I\mathrm{and}s,r\in R.$$

As $\beta$ is an epimorphism of R, we obtain

$$\left[\alpha \right(\mathsf{ı}),\beta (\mathsf{\ell }\left)\right]R\beta (\mathsf{ȷ})Rd(\mathsf{ı})\subseteq P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I.$$

Applying Remark 1, either $d(\mathsf{ı})\in P$ or $\left[\alpha \right(\mathsf{ı}),\beta (\mathsf{ȷ}\left)\right]\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I$. By setting $\Gamma =\{\mathsf{ı}\in I|\left[\alpha \right(\mathsf{ı}),\beta (\mathsf{ȷ}\left)\right]\in Pforall\mathsf{ȷ}\in I\}$ and $\Delta =\{\mathsf{ı}\in I|d(\mathsf{ı})\in P\}$, it is easy to note that $\Gamma$ and $\Delta$ are additive subgroups of I such that $\Gamma \cup \Delta =I$. Applying Brauer’s trick, we obtain either $\Gamma =I$ or $\Delta =I$. If $\Gamma =I$, we have $\left[\alpha \right(\mathsf{ı}),\beta (\mathsf{ȷ}\left)\right]\in P$ for all $\mathsf{ı},\mathsf{ȷ}\in I$. By using Lemma 3, we obtain $R/P$, which is a commutative integral domain.

On the other hand, if $\Delta =I$, then $d(\mathsf{ı})\in P$ for all $\mathsf{ı}\in I$, so $d\left(I\right)\subseteq P$. Therefore, $d\left(R\right)\subseteq P$ by Lemma 1. □

As a special case of Lemma 4, we can easily obtain the following corollary when $F=d$:

Corollary 1.

Consider two ideals, P and I, of a ring R, where P is a prime ideal and $\alpha ,\beta$ are automorphisms provided that $P⊊\alpha \left(I\right)$ and $P⊊\beta \left(I\right)$. If R is equipped with an $(\alpha ,\beta )$-derivation d such that $\left[\alpha \right(\mathsf{ı}),d(\mathsf{ı}\left)\right]\in P$ for all $\mathsf{ı}\in I$, then $R/P$ is a commutative integral domain or d maps R to a prime ideal P.

3. Main Results

In this section, for brevity, $(F,d)$ and $(G,g)$ represent generalized $(\alpha ,\beta )$-derivations associated with $(\alpha ,\beta )$-derivations d and g, respectively, where $\alpha$ and $\beta$ are automorphisms of R. Also, I denotes a non-zero ideal of a ring R, and P denotes a prime ideal of R such that $P⊊\alpha \left(I\right)$ and $P⊊\beta \left(I\right)$.

Theorem 1.

Consider two ideals, I and P, of a ring R, where P is a prime ideal and $\alpha ,\beta$ are automorphisms provided that $P⊊\alpha \left(I\right)$ and $P⊊\beta \left(I\right)$. Suppose R is equipped with generalized $(\alpha ,\beta )$-derivations $(F,d)$ and $(G,g)$ satisfying the identity $G(\mathsf{ı}\mathsf{ȷ})\pm d(\mathsf{ı})F(\mathsf{ȷ})+\alpha (\mathsf{ı}\mathsf{ȷ})\in P$ for all $\mathsf{ı},\mathsf{ȷ}\in I$. Then, it follows that $d\left(R\right)\subseteq P$, $g\left(R\right)\subseteq P$ and $(G+\alpha )\left(R\right)\subseteq P$.

Proof. 

The given hypothesis states that

$$G(\mathsf{ı}\mathsf{ȷ})\pm d(\mathsf{ı})F(\mathsf{ȷ})+\alpha (\mathsf{ı}\mathsf{ȷ})\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I.$$

(9)

Setting $\mathsf{ȷ}=\mathsf{ȷ}\mathsf{\ell }$ in Equation (9) and using it, we obtain

$$\beta (\mathsf{ı}\mathsf{ȷ})g(\mathsf{\ell })\pm d(\mathsf{ı})\beta (\mathsf{ȷ})ð(\mathsf{\ell })\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ},\mathsf{\ell }\in I.$$

(10)

Replacing $\mathsf{ȷ}=\mathsf{ı}\mathsf{ȷ}$ in Equation (10), we obtain $\beta (\mathsf{ı})\beta (\mathsf{ı}\mathsf{ȷ})g(\mathsf{\ell })\pm d(\mathsf{ı})\beta (\mathsf{ı})\beta (\mathsf{ȷ})d(\mathsf{\ell })\in P$. Left multiplying Equation (10) by $\beta (\mathsf{ı})$ and comparing it with the last relation, we find

$$\left[d\right(\mathsf{ı}),\beta (\mathsf{ı}\left)\right]\beta (\mathsf{ȷ})d(\mathsf{\ell })\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ},\mathsf{\ell }\in I.$$

(11)

Taking $\mathsf{ȷ}=\rho \mathsf{ȷ}r$ where $\rho ,r\in R$ in Equation (11) along with the automorphism of $\beta$, we have $\left[d\right(\mathsf{ı}),\beta (\mathsf{ı}\left)\right]R\beta (\mathsf{ȷ})Rd(\mathsf{\ell })\subseteq P$ for all $\mathsf{ı},\mathsf{ȷ},\mathsf{\ell }\in I$. Considering the primeness of P, the last expression implies either $\left[d\right(\mathsf{ı}),\beta (\mathsf{ı}\left)\right]\in P$ for all $\mathsf{ı}\in I$ or $d(\mathsf{\ell })\in P$ for all $\mathsf{\ell }\in I$. Note that $d(\mathsf{\ell })\in P$ for all $\mathsf{\ell }\in I$ implies that $\left[d\right(\mathsf{ı}),\beta (\mathsf{ı}\left)\right]\in P$ for all $\mathsf{ı}\in I$. Thus, we have $\left[d\right(\mathsf{ı}),\beta (\mathsf{ı}\left)\right]\in P$ for all $\mathsf{ı}\in I$ in both cases. By utilizing Corollary 1, we deduce two cases: $R/P$ is a commutative integral domain or $d\left(R\right)\subseteq P$.

