Exploring Commutativity via Generalized (α, β)-Derivations Involving Prime Ideals

Abstract

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.

1. Introduction

Throughout this paper ℑ stands for an associative ring and its center is $Z(\Im )$. It is appropriate to start by recalling some well-known concepts about rings. A ring ℑ is called a prime ring if $\varsigma \Im \iota =0$ for each $\varsigma ,\iota \in \Im$, then either $\varsigma =0$ or $\iota =0$. An ideal ℘ of a ring ℑ with $\wp \ne \Im$ is called prime if $\varsigma \Im \iota \subseteq \wp$ ($\varsigma \iota \in \wp$) for $\varsigma ,\iota \in \Im$, which implies that either $\varsigma \in \wp$ or $\iota \in \wp$. Consequently, ℑ is a prime ring if and only if $\left\{0\right\}$ is a prime ideal of ℑ. We recall that a ring without non-zero divisors is a domain, and the integral domain is a commutative domain with identity. It is known that every integral domain is a prime ring and the converse needs not to be true in general. It is also known that ℘ is a prime ideal if and only if $\Im /\wp$ is an integral domain. Additionally, if ℘ is an ideal in a commutative ring ℑ, then $\Im /\wp$ is commutative. It is worth mentioning prime ideal would make an interesting fertile topic to research, not only in rings, but also in algebras such as $BCI-$algebras, $C\ast$-algebra and Lie algebra (for more details, see refs. [1,2,3,4]). An additive map $\partial :\Im ⟶\Im$ that satisfies $\partial \left(\varsigma \iota \right)=\partial \left(\varsigma \right)\iota +\varsigma \partial \left(\iota \right)$ for all $\varsigma ,\iota \in \Im$ is called an ordinary derivation, while an additive map $\vartheta :\Im ⟶\Im$ which satisfies $\vartheta \left(\varsigma \iota \right)=\vartheta \left(\varsigma \right)\iota +\varsigma \partial \left(\iota \right)$ for every $\varsigma ,\iota \in \Im$ is called a generalized derivation, where ∂ is just an associated derivation map.

Suppose that $\alpha ,\beta :\Im ⟶\Im$ are automorphisms on ℑ, then an additive map $\partial :\Im ⟶\Im$ is called an $(\alpha ,\beta )$-derivation if it satisfies $\partial \left(\varsigma \iota \right)=\partial \left(\varsigma \right)\alpha \left(\iota \right)+\beta \left(\varsigma \right)\partial \left(\iota \right)$ for any two elements $\varsigma ,\iota \in \Im$. Afterwards, this concept was expanded to a generalized $(\alpha ,\beta )-$ derivation as follows: $\vartheta \left(\varsigma \iota \right)=\vartheta \left(\varsigma \right)\alpha \left(\iota \right)+\beta \left(\varsigma \right)\partial \left(\iota \right)$ for any two elements $\varsigma ,\iota \in \Im$. Without any controversy, this concept covers the generalized derivation when $\alpha =\beta =I$ as well as the ordinary derivation when $\vartheta =\partial$ and $\alpha =\beta =I$, where I the identity map on ℑ. One of the basic problems in ring theory is to investigate the various conditions under which a ring ℑ becomes commutative. For this purpose, there has been a great deal of effort to link the commutativity of a prime or a semiprime ring ℑ with the existence of additive maps defined on it, such as a generalized $(\alpha ,\beta )$-derivation and an $(\alpha ,\beta )$-derivation that satisfy differential identities over the entire ring or any appropriate subset of it. For more details, the reader can refer—for example—to refs. [5,6,7,8]. As an extension of these studies, instead of proving commutativity on a prime or a semiprime ring, Almahdi et al. [9] strengthened it without imposing any restrictions on the ring ℑ. They proved that either $\Im /\wp$ is a commutative integral domain or $\partial (\Im )\subseteq \wp$, if ℑ admits a derivation ∂ that satisfies $\left[\right[\partial \left(\varsigma \right),\varsigma ],\iota ]\in \wp$ for any $\varsigma ,\iota \in \Im$, where ℘ is a prime ideal on ℑ, which is generalized by the second Posner’s Theorem. Before these authors, in ref. [10], Creedon generalized the first Posner’s Theorem in prime ideal with two iterates of derivations when a ring is restricted by a characteristic two. In this direction, studies and interests have been continued by many researchers; see for example refs. [11,12,13,14]. In this article, instead of considering a generalized derivation, we examine differential identities involving a generalized $(\alpha ,\beta )$-derivation $\vartheta$ associated with an $(\alpha ,\beta )$-derivation ∂. Consequently, we prove that either $\Im /\wp$ is a commutative integral domain or $\partial (\Im )\subseteq \wp$, where ℘ is a prime ideal of an arbitrary ring ℑ. Furthermore, we explore several sequels and special cases as corollaries of our results. Finally, we devote several examples to emphasize the necessity of the various hypotheses imposed in our theorems.