Case 1: Suppose that $R/P$ is a commutative integral domain. Equation (10) is reduced to

$$\beta (\mathsf{ı})g(\mathsf{\ell })\pm d(\mathsf{ı})d(\mathsf{\ell })\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{\ell }\in I.$$

(12)

Setting $\mathsf{ı}=\mu \mathsf{ı}$ in Equation (12), we obtain $\beta \left(\mu \right)\beta (\mathsf{ı})g(\mathsf{\ell })\pm d\left(\mu \right)\alpha (\mathsf{ı})d(\mathsf{\ell })\pm \beta \left(\mu \right)d(\mathsf{ı})d(\mathsf{\ell })\in P$ for all $\mathsf{ı},\mathsf{\ell },\mu \in I$. Left multiplying Equation (12) by $\beta \left(\mu \right)$ and comparing it with the last relation, we obtain

$$d\left(\mu \right)\alpha (\mathsf{ı})d(\mathsf{\ell })\in P\mathrm{for}\mathrm{all}\mathsf{\ell },\mu \in I.$$

Therefore,

$$d\left(\mu \right)R\alpha (\mathsf{ı})Rd(\mathsf{\ell })\subseteq P\mathrm{for}\mathrm{all}\mathsf{\ell },\mu \in I.$$

From this and Remark 1, we can conclude that $d\left(I\right)\subseteq P$. Using Lemma 1, we find that $d\left(R\right)\subseteq P$. Therefore, Equation (12) simplifies to

$$\beta (\mathsf{ı})g(\mathsf{\ell })\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{\ell }\in I.$$

(13)

Since $R/P$ is a commutative integral domain, we obtain $\beta (\mathsf{ı})Rg(\mathsf{\ell })\subseteq P$ for all $\mathsf{ı},\mathsf{\ell }\in I$. The primeness of P implies that $g(\mathsf{\ell })\in P$ for all $\mathsf{\ell }\in I$. Hence, $g\left(I\right)\subseteq P$, and using Lemma 1, we obtain $g\left(R\right)\subseteq P$. Thus, Equation (9) is reduced to $G(\mathsf{ı})\alpha (\mathsf{ȷ})+\alpha (\mathsf{ı}\mathsf{ȷ})=\left(G\right(\mathsf{ı})+\alpha (\mathsf{ı}\left)\right)\alpha (\mathsf{ȷ})\in P$ for all $\mathsf{ı},\mathsf{ȷ}\in I$. Since $R/P$ is a commutative integral domain and using the primeness of P along with $P⊊\alpha \left(I\right)$, we can deduce that $G(\mathsf{ı})+\alpha (\mathsf{ı})\in P$ for all $\mathsf{ı}\in I$. Hence, $(G+\alpha )\left(R\right)\subseteq P.$

Case 2: Supposing that $d\left(R\right)\subseteq P$, we obtain Equation (13). For any $\rho \in R$, $\beta (\mathsf{ı})\beta \left(\rho \right)g(\mathsf{\ell })\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{\ell }\in I$. It means that $\beta (\mathsf{ı})Rg(\mathsf{\ell })\subseteq P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{\ell }\in I$. Hence, $g(\mathsf{\ell })\subseteq P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{\ell }\in I$. Thus, Equation (9) is reduced to $G(\mathsf{ı})\alpha (\mathsf{ȷ})+\alpha (\mathsf{ı}\mathsf{ȷ})=\left(G\right(\mathsf{ı})+\alpha (\mathsf{ı}\left)\right)\alpha (\mathsf{ȷ})\in P$ for all $\mathsf{ı},\mathsf{ȷ}\in I$. For any $\rho \in R$, $\left(G\right(\mathsf{ı})+\alpha (\mathsf{ı}\left)\right)\alpha \left(\rho \right)\alpha (\mathsf{ȷ})\in P$ for all $\mathsf{ı},\mathsf{ȷ}\in I$. We can deduce that $G(\mathsf{ı})+\alpha (\mathsf{ı})\in P$ for all $\mathsf{ı}\in I$. Hence, $(G+\alpha )\left(R\right)\subseteq P.$ Then, similar to the previous discussion, we can conclude that $(G+\alpha )\left(R\right)$ is also a subset of P. □

Without difficulty, we can follow arguments similar to those used in the previous theorem to prove the following statement:

Theorem 2.

Consider two ideals, I and P, of a ring R, where P is a prime ideal and $\alpha ,\beta$ are automorphisms provided that $P⊊\alpha \left(I\right)$ and $P⊊\beta \left(I\right)$. Suppose R is equipped with generalized $(\alpha ,\beta )$-derivations $(F,d)$ and $(G,g)$ satisfying the identity $G(\mathsf{ı}\mathsf{ȷ})\pm d(\mathsf{ı})F(\mathsf{ȷ})-\alpha (\mathsf{ı}\mathsf{ȷ})\in P$ for all $\mathsf{ı},\mathsf{ȷ}\in I$. Then, it follows that $d\left(R\right)\subseteq P$, $g\left(R\right)\subseteq P$ and $(G-\alpha )\left(R\right)\subseteq P$.

If we replace G with $G-\alpha$ in Theorem 1 and $G+\alpha$ in Theorem 2, respectively, we can easily derive the following:

Corollary 2.

Consider two ideals, I and P, of a ring R, where P is a prime ideal and $\alpha ,\beta$ are automorphisms provided that $P⊊\alpha \left(I\right)$ and $P⊊\beta \left(I\right)$. Suppose R is equipped with generalized $(\alpha ,\beta )$-derivations $(F,d)$ and $(G,g)$ satisfying the identity $G(\mathsf{ı}\mathsf{ȷ})\pm d(\mathsf{ı})F(\mathsf{ȷ})\in P$ for all $\mathsf{ı},\mathsf{ȷ}\in I$. Then, it follows that $d\left(R\right)\subseteq P$, $g\left(R\right)\subseteq P$ and $G\left(R\right)\subseteq P$.

Furthermore, if we consider $F=G$ in Theorems 1 and 2, and follow similar arguments, the following corollary can be obtained:

Corollary 3.

Consider two ideals, I and P, of a ring R, where P is a prime ideal and $\alpha ,\beta$ are automorphisms provided that $P⊊\alpha \left(I\right)$ and $P⊊\beta \left(I\right)$. Suppose R is equipped with a generalized $(\alpha ,\beta )$-derivation $(F,d)$ satisfying the identity $F(\mathsf{ı}\mathsf{ȷ})\pm d(\mathsf{ı})F(\mathsf{ȷ})\pm \alpha (\mathsf{ı}\mathsf{ȷ})\in P$ for all $\mathsf{ı},\mathsf{ȷ}\in I$. Then, it follows that $d\left(R\right)\subseteq P$ and $(F\pm \alpha )\left(R\right)\subseteq P$.

Theorem 3.

Consider two ideals, I and P, of a ring R, where P is a prime ideal and $\alpha ,\beta$ are automorphisms provided that $P⊊\alpha \left(I\right)$ and $P⊊\beta \left(I\right)$. Suppose R is equipped with generalized $(\alpha ,\beta )$-derivations $(F,d)$ and $(G,g)$ satisfying the identity $G(\mathsf{ı}\mathsf{ȷ})\pm d(\mathsf{ı})F(\mathsf{ȷ})\pm \alpha (\mathsf{ȷ}\mathsf{ı})\in P$ for all $\mathsf{ı},\mathsf{ȷ}\in I$. Then, it follows that $d\left(R\right)\subseteq P$, $g\left(R\right)\subseteq P$, and $(G\pm \alpha )\left(R\right)\subseteq P$.

Proof. 

The initial hypothesis can be restated as follows:

$$G(\mathsf{ı})\alpha (\mathsf{ȷ})+\beta (\mathsf{ı})g(\mathsf{ȷ})+d(\mathsf{ı})F(\mathsf{ȷ})+\alpha (\mathsf{ȷ}\mathsf{ı})\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I.$$

(14)

By setting $\mathsf{ȷ}=\mathsf{ȷ}\mathsf{\ell }$ in Equation (14) and utilizing it, we arrive at

$$\beta (\mathsf{ı})\beta (\mathsf{ȷ})g(\mathsf{\ell })+d(\mathsf{ı})\beta (\mathsf{ȷ})d(\mathsf{\ell })+\alpha (\mathsf{ȷ})\alpha \left(\right[\mathsf{\ell },\mathsf{ı}\left]\right)\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ},\mathsf{\ell }\in I.$$

(15)

In Equation (15), substituting ı with $\mathsf{\ell }\mathsf{ı}$ and utilizing it, we obtain

$$d(\mathsf{\ell })\alpha (\mathsf{ı})\beta (\mathsf{ȷ})d(\mathsf{\ell })+\alpha (\mathsf{ȷ})\alpha (\mathsf{\ell })\alpha \left(\right[\mathsf{\ell },\mathsf{ı}\left]\right)-\beta (\mathsf{\ell })\alpha (\mathsf{ȷ})\alpha \left(\right[\mathsf{\ell },\mathsf{ı}\left]\right)\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ},\mathsf{\ell }\in I.$$

(16)

In particular, let $\mathsf{\ell }=\mathsf{ı}$ in the previous equation to obtain $d(\mathsf{ı})\alpha (\mathsf{ı})\beta (\mathsf{ȷ})d(\mathsf{ı})\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I$. It follows that

$$d(\mathsf{ı})\alpha (\mathsf{ı})\beta (\mathsf{ȷ})d(\mathsf{ı})\alpha (\mathsf{ı})\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I.$$

Since $\mathsf{ȷ}\rho \in I$ for all $\mathsf{ȷ}\in I$ and $\rho \in R$, we obtain

$$d(\mathsf{ı})\alpha (\mathsf{ı})\beta (\mathsf{ȷ})\beta \left(\rho \right)d(\mathsf{ı})\alpha (\mathsf{ı})\beta (\mathsf{ȷ})\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I\mathrm{and}\rho \in R.$$

Hence,

$$d(\mathsf{ı})\alpha (\mathsf{ı})\beta (\mathsf{ȷ})Rd(\mathsf{ı})\alpha (\mathsf{ı})\beta (\mathsf{ȷ})\subseteq P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I.$$

Therefore, $d(\mathsf{ı})\alpha (\mathsf{ı})\beta (\mathsf{ȷ})\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I$. By using the above in a similar way, we conclude that $d(\mathsf{ı})\alpha (\mathsf{ı})R\beta (\mathsf{ȷ})\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I$. Thus, we have

$$d(\mathsf{ı})\alpha (\mathsf{ı})\in P\mathrm{for}\mathrm{all}\mathsf{ı}\in I.$$

(17)

By linearizing the last relation, we obtain $d(\mathsf{ı})\alpha (\mathsf{ȷ})+d(\mathsf{ȷ})\alpha (\mathsf{ı})\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I$. By replacing ı with $\mathsf{ı}\mu$ in the last equation and using it, we obtain $d(\mathsf{ı})\alpha \left(\mu \right)\alpha (\mathsf{ȷ})-d(\mathsf{ı})\alpha (\mathsf{ȷ})\alpha \left(\mu \right)+\beta (\mathsf{ı})d\left(\mu \right)\alpha (\mathsf{ȷ})\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ},\mu \in I$. Again, replacing ȷ with $\mathsf{ȷ}\mathsf{\ell }$ in the last equation and using it, we obtain $d(\mathsf{ı})\alpha (\mathsf{ȷ})\left[\alpha \right(\mu ),\alpha (\mathsf{\ell }\left)\right]\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ},\mu ,\mathsf{\ell }\in I$. Putting $\mathsf{ȷ}={\alpha }^{-1}\left(r\right)\mathsf{ȷ}{\alpha }^{-1}\left(s\right)$, where $r,s\in R$, we obtain $d(\mathsf{ı})R\alpha (\mathsf{ȷ})R\left[\alpha \right(\mu ),\alpha (\mathsf{\ell }\left)\right]\subseteq P\mathrm{for}\mathrm{all}\mathsf{ı},\mu ,\mathsf{\ell }\in I$. Remark 1 together with Brauer’s trick implies that $d(\mathsf{ı})\in P$ or $\left[\alpha \right(\mu ),\alpha (\mathsf{\ell }\left)\right]\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mu ,\mathsf{\ell }\in I$. Firstly, if $d(\mathsf{ı})\in P$ for all $\mathsf{ı}\in I$, then Equation (16) is reduced to $\alpha (\mathsf{ȷ})\alpha (\mathsf{\ell })\alpha \left(\right[\mathsf{\ell },\mathsf{ı}\left]\right)-\beta (\mathsf{\ell })\alpha (\mathsf{ȷ})\alpha \left(\right[\mathsf{\ell },\mathsf{ı}\left]\right)\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ},\mathsf{\ell }\in I$. By replacing ȷ with $\mathsf{ı}\mathsf{ȷ}$ in the previous expression and using it with some simple calculations, we arrive at $\left[\alpha \right(\mathsf{ı}),\beta (\mathsf{\ell }\left)\right]\alpha (\mathsf{ȷ})\alpha \left(\right[\mathsf{\ell },\mathsf{ı}\left]\right)\in P$ for all $\mathsf{ı},\mathsf{ȷ},\mathsf{\ell }\in I$. By utilizing the fact that $\alpha$ is an automorphism, we obtain $\left[\alpha \right(\mathsf{ı}),\beta (\mathsf{\ell }\left)\right]I\alpha \left(\right[\mathsf{\ell },\mathsf{ı}\left]\right)\subseteq P$ for all $\mathsf{ı},\mathsf{\ell }\in I$. Once again, using Remark 1 together with Brauer’s trick, we obtain either $\left[\alpha \right(\mathsf{ı}),\beta (\mathsf{\ell }\left)\right]\in P$ for all $\mathsf{ı},\mathsf{\ell }\in I$ or $\alpha \left(\right[\mathsf{\ell },\mathsf{ı}\left]\right)\in P$ for all $\mathsf{ı},\mathsf{\ell }\in I$. Therefore, in both cases, $R/P$ is a commutative integral domain. So, our hypothesis is equivalent to the hypothesis of Theorem 1.

Secondly, if $\left[\alpha \right(\mu ),\alpha (\mathsf{\ell }\left)\right]\in P\mathrm{for}\mathrm{all}\mu ,\mathsf{\ell }\in I$, then use Lemma 3 to obtain $R/P$, which is a commutative integral domain. Hence, from Theorem 1, we obtain $d\left(R\right)\subseteq P$, $g\left(R\right)\subseteq P$, and $(G+\alpha )\left(R\right)\subseteq P$, as desired.