2. Preliminaries

For any pair of elements $\varsigma ,\iota \in \Im$, the symbol $[\varsigma ,\iota ]$ indicates the commutator $\varsigma \iota -\iota \varsigma$ while $\varsigma \circ \iota$ indicates the anticommutator $\varsigma \iota +\iota \varsigma$. The following identities will be used vastly throughout this paper to make access easier to the proofs of our theorems that hold for all $\varsigma ,\iota ,\kappa \in \Im$:

$$\begin{gathered} [\varsigma \iota ,\kappa ]=\varsigma [\iota ,\kappa ]+[\varsigma ,\kappa ]\iota \\ [\varsigma ,\iota \kappa ]=\iota [\varsigma ,\kappa ]+[\varsigma ,\iota ]\kappa \\ \varsigma \circ \left(\iota \kappa \right)=(\varsigma \circ \iota )\kappa -\iota [\varsigma ,\kappa ]=\iota (\varsigma \circ \kappa )+[\varsigma ,\iota ]\kappa \\ \left(\varsigma \iota \right)\circ \kappa =\varsigma (\iota \circ \kappa )-[\varsigma ,\kappa ]\iota =(\varsigma \circ \kappa )\iota +\varsigma [\iota ,\kappa ]. \end{gathered}$$

To develop our results, we exhibit the following important lemma:

Lemma 1 

([14], Lemma 1). Let ℑ be a ring. If ℘ is a prime ideal of ℑ, then $\Im /\wp$ is a commutative integral domain if any of the following holds, for every $\varsigma ,\iota \in \Im$:

(i) 

$[\varsigma ,\iota ]\in \wp ,$

(ii) 

$\varsigma \circ \iota \in \wp .$

3. Main Results

In the context of this paper, the pair $(\vartheta ,\partial )$ stands for a generalized $(\alpha ,\beta )$-derivation associated with an $(\alpha ,\beta )$-derivation ∂, where the two maps $\alpha ,\beta :\Im ⟶\Im$ are automorphisms on ℑ, unless we mention otherwise. Moreover, the map $I:\Im ⟶\Im$, defined by $I\left(\varsigma \right)=\varsigma$ for any $\varsigma \in \Im$, expresses the identity map on ℑ.

In ref. [15] (Lemma 2.1), Bera et al. showed that ∂ maps ℑ to $Z(\Im )$, if a semiprime ring ℑ admits a generalized ($\alpha$, $\beta$)-derivation $\vartheta$ associated with an ($\alpha$, $\beta$)-derivation ∂ such that $[\partial (\varsigma ),\beta (\varsigma \left)\right]\in Z(\Im )$ for every $\varsigma \in \Im$. Here, we will verify a similar result without imposing any restrictions on ℑ, as shown below:

Theorem 1. 

Assume that ∂ is an $(\alpha$, β)-derivation on an arbitrary ring ℑ such that $[\partial (\varsigma ),\beta (\varsigma \left)\right]\in \wp$ for every $\varsigma \in \Im$, where ℘ is a prime ideal of ℑ. Then $\Im /\wp$ is a commutative integral domain or the associated $(\alpha ,\beta )$-derivation ∂ maps ℑ to ℘.