By applying similar techniques and arguments, we can arrive at the desired conclusion for the case $G(\mathsf{ı}\mathsf{ȷ})-d(\mathsf{ı})F(\mathsf{ȷ})-\alpha (\mathsf{ȷ}\mathsf{ı})\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I$ using Theorem 2. □

If we substitute G with $G-\alpha$ and $G+\alpha$, respectively, in the identity $G(\mathsf{ı}\mathsf{ȷ})+d(\mathsf{ı})F(\mathsf{ȷ})+\alpha (\mathsf{ȷ}\mathsf{ı})\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I$ imposed in Theorem 3, and follow similar arguments with necessary modifications, then we can prove the following:

Corollary 4.

Consider two ideals, I and P, of a ring R, where P is a prime ideal and $\alpha ,\beta$ are automorphisms provided that $P⊊\alpha \left(I\right)$ and $P⊊\beta \left(I\right)$. If R is equipped with generalized $(\alpha ,\beta )$-derivations $(F,d)$ and $(G,g)$ such that

$\left(i\right)$$G(\mathsf{ı}\mathsf{ȷ})+d(\mathsf{ı})F(\mathsf{ȷ})-\alpha \left(\right[\mathsf{ı},\mathsf{ȷ}\left]\right)\in P$ for all $\mathsf{ı},\mathsf{ȷ}\in I$, then $g\left(R\right)\subseteq P$ and $G\left(R\right)\subseteq P$.

$\left(ii\right)$$G(\mathsf{ı}\mathsf{ȷ})+d(\mathsf{ı})F(\mathsf{ȷ})+\alpha (\mathsf{ı}\circ \mathsf{ȷ})\in P$ for all $\mathsf{ı},\mathsf{ȷ}\in I$, then $g\left(R\right)\subseteq P$ and $(G+2\alpha )\left(R\right)\subseteq P$.

Moreover, by setting $F=G$ in Theorem 3, the proof of the following corollary can be easily verified:

Corollary 5.

Consider two ideals, I and P, of a ring R, where P is a prime ideal and $\alpha ,\beta$ are automorphisms provided that $P⊊\alpha \left(I\right)$ and $P⊊\beta \left(I\right)$. Suppose R is equipped with a generalized $(\alpha ,\beta )$-derivation $(F,d)$ satisfying the identity $F(\mathsf{ı}\mathsf{ȷ})\pm d(\mathsf{ı})F(\mathsf{ȷ})\pm \alpha (\mathsf{ȷ}\mathsf{ı})\in P$ for all $\mathsf{ı},\mathsf{ȷ}\in I$. Then, it follows that $d\left(R\right)\subseteq P$ and $(F\pm \alpha )\left(R\right)\subseteq P$.

Theorem 4.

Consider two ideals, I and P, of a ring R, where P is a prime ideal and $\alpha ,\beta$ are automorphisms provided that $P⊊\alpha \left(I\right)$ and $P⊊\beta \left(I\right)$. Suppose R is equipped with generalized $(\alpha ,\beta )$-derivations $(F,d)$ and $(G,g)$ satisfying the identity $G(\mathsf{ı}\mathsf{ȷ})\pm d(\mathsf{ȷ})F(\mathsf{ı})\pm \alpha (\mathsf{ȷ}\mathsf{ı})\in P$ for all $\mathsf{ı},\mathsf{ȷ}\in I$. Then, it follows that $d\left(R\right)\subseteq P$, $g\left(R\right)\subseteq P$ and $(G\pm \alpha )\left(R\right)\subseteq P$.

Proof. 

We have

$$G(\mathsf{ı}\mathsf{ȷ})\pm d(\mathsf{ȷ})F(\mathsf{ı})+\alpha (\mathsf{ȷ}\mathsf{ı})\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I.$$

(18)

Setting $\mathsf{ı}=\mathsf{ı}\mathsf{ȷ}$ in Equation (18) and then using it, we obtain

$$\beta (\mathsf{ı}\mathsf{ȷ})g(\mathsf{ȷ})\pm d(\mathsf{ȷ})\beta (\mathsf{ı})d(\mathsf{ȷ})\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I.$$

(19)

For any $\mathsf{\ell }\in I$, if we set $\mathsf{ı}=\mathsf{\ell }\mathsf{ı}$ in Equation (19), then, by using it, we obtain

$$\left[d\right(\mathsf{ȷ}),\beta (\mathsf{\ell }\left)\right]\beta (\mathsf{ı})d(\mathsf{ȷ})\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I.$$

(20)

Similarly, if we set $\mathsf{ı}=\mathsf{ı}{\beta }^{-1}\left(r\right)$ in Equation (20) and use it, where $r\in R$, we obtain

$$\left[d\right(\mathsf{ȷ}),\beta (\mathsf{\ell }\left)\right]\beta (\mathsf{ı})rd(\mathsf{ȷ})\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ},\mathsf{\ell }\in I,\mathrm{and}r\in R.$$

Hence,

$$\left[d\right(\mathsf{ȷ}),\beta (\mathsf{\ell }\left)\right]\beta (\mathsf{ı})r\left[d\right(\mathsf{ȷ}),\beta (\mathsf{\ell }\left)\right]\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ},\mathsf{\ell }\in I\mathrm{and}r\in R.$$

Therefore,

$$\left[d\right(\mathsf{ȷ}),\beta (\mathsf{\ell }\left)\right]\beta (\mathsf{ı})R\left[d\right(\mathsf{ȷ}),\beta (\mathsf{\ell }\left)\right]\beta (\mathsf{ı})\subseteq P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ},\mathsf{\ell }\in I.$$

By the primeness of P, we obtain

$$\left[d\right(\mathsf{ȷ}),\beta (\mathsf{\ell }\left)\right]\beta (\mathsf{ı})\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ},\mathsf{\ell }\in I.$$

By replacing ı with $\mu \mathsf{ı}$ in the last relation and using it, where $\mu \in R$, we obtain

$$\left[d\right(\mathsf{ȷ}),\beta (\mathsf{\ell }\left)\right]R\beta (\mathsf{ı})\subseteq P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ},\mathsf{\ell }\in I.$$

It follows that

$$\left[d\right(\mathsf{ȷ}),\beta (\mathsf{\ell }\left)\right]R\left[d\right(\mathsf{ȷ}),\beta (\mathsf{\ell }\left)\right]\subseteq P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ},\mathsf{\ell }\in I.$$