Proof. 

The given hypothesis states

$$[\partial (\varsigma ),\beta (\varsigma \left)\right]\in \wp ,\mathrm{for}\mathrm{all}\varsigma \in \Im ,$$

(1)

applying the linearity in the previous equation, we obtain

$$[\partial (\varsigma ),\beta (\iota \left)\right]+[\partial (\iota ),\beta (\varsigma \left)\right]\in \wp ,\mathrm{for}\mathrm{all}\varsigma ,\iota \in \Im ,$$

(2)

if we set $\varsigma \iota$ instead of $\iota$ in Equation (2) and use it in Equation (1), we have

$$\partial \left(\varsigma \right)\left[\alpha \right(\iota ),\beta (\varsigma \left)\right]\in \wp ,\mathrm{for}\mathrm{all}\varsigma ,\iota \in \Im ,$$

again, we set $\iota \kappa$ instead of $\iota$ in the above relation to obtain

$$\partial \left(\varsigma \right)\alpha \left(\iota \right)\left[\alpha \right(\kappa ),\beta (\varsigma \left)\right]\in \wp ,\mathrm{for}\mathrm{all}\varsigma ,\iota ,\kappa \in \Im ,$$

(3)

according to the assumption that ℘ is prime and $\alpha$ is an automorphism on ℑ, we deduce either $\partial \left(\varsigma \right)\in \wp$ or $\left[\alpha \right(\kappa ),\beta (\varsigma \left)\right]\in \wp ,\mathrm{for}\mathrm{all}\varsigma ,\kappa \in \Im$. Let $\mathsf{\Gamma }=\{\varsigma \in \Im :\partial (\varsigma )\in \wp \}$ and $\mathsf{\Lambda }=\{\varsigma \in \Im :[\alpha \left(\kappa \right),\beta \left(\varsigma \right)]\in \wp ,\mathrm{for}\mathrm{all}\kappa \in \Im \}$. Then, it can be easily verified that both $\mathsf{\Gamma }$ and $\mathsf{\Lambda }$ are additive subgroups of ℑ and their union equals ℑ. Applying Brauer’s trick, we obtain either $\mathsf{\Gamma }=\Im$ or $\mathsf{\Lambda }=\Im$. If $\mathsf{\Gamma }=\Im$, then $\partial \left(\varsigma \right)\in \wp$, for all $\varsigma \in \Im$ and hence $\partial (\Im )\subseteq \wp$. On the other hand, if $\mathsf{\Lambda }=\Im$, then $\left[\alpha \right(\kappa ),\beta (\varsigma \left)\right]\in \wp ,\mathrm{for}\mathrm{all}\varsigma ,\kappa \in \Im$. In the previous relation, as $\alpha ,\beta$ are automorphisms on ℑ, it is possible to set $\kappa ={\alpha }^{-1}\left(\nu \right)$ and $\varsigma ={\beta }^{-1}\left(\mu \right)$ to obtain $[\nu ,\mu ]\in \wp ,\mathrm{for}\mathrm{all}\nu ,\mu \in \Im .$ Thus, $\Im /\wp$ is a commutative integral domain, according to Lemma 1. □

Remark 1. 

Lemma 2.1 of ref. [9] will be a special case of Theorem 1 by putting $\alpha =\beta =I$.

The conclusion of ref. [14], Proposition 1.3, is that either $\Im /\wp$ or $\partial (\Im )\subseteq \wp$, when ℑ admits a generalized derivation that satisfies $\left[\vartheta \right(\varsigma ),\varsigma ]\in \wp$, for every $\varsigma \in \Im$, where ℘ is a prime ideal of ℑ. Also, in ref. [11], Theorem 3.1, the same conclusion is obtained, when ℑ admits a multiplicative left-generalized $(\alpha ,\beta )$-derivation associated with an $(\alpha ,\beta )$-derivation ∂ that satisfies $\left[\alpha \right(\varsigma ),\vartheta (\iota \left)\right]\in \wp$ for each $\varsigma ,\iota \in \Im$, where $\alpha$ and $\beta$ are automorphisms of ℑ. The following theorem aims to discuss the effect of the identity $\left[\vartheta \right(\varsigma ),\alpha (\varsigma \left)\right]\in \wp$ for any $\varsigma \in \Im$ on the behavior of the ring ℑ.