By the primeness of P, we obtain $\left[d\right(\mathsf{ȷ}),\beta (\mathsf{\ell }\left)\right]\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ},\mathsf{\ell }\in I$. By utilizing Corollary 1, we deduce two cases: $R/P$ is a commutative integral domain or $d\left(R\right)\subseteq P$. In the first case, using Theorem 3, we obtain $d\left(R\right)\subseteq P$, $g\left(R\right)\subseteq P$, and $(G\pm \alpha )\left(R\right)\subseteq P$. In the second case, Equation (19) is reduced to $\beta (\mathsf{ı}\mathsf{ȷ})g(\mathsf{ȷ})\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I$. Hence, $g\left(R\right)\subseteq P$. Using the last relation and the fact that $d\left(R\right)\subseteq P$ in our hypothesis, we obtain $G(\mathsf{ı})\alpha (\mathsf{ȷ})\pm \alpha (\mathsf{ȷ})\alpha (\mathsf{ı})\in P$ for all $\mathsf{ı},\mathsf{ȷ}\in I$. Replacing ȷ with $\mathsf{ȷ}\mu$ in the last relation, where $\mu \in R$, and using it, we obtain $\left[\alpha \right(\mathsf{ȷ}),\alpha (\mu \left)\right]\alpha (\mathsf{ı})\in P$ for all $\mathsf{ı},\mathsf{ȷ},\mu \in I$. As in the proof of Theorem 3, we can deduce that $R/P$ is a commutative integral domain. So, by Theorem 3, we obtain $d\left(R\right)\subseteq P$, $g\left(R\right)\subseteq P$, and $(G\pm \alpha )\left(R\right)\subseteq P$. □

By substituting G with $G-\alpha$ in the identity $G(\mathsf{ı}\mathsf{ȷ})\pm d(\mathsf{ȷ})F(\mathsf{ı})+\alpha (\mathsf{ȷ}\mathsf{ı})\in P$ for all $\mathsf{ı},\mathsf{ȷ}\in I$, as assumed in Theorem 4, we can proceed to prove the following corollary using similar arguments:

Corollary 6.

Consider two ideals, I and P, of a ring R, where P is a prime ideal and $\alpha ,\beta$ are automorphisms provided that $P⊊\alpha \left(I\right)$ and $P⊊\beta \left(I\right)$. Suppose R is equipped with generalized $(\alpha ,\beta )$-derivations $(F,d)$ and $(G,g)$ satisfying the identity $G(\mathsf{ı}\mathsf{ȷ})\pm d(\mathsf{ȷ})F(\mathsf{ı})-\alpha \left(\right[\mathsf{ı},\mathsf{ȷ}\left]\right)\in P$ for all $\mathsf{ı},\mathsf{ȷ}\in I$. Then, it follows that $d\left(R\right)\subseteq P$, $g\left(R\right)\subseteq P$, $R/P$ is a commutative integral domain and $G\left(R\right)\subseteq P$.

Theorem 5.

Consider two ideals, I and P, of a ring R, where P is a prime ideal and $\alpha ,\beta$ are automorphisms provided that $P⊊\alpha \left(I\right)$ and $P⊊\beta \left(I\right)$. Suppose R is equipped with generalized $(\alpha ,\beta )$-derivations $(F,d)$ and $(G,g)$ satisfying the identity $F(\mathsf{ı}\mathsf{ȷ})\pm G(\mathsf{ı})\alpha (\mathsf{ȷ})\pm \alpha (\mathsf{ȷ}\mathsf{ı})\in P$ for all $\mathsf{ı},\mathsf{ȷ}\in I$. Then, it follows that $d\left(R\right)\subseteq P$, $g\left(R\right)\subseteq P$, and $(F\pm G\pm \alpha )\left(R\right)\subseteq P$.

Proof. 

The initial assumption is as follows:

$$F(\mathsf{ı}\mathsf{ȷ})+G(\mathsf{ı})\alpha (\mathsf{ȷ})+\alpha (\mathsf{ȷ}\mathsf{ı})\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I.$$

(21)

Letting $\mathsf{ȷ}=\mathsf{ȷ}\mathsf{ı}$ in Equation (21) and using it, we obtain

$$\beta (\mathsf{ı}\mathsf{ȷ})d(\mathsf{ı})\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I.$$

(22)

Hence, $d\left(R\right)\subseteq P$. Using the last relation in Equation (21), we obtain

$$(F+G)(\mathsf{ı})\alpha (\mathsf{ȷ})+\alpha (\mathsf{ȷ})\alpha (\mathsf{ı})\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I.$$

(23)

Taking $\mathsf{ȷ}=\mathsf{ȷ}\mu$ in the last relation and using it, we obtain $\left[\alpha \right(\mathsf{ȷ}),\alpha (\mu \left)\right]\alpha (\mathsf{ı})\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ},\mu \in I$. As in the proof of Theorem 3, we can deduce that $R/P$ is a commutative integral domain. Using the last fact in Equation (23), we obtain $(F+G+\alpha )(\mathsf{ı})\alpha (\mathsf{ȷ})\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I$. Hence, $(F+G+\alpha )\left(R\right)\subseteq P$. On the other hand, $(F+G+\alpha )(\mathsf{ı})\mathrm{for}\mathrm{all}\mathsf{ı}\in I$. Replacing ı with $\mathsf{ı}\mathsf{ȷ}$ in the last relation and using it, and the fact that $d\left(R\right)\subseteq P$, we obtain $\beta (\mathsf{ı})g(\mathsf{ȷ})\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I$. Therefore, $g\left(R\right)\subseteq P$. Following similar arguments, we can easily reach the desired conclusion in the case of $F(\mathsf{ı}\mathsf{ȷ})-G(\mathsf{ı})\alpha (\mathsf{ȷ})-\alpha (\mathsf{ȷ}\mathsf{ı})\in P$ for all $\mathsf{ı},\mathsf{ȷ}\in I$. □

If we replace F with $F-\alpha$ and $F+\alpha$, respectively, in the identity $F(\mathsf{ı}\mathsf{ȷ})\pm G(\mathsf{ı})\alpha (\mathsf{ȷ})+\alpha (\mathsf{ȷ}\mathsf{ı})\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I$, as assumed in Theorem 5, and follow similar arguments with some necessary modifications, we can prove the following:

Corollary 7.

Consider two ideals, I and P, of a ring R, where P is a prime ideal and $\alpha ,\beta$ are automorphisms provided that $P⊊\alpha \left(I\right)$ and $P⊊\beta \left(I\right)$. If R is equipped with generalized $(\alpha ,\beta )$-derivations $(F,d)$ and $(G,g)$ such that

$\left(i\right)$$F(\mathsf{ı}\mathsf{ȷ})\pm G(\mathsf{ı})\alpha (\mathsf{ȷ})+\alpha \left(\right[\mathsf{ȷ},\mathsf{ı}\left]\right)\in P$ for all $\mathsf{ı},\mathsf{ȷ}\in I$, then $R/P$ is a commutative integral domain, $d\left(R\right)\subseteq P$ and $(F\pm G)\left(R\right)\subseteq P$.