Theorem 2. 

Assume that $(\vartheta ,\partial )$ is a generalized $(\alpha$, β)-derivation on an arbitrary ring ℑ such that $\left[\vartheta \right(\varsigma ),\alpha (\varsigma \left)\right]\in \wp$ for every $\varsigma \in \Im$, where ℘ is a prime ideal of ℑ. Then, $\Im /\wp$ is a commutative integral domain or the associated $(\alpha ,\beta )$-derivation ∂ maps ℑ to ℘.

Proof. 

For each $\varsigma \in \Im$, we have the following assumption:

$$\left[\vartheta \right(\varsigma ),\alpha (\varsigma \left)\right]\in \wp .$$

(4)

Using the linearity in the previous equation, we obtain

$$\left[\vartheta \right(\varsigma ),\alpha (\iota \left)\right]+\left[\vartheta \right(\iota ),\alpha (\varsigma \left)\right]\in \wp ,\mathrm{for}\mathrm{all}\varsigma ,\iota \in \Im .$$

Replacing $\iota$ with $\iota \varsigma$ in the last expression and using it in Equation (4) gives

$$\beta \left(\iota \right)[\partial (\varsigma ),\alpha (\varsigma \left)\right]+\left[\beta \right(\iota ),\alpha (\varsigma \left)\right]\partial \left(\varsigma \right)\in \wp ,\mathrm{for}\mathrm{all}\varsigma ,\iota \in \Im .$$

(5)

For each $\kappa \in \Im$, we put $\kappa \iota$ in the place of $\iota$ in Equation (5) to obtain

$$\begin{array}{c}\hfill \beta \left(\kappa \right)\beta \left(\iota \right)[\partial (\varsigma ),\alpha (\varsigma \left)\right]+\beta \left(\kappa \right)\left[\beta \right(\iota ),\alpha (\varsigma \left)\right]\partial \left(\varsigma \right)+\left[\beta \right(\kappa ),\alpha (\varsigma \left)\right]\beta \left(\iota \right)\partial \left(\varsigma \right)\in \wp .\end{array}$$

Multiplying Equation (5) from the left by $\beta \left(\kappa \right)$ and subtracting it from the previous equation, we obtain

$$\left[\beta \right(\kappa ),\alpha (\varsigma \left)\right]\beta \left(\iota \right)\partial \left(\varsigma \right)\in \wp ,\mathrm{for}\mathrm{all}\varsigma ,\iota ,\kappa \in \Im .$$

(6)

Now, applying a similar argument as that after Equation (3), we obtain the desired conclusion. □

In ref. [8], Rehman et al. showed that L is contained in the center of a prime ring $(\Im ,\ast )$ admitting a generalized $(\alpha ,\beta )$-derivation $\vartheta$ associated with an $(\alpha ,\beta )$-derivation ∂, that satisfies $\vartheta [\varsigma ,\iota ]-\alpha (\varsigma \circ \iota )=0$ for all $\varsigma ,\iota \in L$, where L is a Lie ideal and ∗ is an involution on ℑ. In the following theorem, we will see what happens when the prior identity involves a prime ideal ℘ of a ring ℑ that is neither prime nor equipped with ∗.

Theorem 3. 

Assume that $(\vartheta ,\partial )$ is a generalized $(\alpha$, β)-derivation on an arbitrary ring ℑ such that $\vartheta \left(\right[\varsigma ,\iota \left]\right)\pm \alpha (\varsigma \circ \iota )\in \wp$ for every $\varsigma ,\iota \in \Im$, where ℘ is a prime ideal of ℑ. Then, $\Im /\wp$ is a commutative integral domain or the associated $(\alpha ,\beta )$-derivation ∂ maps ℑ to ℘.

Proof. 

From the given assumption, we have

$$\vartheta \left(\right[\varsigma ,\iota \left]\right)\pm \alpha (\varsigma \circ \iota )\in \wp ,\mathrm{for}\mathrm{all}\varsigma ,\iota \in \Im ,$$

(7)

if $\vartheta =0$, then $\alpha (\varsigma \circ \iota )\in \wp$, for all $\varsigma ,\iota \in \Im$. The automorphism property of $\alpha$ implies that $\alpha \left(\varsigma \right)\circ \alpha \left(\iota \right)\in \wp$, for all $\varsigma ,\iota \in \Im$. We set $\varsigma ={\alpha }^{-1}\left(\nu \right)$ and $\iota ={\alpha }^{-1}\left(\mu \right)$ to obtain $\nu \circ \mu \in \wp$, for all $\nu ,\mu \in \Im$. Hence, using Lemma 1, $\Im /\wp$ is a commutative integral domain.

From now on, let $\vartheta \ne 0$. Then, for all $\varsigma ,\iota \in \Im$, we have

$$\vartheta \left(\right[\varsigma ,\iota \left]\right)\pm \alpha (\varsigma \circ \iota )\in \wp ,$$

(8)

setting $\iota \varsigma$ instead of $\iota$ in Equation (8) gives

$$\begin{array}{c}\hfill \vartheta [\varsigma ,\iota ]\alpha \left(\varsigma \right)+\beta [\varsigma ,\iota ]\partial \left(\varsigma \right)\pm \alpha (\varsigma \circ \iota )\alpha \left(\varsigma \right)\in \wp ,\mathrm{for}\mathrm{all}\varsigma ,\iota \in \Im ,\end{array}$$

if we multiply Equation (8) by $\alpha \left(\varsigma \right)$ from the right and comparing it with the previous equation, we obtain

$$\beta [\varsigma ,\iota ]\partial \left(\varsigma \right)\in \wp ,\mathrm{for}\mathrm{all}\varsigma ,\iota \in \Im ,$$

but $\beta$ is an automorphism on ℑ, so the previous equation can be rewritten as $\left[\beta \right(\varsigma ),\beta (\iota \left)\right]\partial \left(\varsigma \right)\in \wp \mathrm{for}\mathrm{all}\varsigma ,\iota \in \Im$. If we change $\iota$ by $\kappa \iota$ in the last relation and apply it, we find $\left[\beta \right(\varsigma ),\beta (\kappa \left)\right]\beta \left(\iota \right)\partial \left(\varsigma \right)\in \wp \mathrm{for}\mathrm{all}\varsigma ,\iota ,\kappa \in \Im$. By repeating the similar arguments and techniques after Equation (3), we obtain the desired result. □

Remark 2. 

Corollary 11(1) of ref. [13], will be an immediate consequence of Theorem 3 by putting $\alpha =\beta =I$.

As an application of the previous theorem, if ℑ is a prime ring, then we have the following corollary:

Corollary 1. 

Assume that $(\vartheta ,\partial )$ is a generalized $(\alpha$, β)-derivation on a prime ring ℑ such that $\vartheta \left(\right[\varsigma ,\iota \left]\right)\pm \alpha (\varsigma \circ \iota )=0$ for every $\varsigma ,\iota \in \Im$. Then, ℑ is either commutative or the $(\alpha ,\beta )$-associated derivation ∂ is zero (in this case, ϑ outputs a left centralizer).

The following theorem is an extension of ref. [7], Theorem 3.5.

Theorem 4. 

Assume that $(\vartheta ,\partial )$ is a generalized $(\alpha$, β)-derivation on an arbitrary ring ℑ such that $\vartheta (\varsigma \circ \iota )\in \wp$, for every $\varsigma ,\iota \in \Im$, where ℘ is a prime ideal of ℑ. Then, $\Im /\wp$ is a commutative integral domain or the associated $(\alpha ,\beta )$-derivation ∂ maps ℑ to ℘.

Proof. 