$\left(ii\right)$$F(\mathsf{ı}\mathsf{ȷ})\pm G(\mathsf{ı})\alpha (\mathsf{ȷ})+\alpha (\mathsf{ı}\circ \mathsf{ȷ})\in P$ for all $\mathsf{ı},\mathsf{ȷ}\in I$, then $R/P$ is a commutative integral domain, $d\left(R\right)\subseteq P$, and $(F\pm G+2\alpha )\left(R\right)\subseteq P$.

Theorem 6.

Consider two ideals, I and P, of a ring R, where P is a prime ideal and $\alpha ,\beta$ are automorphisms provided that $P⊊\alpha \left(I\right)$ and $P⊊\beta \left(I\right)$. Suppose R is equipped with generalized $(\alpha ,\beta )$-derivations $(F,d)$ and $(G,g)$ satisfying the identity $F(\mathsf{ı}\mathsf{ȷ})\pm G(\mathsf{ȷ})\alpha (\mathsf{ı})\pm \alpha (\mathsf{ı}\mathsf{ȷ})\in P$ for all $\mathsf{ı},\mathsf{ȷ}\in I$. Then, one of the following holds:

$\left(i\right)$$d\left(R\right)\subseteq P$, $g\left(R\right)\subseteq P$, and $(F\pm G\pm \alpha )\left(R\right)\subseteq P$.

$\left(ii\right)$$d\left(R\right)\subseteq P$, $g\left(R\right)\subseteq P$, $G\left(R\right)\subseteq P$, and $(F+\alpha )\left(R\right)\subseteq P$.

Proof. 

The initial assumption is as follows:

$$F(\mathsf{ı}\mathsf{ȷ})+G(\mathsf{ȷ})\alpha (\mathsf{ı})+\alpha (\mathsf{ı}\mathsf{ȷ})\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I.$$

(24)

By letting $\mathsf{ı}=\mathsf{ı}\mathsf{ȷ}$ in Equation (24) and using it, we obtain $\beta (\mathsf{ı}\mathsf{ȷ})d(\mathsf{ȷ})\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I$. Hence, $d\left(R\right)\subseteq P$. Equation (24) is reduced to

$$(F+\alpha )(\mathsf{ı})\alpha (\mathsf{ȷ})+G(\mathsf{ȷ})\alpha (\mathsf{ı})\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I.$$

(25)

Taking $\mathsf{ȷ}=\mathsf{ȷ}\mathsf{ı}$ in Equation (25) and using it, we obtain

$$\beta (\mathsf{ȷ})g(\mathsf{ı})\alpha (\mathsf{ı})\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I.$$

(26)

Therefore, $g(\mathsf{ı})\alpha (\mathsf{ı})\in P\mathrm{for}\mathrm{all}\mathsf{ı}\in I$. As in Equation (17), we obtain $g\left(R\right)\subseteq P$ or $R/P$ is a commutative integral domain. Note that the second case implies the first case when we use Equation (26). Therefore, we suppose that $g\left(R\right)\subseteq P$. By replacing ȷ with $\mathsf{ȷ}{\alpha }^{-1}\left(\mu \right)$ in Equation (25) and using the last relation, we obtain $G(\mathsf{ȷ})[\mu ,\alpha (\mathsf{ı}\left)\right]\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I,\mathrm{and}\mu \in R$. Hence, $G(\mathsf{ȷ})R[\mu ,\alpha (\mathsf{ı}\left)\right]\subseteq P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I,\mathrm{and}\mu \in R$. By the primeness of P, we obtain $G\left(R\right)\subseteq P$ or $[\mu ,\alpha (\mathsf{ı}\left)\right]\in P\mathrm{for}\mathrm{all}\mathsf{ı}\in I,\mathrm{and}\mu \in R$.

The first case reduces Equation (25) to $(F+\alpha )(\mathsf{ı})\alpha (\mathsf{ȷ})\in P\mathrm{for}\mathrm{all}\mathsf{ı},\mathsf{ȷ}\in I$. Therefore, it is evident that $(F+\alpha )\left(R\right)\subseteq P$.

The second case reduces our hypothesis to the hypothesis of Theorem 5. Therefore, we obtain $\left(i\right)$.

Similarly, the theorem can be proven in the case where $F(\mathsf{ı}\mathsf{ȷ})-G(\mathsf{ȷ})\alpha (\mathsf{ı})-\alpha (\mathsf{ı}\mathsf{ȷ})\in P$ for all $\mathsf{ı},\mathsf{ȷ}\in I$. □

In Theorem 6, substituting F with $F-\alpha$ in the identity $F(\mathsf{ı}\mathsf{ȷ})\pm G(\mathsf{ȷ})\alpha (\mathsf{ı})+\alpha (\mathsf{ı}\mathsf{ȷ})\in P$ for all $\mathsf{ı},\mathsf{ȷ}\in I$ and following similar arguments with minor modifications, we can easily prove the following:

Corollary 8.

Consider two ideals, I and P, of a ring R, where P is a prime ideal and $\alpha ,\beta$ are automorphisms provided that $P⊊\alpha \left(I\right)$ and $P⊊\beta \left(I\right)$. Suppose R is equipped with generalized $(\alpha ,\beta )$-derivations $(F,d)$ and $(G,g)$ satisfying the identity $F(\mathsf{ı}\mathsf{ȷ})\pm G(\mathsf{ȷ})\alpha (\mathsf{ı})\in P$ for all $\mathsf{ı},\mathsf{ȷ}\in I$. Then, one of the following holds:

$\left(i\right)$$d\left(R\right)\subseteq P$, $g\left(R\right)\subseteq P$, and $(F\pm G)\left(R\right)\subseteq P$.

$\left(ii\right)$$d\left(R\right)\subseteq P$, $g\left(R\right)\subseteq P$, $G\left(R\right)\subseteq P$, and $F\left(R\right)\subseteq P$.

Before we provide examples to illustrate the importance of the assumptions in the given theorems, first we present the following definition. For more details, refer to [15,16].

Definition 1

([17]). Define $K_{2^n}$ as follows:

$$K_{2^n}:=〈{\mathsf{ı}}_1,\dots ,{\mathsf{ı}}_n|{\mathsf{ı}}_i{\mathsf{ı}}_j={\mathsf{ı}}_i,2{\mathsf{ı}}_i=0,\forall i=1,2,\dots n〉.$$

It is evident that $K_{2^n}$ is not a commutative ring and lacks a multiplicative identity. Additionally, the characteristic of $K_{2^n}$ is two, and every element within it is a zero divisor. Especially when $n=3$, $K_{2^3}=〈a,b,c〉=\{0,a,b,c,h,e,f,g\}$ with the following relations:

$$\begin{array}{c}\hfill 2a=2b=2c=0,\hfill \\ \hfill a^2=ab=ac=a,b^2=ba=bc=b,c^2=ca=cb=c,\hfill \\ \hfill h=a+b,e=a+c,f=b+c,g=a+b+c\hfill \end{array}$$

The opposite ring of $K_{2^n}$ is defined as follows:

$${\left(K_{2^n}\right)}^{op}:=〈{\mathsf{ı}}_1,\dots ,{\mathsf{ı}}_n|{\mathsf{ı}}_i{\mathsf{ı}}_j={\mathsf{ı}}_j,2{\mathsf{ı}}_i=0,\forall i=1,2,\dots n〉.$$

This ring has the same properties as $K_{2^n}$. The additive and multiplicative tables (

Table 1. Table of additive.

+	0	a	b	c	h	e	f	g
0	0	a	b	c	h	e	f	g
a	a	0	h	e	b	c	g	f
b	b	h	0	f	a	g	c	e
c	c	e	f	0	g	a	b	h
h	h	b	a	g	0	f	e	c
e	e	c	g	a	f	0	h	b
f	f	g	c	b	e	h	0	a
g	g	f	e	h	c	b	a	0

and

Table 2. Table of multiplicative.

.	0	a	b	c	h	e	f	g
0	0	0	0	0	0	0	0	0
a	0	a	a	a	0	0	0	a
b	0	b	b	b	0	0	0	b
c	0	c	c	c	0	0	0	c
h	0	h	h	h	0	0	0	h
e	0	e	e	e	0	0	0	e
f	0	f	f	f	0	0	0	f
g	0	g	g	g	0	0	0	g

) of $K_{2^3}$ are as follows:

Note that the concrete forms of $K_{2^3}$ and ${\left(K_{2^3}\right)}^{op}$ in matrix notation are as follows:

$$\begin{array}{c}\hfill K_{2^3}=\{0=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],a=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],b=\left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right],c=\left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 0\\ 1& 0& 0\end{array}\right],\\ \hfill h=\left[\begin{array}{ccc}0& 0& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right],e=\left[\begin{array}{ccc}0& 0& 0\\ 1& 0& 0\\ 1& 0& 0\end{array}\right],f=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 1& 0& 0\end{array}\right],g=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 1& 0& 0\end{array}\right],0,1\in {\mathbb{Z}}_2\}.\end{array}$$

And

$$\begin{array}{c}\hfill {\left(K_{2^3}\right)}^{op}=\{0=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],a=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],b=\left[\begin{array}{ccc}1& 1& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],c=\left[\begin{array}{ccc}1& 1& 1\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],\\ \hfill h=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],e=\left[\begin{array}{ccc}0& 1& 1\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],f=\left[\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],g=\left[\begin{array}{ccc}1& 0& 1\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],0,1\in {\mathbb{Z}}_2\}.\end{array}$$

Example 1.

Let $R=K_{2^3}$ be as above, $I=\{0,h,e,g\}$, and $\P =\left\{0\right\}$. Define $\alpha ,\beta ,(F,d),(G,g):R⟶R$ by

$$\alpha (\mathsf{ı})=\left\{\begin{array}{ccc}\hfill 0& if\mathsf{ı}=0,h,e,f,& \\ \hfill b& if\mathsf{ı}=a,b,c,g.\end{array}\right.;\beta (\mathsf{ı})=\left\{\begin{array}{ccc}\hfill 0& if\mathsf{ı}=0,h,e,f,& \\ \hfill a& if\mathsf{ı}=a,b,c,g.\end{array}\right.$$

$$F(\mathsf{ı})=\left\{\begin{array}{ccc}\hfill 0& if\mathsf{ı}=0,h,e,f& \\ \hfill b& if\mathsf{ı}=a,b,c,g.\end{array}\right.;d(\mathsf{ı})=\left\{\begin{array}{ccc}\hfill 0& if\mathsf{ı}=0,h,e,f,& \\ \hfill h& if\mathsf{ı}=a,b,c,g.\end{array}\right.$$

$$G(\mathsf{ı})=\left\{\begin{array}{ccc}\hfill 0& if\mathsf{ı}=0,h,e,f,& \\ \hfill a& if\mathsf{ı}=a,b,c,g.\end{array}\right.\mathit{and}g(\mathsf{ı})=\left\{\begin{array}{ccc}\hfill 0& if\mathsf{ı}=0,h,e,f,& \\ \hfill e& if\mathsf{ı}=a,b,c,g.\end{array}\right.$$

It is easy to verify that α and β are endomorphisms on R, I is a non-zero ideal of R, and P is a trivial ideal of R that satisfies $P⊊\alpha \left(I\right)$ and $P⊊\beta \left(I\right)$. Clearly, R is not prime because P is not prime (it is evident that, for a pair of elements $c,h\in I$, $cRh\subseteq P$ but neither $c\in P$ nor $h\in P$). Moreover, it can be verified that $(F,d)$ and $(G,g)$ are generalized $(\alpha ,\beta )$-derivations associated with $(\alpha ,\beta )$-derivations d and g, respectively. Obviously, $d\left(R\right)⊈P$, $g\left(R\right)⊈P$, $G\left(R\right)⊈P$, $(F+G+\alpha )\left(R\right)⊈P$, $(F+\alpha )\left(R\right)⊈P$, and $R/P$ is not commutative. Furthermore, $(F,d)$ and $(G,g)$ satisfy the assumptions of our theorems. Therefore, the hypothesis that P is prime in Theorems 1–6 is mandatory.

Example 2.

Let $R=M_2\left(\mathbb{Z}\right)\times \mathbb{Z}$ be a ring, $I=\left\{0\right\}\times \mathbb{Z}$, and $P=\left\{0\right\}\times 4\mathbb{Z}$. Define $\alpha ,\beta ,(F,d),(G,g):R⟶R$ by $\alpha =\beta =I{d}_R$, $F(\mathsf{ı},\mathsf{ȷ})=G(\mathsf{ı},\mathsf{ȷ})=(\mathsf{ı},4\mathsf{ȷ})$ for all $\mathsf{ı},\mathsf{ȷ}\in R$, and $d=g=0$.

It is easy to verify that α and β are automorphisms on R, I is a non-zero ideal of R, and P is a non-trivial ideal of R that satisfies $P⊊\alpha \left(I\right)$ and $P⊊\beta \left(I\right)$. Clearly, P is not a prime ideal because $(0,2\mathbb{Z})R(0,2\mathbb{Z})\subseteq P$; however, $(0,2\mathbb{Z})\notin P$. Moreover, it can be verified that $(F,d)$ and $(G,g)$ are generalized $(\alpha ,\beta )$-derivations associated with $(\alpha ,\beta )$-derivations d and g, respectively.

Obviously, $G\left(R\right)⊈P$, $(F\pm G\pm \alpha )\left(R\right)⊈P$, and $(F+\alpha )\left(R\right)⊈P$. Furthermore, $(F,d)$ and $(G,g)$ satisfy the identities of our theorems. Therefore, the hypothesis that P is a prime ideal in Theorems 1–6 is mandatory.

Example 3.