We start with the given assumption

$$\vartheta (\varsigma \circ \iota )\in \wp ,\mathrm{for}\mathrm{all}\varsigma ,\iota \in \Im ,$$

(9)

and set $\iota \varsigma$ instead of $\iota$ to have

$$\begin{array}{c}\hfill \vartheta (\varsigma \circ \iota )\alpha \left(\varsigma \right)+\beta (\varsigma \circ \iota )\partial \left(\varsigma \right)\in \wp ,\mathrm{for}\mathrm{all}\varsigma ,\iota \in \Im .\end{array}$$

We multiply Equation (9) by $\alpha \left(\varsigma \right)$ from the right and subtract it from the previous equation to obtain

$$\beta (\varsigma \circ \iota )\partial \left(\varsigma \right)\in \wp ,\mathrm{for}\mathrm{all}\varsigma ,\iota \in \Im .$$

Putting $\kappa \iota$ instead of $\iota$ in the previous equation and using it give $\beta \left(\right[\varsigma ,\kappa \left]\right)\beta \left(\iota \right)\partial \left(\varsigma \right)\in \wp$ for each $\varsigma ,\iota ,\kappa \in \Im$, this equation is similar to Equation (3); so, following similar arguments and techniques with some necessary modifications leads to the desired result. □

Theorem 5. 

Assume that $(\vartheta ,\partial )$ is a generalized $(\alpha$, β)-derivation on an arbitrary ring ℑ such that $\vartheta \left({\varsigma }^2\right)\in \wp$ for every $\varsigma \in \Im$, where ℘ is a prime ideal of ℑ. Then, $\Im /\wp$ is a commutative integral domain or otherwise the associated $(\alpha ,\beta )$-derivation ∂ maps ℑ to ℘.

Proof. 

The given assumption states that

$$\vartheta \left({\varsigma }^2\right)\in \wp ,\mathrm{for}\mathrm{all}\varsigma \in \Im ,$$

Linearizing the previous equation and then applying it give $\vartheta (\varsigma \iota +\iota \varsigma )\in \wp$ for all $\varsigma ,\iota \in \Im$, that is, $\vartheta (\varsigma \circ \iota )\in \wp$ for all $\varsigma ,\iota \in \Im$ which is the same as the identity in Theorem 4. Therefore, following it induces the desired conclusion. □

Remark 3. 

It is easy to verify that, if ϑ is a generalized (α, β)-derivation associated with an (α, β)-derivation ∂, then $\vartheta \pm \alpha$ is also a generalized (α, β)-derivation associated with an (α, β)-derivation ∂.

Applying the previous remark in Theorem 4 leads to the following result:

Theorem 6. 

Assume that $(\vartheta ,\partial )$ is a generalized $(\alpha$, β)-derivation on an arbitrary ring ℑ such that $\vartheta (\varsigma \circ \iota )\pm \alpha (\varsigma \circ \iota )\in \wp$ for every $\varsigma ,\iota \in \Im$, where ℘ is a prime ideal of ℑ. Then, $\Im /\wp$ is a commutative integral domain or ∂ maps ℑ to ℘.

Proof. 

Given that $\vartheta$ is a generalized ($\alpha$, $\beta$)-derivation with an ($\alpha$, $\beta$)-derivation ∂, hence, according to Remark 3, $\vartheta \pm \alpha$ is also a generalized ($\alpha$, $\beta$)-derivation that satisfies Identity 9. Thus, $(\vartheta \pm \alpha )(\varsigma \circ \iota )=\vartheta (\varsigma \circ \iota )\pm \alpha (\varsigma \circ \iota )\in \wp$ for each $\varsigma ,\iota \in \Im$. Therefore, by employing similar arguments as those mentioned above, we can achieve the desired outcome. □

The question which arises here is whether Theorem 6 is still valid in the case of a commutator. The following theorem provides the answer:

Theorem 7. 

Assume that $(\vartheta ,\partial )$ is a generalized $(\alpha$, β)-derivation on an arbitrary ring ℑ such that $\vartheta \left(\right[\varsigma ,\iota \left]\right)\pm \alpha [\varsigma ,\iota ]\in \wp$ for every $\varsigma ,\iota \in \Im$, where ℘ is a prime ideal of ℑ. Then, $\Im /\wp$ is a commutative integral domain or ∂ maps ℑ to ℘.

Proof. 

Applying arguments and techniques similar to those used to prove Theorem 3 with a few necessary modifications yields the required proof. □

As an application of the previous theorem, we present the following corollary, which is a generalization of ref. [13], Theorem 6:

Corollary 2. 

Assume that $(\vartheta ,\partial )$ is a generalized $(\alpha$, β)-derivation on an arbitrary ring ℑ such that $\vartheta \left(\right[\varsigma ,\iota \left]\right)\in \wp$ for every $\varsigma ,\iota \in \Im$, where ℘ is a prime ideal of ℑ. Then, $\Im /\wp$ is a commutative integral domain or ∂ maps ℑ to ℘.

Proof. 

Note that Remark 3 states that $\vartheta \pm \alpha$ is a generalized ($\alpha$, $\beta$)-derivation associated with an ($\alpha$, $\beta$)-derivation ∂. Hence, we can immediately derive the proof by applying the identity $\vartheta \left(\right[\varsigma ,\iota \left]\right)\pm \alpha [\varsigma ,\iota ]\in \wp$ for every $\varsigma ,\iota \in \Im$ in Theorem 7. □

Remark 4. 

In the previous corollary, if we choose both α and β to be equal to the identity map, then ref. [13], Corollary 11(2), is directly taken as a special case.

In ref. [15], Theorem 3.1, Bera et al. discuss the identities $\varrho \left(\varsigma \iota \right)+\partial \left(\varsigma \right)\vartheta \left(\iota \right)+\alpha \left(\varsigma \iota \right)=0$, $\varrho \left(\varsigma \iota \right)+\partial \left(\iota \right)\vartheta \left(\varsigma \right)+\alpha \left(\iota \varsigma \right)=0$ and $\varrho \left(\varsigma \iota \right)+\partial \left(\varsigma \right)\vartheta \left(\iota \right)+\alpha \left(\iota \varsigma \right)=0$ for all $\varsigma ,\iota \in \aleph$, where ℵ is a left ideal of a semiprime ring ℑ and $\varrho ,\vartheta$ are two generalized ($\alpha$, $\beta$)-derivations associated with ($\alpha$, $\beta$)-derivations $\zeta ,\partial$, respectively. In the following theorem, without imposing any restrictions on the ring ℑ, we will discuss analog identities in a prime ideal for one generalized ($\alpha$, $\beta$)-derivation $\vartheta$ associated with an ($\alpha$, $\beta$)-derivation ∂.

Theorem 8. 

Assume that $(\vartheta ,\partial )$ is a generalized $(\alpha$, β)-derivation on an arbitrary ring ℑ such that $\vartheta \left(\varsigma \iota \right)+\partial \left(\varsigma \right)\vartheta \left(\iota \right)+\alpha \left(\varsigma \iota \right)\in \wp$ for every $\varsigma ,\iota \in \Im$, where ℘ is a prime ideal of ℑ. Then, the associated $(\alpha ,\beta )$-derivation ∂ maps ℑ to ℘ and $(\vartheta +\alpha )(\Im )\subseteq \wp .$

Proof. 

We have

$$\vartheta \left(\varsigma \iota \right)+\partial \left(\varsigma \right)\vartheta \left(\iota \right)+\alpha \left(\varsigma \iota \right)\in \wp ,\mathrm{for}\mathrm{all}\varsigma ,\iota \in \Im .$$

(10)

For each $\kappa \in \Im$, we put $\iota \kappa$ instead of $\iota$ in Equation (10) to obtain

$$\vartheta \left(\varsigma \iota \right)\alpha \left(\kappa \right)+\beta \left(\varsigma \iota \right)\partial \left(\kappa \right)+\partial \left(\varsigma \right)\left(\vartheta \right(\iota \left)\alpha \right(\kappa )+\beta (\iota )\partial (\kappa \left)\right)+\alpha \left(\varsigma \iota \right)\alpha \left(\kappa \right)\in \wp .$$