Let $R=\{\left(\begin{array}{cccc}0& \mathsf{ı}& \mathsf{ȷ}& \tau \\ 0& 0& 0& \mathsf{ȷ}\\ 0& 0& 0& \mathsf{ı}\\ 0& 0& 0& 0\end{array}\right)|\mathsf{ı},\mathsf{ȷ},\tau \in \mathbb{R}\}$ be a ring, $I=\{\left(\begin{array}{cccc}0& 0& 0& \tau \\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)\}$, and $P=\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$. Define $\alpha ,\beta ,(F,d),(G,g):R⟶R$ by

$$\alpha \left(\begin{array}{cccc}0& \mathsf{ı}& \mathsf{ȷ}& \tau \\ 0& 0& 0& \mathsf{ȷ}\\ 0& 0& 0& \mathsf{ı}\\ 0& 0& 0& 0\end{array}\right)=\left(\begin{array}{cccc}0& 0& 0& \tau \\ 0& 0& 0& \mathsf{ȷ}\\ 0& 0& 0& \mathsf{ı}\\ 0& 0& 0& 0\end{array}\right),\beta \left(\begin{array}{cccc}0& \mathsf{ı}& \mathsf{ȷ}& \tau \\ 0& 0& 0& \mathsf{ȷ}\\ 0& 0& 0& \mathsf{ı}\\ 0& 0& 0& 0\end{array}\right)=\left(\begin{array}{cccc}0& \mathsf{ı}& 0& \tau \\ 0& 0& 0& 0\\ 0& 0& 0& \mathsf{ȷ}\\ 0& 0& 0& 0\end{array}\right),$$

$$F\left(\begin{array}{cccc}0& \mathsf{ı}& \mathsf{ȷ}& \tau \\ 0& 0& 0& \mathsf{ȷ}\\ 0& 0& 0& \mathsf{ı}\\ 0& 0& 0& 0\end{array}\right)=\left(\begin{array}{cccc}0& \mathsf{ı}& \mathsf{ȷ}& \tau \\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right),d\left(\begin{array}{cccc}0& \mathsf{ı}& \mathsf{ȷ}& \tau \\ 0& 0& 0& \mathsf{ȷ}\\ 0& 0& 0& \mathsf{ı}\\ 0& 0& 0& 0\end{array}\right)=\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \mathsf{ı}\\ 0& 0& 0& 0\end{array}\right),$$

$$G\left(\begin{array}{cccc}0& \mathsf{ı}& \mathsf{ȷ}& \tau \\ 0& 0& 0& \mathsf{ȷ}\\ 0& 0& 0& \mathsf{ı}\\ 0& 0& 0& 0\end{array}\right)=\left(\begin{array}{cccc}0& 2\mathsf{ı}& 2\mathsf{ȷ}& 2\tau \\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right),g\left(\begin{array}{cccc}0& \mathsf{ı}& \mathsf{ȷ}& \tau \\ 0& 0& 0& \mathsf{ȷ}\\ 0& 0& 0& \mathsf{ı}\\ 0& 0& 0& 0\end{array}\right)=\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \mathsf{ȷ}\\ 0& 0& 0& 0\end{array}\right),$$

for all $\mathsf{ı},\mathsf{ȷ},\tau \in R$. It is easy to verify that α and β are automorphisms on R and I and P are ideals of R that satisfy $P⊊\alpha \left(I\right)$ and $P⊊\beta \left(I\right)$. Clearly, P is not a prime ideal because

$\left(\begin{array}{cccc}0& 0& 0& \tau \\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)R\left(\begin{array}{cccc}0& \mathsf{ı}& \mathsf{ȷ}& 0\\ 0& 0& 0& \mathsf{ȷ}\\ 0& 0& 0& \mathsf{ı}\\ 0& 0& 0& 0\end{array}\right)\subseteq P$, but $\left(\begin{array}{cccc}0& 0& 0& \tau \\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)\notin P$ and $\left(\begin{array}{cccc}0& \mathsf{ı}& \mathsf{ȷ}& 0\\ 0& 0& 0& \mathsf{ȷ}\\ 0& 0& 0& \mathsf{ı}\\ 0& 0& 0& 0\end{array}\right)\notin P$.

• Moreover, it can be verified that $(F,d)$ and $(G,g)$ are generalized $(\alpha ,\beta )$-derivations associated with $(\alpha ,\beta )$-derivations d and g, respectively.

Obviously, $G\left(R\right)⊈P$, $(F\pm G\pm \alpha )\left(R\right)⊈P$, and $(F+\alpha )\left(R\right)⊈P$. Furthermore, $(F,d)$ and $(G,g)$ satisfy the identities of our theorems. Therefore, the hypothesis that P is a prime ideal in Theorems 1–6 is mandatory.

4. Conclusions

Our article aimed to further explore the behavior of generalized $(\alpha ,\beta )$-derivations associated with $(\alpha ,\beta )$-derivations through a contemporary approach. To achieve our goal, the ring R was imposed without any restrictions, and the identities considered connected a non-zero ideal I to a prime ideal P. These identities were as follows: $\left(i\right)$$G(\mathsf{ı}\mathsf{ȷ})\pm d(\mathsf{ı})F(\mathsf{ȷ})+\alpha (\mathsf{ı}\mathsf{ȷ})\in P$, $\left(ii\right)$$G(\mathsf{ı}\mathsf{ȷ})\pm d(\mathsf{ı})F(\mathsf{ȷ})-\alpha (\mathsf{ı}\mathsf{ȷ})\in P$, $\left(iii\right)$$G(\mathsf{ı}\mathsf{ȷ})\pm d(\mathsf{ı})F(\mathsf{ȷ})\pm \alpha (\mathsf{ȷ}\mathsf{ı})\in P$, $\left(iv\right)$$G(\mathsf{ı}\mathsf{ȷ})\pm d(\mathsf{ȷ})F(\mathsf{ı})\pm \alpha (\mathsf{ȷ}\mathsf{ı})\in P$, $\left(v\right)$$F(\mathsf{ı}\mathsf{ȷ})\pm G(\mathsf{ı})\alpha (\mathsf{ȷ})\pm \alpha (\mathsf{ȷ}\mathsf{ı})\in P$, $\left(vi\right)$$F(\mathsf{ı}\mathsf{ȷ})\pm G(\mathsf{ȷ})\alpha (\mathsf{ı})\pm \alpha (\mathsf{ı}\mathsf{ȷ})\in P$, for all $\mathsf{ı},\mathsf{ȷ}\in I$. Moreover, several corollaries are listed as important applications of our theorems. To emphasize the necessity of primeness in our hypotheses, various examples have been included.

5. Future Studies

In future studies, the current study can be expanded in various directions. One possible direction is to utilize appropriate algebraic identities to further investigate the relationship between the commutativity of the ring and various additive mappings. These mappings may include reverse derivations, multiplicative derivations, homoderivations, or multipliers.