(11)

Multiplying Equation (10) from the right by $\alpha \left(\kappa \right)$ and then comparing with Equation (11) yields

$$\beta \left(\varsigma \iota \right)\partial \left(\kappa \right)+\partial \left(\varsigma \right)\beta \left(\iota \right)\partial \left(\kappa \right)\in \wp ,\mathrm{for}\mathrm{all}\varsigma ,\iota ,\kappa \in \Im .$$

(12)

Once again, if we put $\varsigma \iota$ instead of $\iota$ in Equation (12), we obtain

$$\beta \left(\varsigma \right)\beta \left(\varsigma \iota \right)\partial \left(\kappa \right)+\partial \left(\varsigma \right)\beta \left(\varsigma \right)\beta \left(\iota \right)\partial \left(\kappa \right)\in \wp ,\mathrm{for}\mathrm{all}\varsigma ,\iota ,\kappa \in \Im .$$

(13)

Multiplying Equation (12) from the left by $\beta \left(\varsigma \right)$ and then comparing with Equation (13) yields

$$\left[\beta \right(\varsigma ),\partial (\varsigma \left)\right]\beta \left(\iota \right)\partial \left(\kappa \right)\in \wp ,\mathrm{for}\mathrm{all}\varsigma ,\iota ,\kappa \in \Im ,$$

that is

$$\left[\beta \right(\varsigma ),\partial (\varsigma \left)\right]\Im \partial \left(\kappa \right)\subseteq \wp ,\mathrm{for}\mathrm{all}\varsigma ,\kappa \in \Im .$$

Applying the hypothesis that ℘ is prime together with Brauer’s trick, we obtain either $\left[\beta \right(\varsigma ),\partial (\varsigma \left)\right]\in \wp$ for all $\varsigma \in \Im$, or $\partial \left(\kappa \right)\in \wp$ for all $\kappa \in \Im$.

We begin by assuming that for each $\kappa \in \Im$, $\partial \left(\kappa \right)\in \wp$. Hence, Equation (10) is reduced to $\vartheta \left(\varsigma \right)\alpha \left(\iota \right)+\alpha \left(\varsigma \iota \right)\in \wp$ for every $\varsigma ,\iota \in \Im$. Thus, $\left(\vartheta \right(\varsigma )+\alpha (\varsigma \left)\right)\alpha \left(\iota \right)\in \wp$ for every $\varsigma ,\iota \in \Im$. Since ℘ is a prime ideal and $\alpha$ is an automorphism, then $\vartheta \left(\varsigma \right)+\alpha \left(\varsigma \right)\in \wp$ for every $\varsigma \in \Im$. Therefore, $(\vartheta +\alpha )(\Im )\subseteq \wp$.

On the other hand, if $\left[\beta \right(\varsigma ),\partial (\varsigma \left)\right]\in \wp$, $\mathrm{for}\mathrm{all}\varsigma \in \Im$, then by Theorem 1, either $\Im /\wp$ is a commutative integral domain or ∂ maps ℑ to ℘. The second case was discussed above, so we consider the case that $\Im /\wp$ is a commutative integral domain. Hence, Equation (12): $\beta \left(\varsigma \iota \right)\partial \left(\kappa \right)+\partial \left(\varsigma \right)\beta \left(\iota \right)\partial \left(\kappa \right)\in \wp ,$ $\mathrm{for}\mathrm{all}\varsigma ,\iota ,\kappa \in \Im$ can be rewritten as $\beta \left(\iota \right)\left(\beta \right(\varsigma )+\partial (\varsigma \left)\right)\partial \left(\kappa \right)\in \wp ,$$\mathrm{for}\mathrm{all}\varsigma ,\iota ,\kappa \in \Im$. Using the two assumptions that $\beta$ is an automorphism and $\wp \ne \Im$ is a prime ideal gives $\left(\beta \right(\varsigma )+\partial (\varsigma \left)\right)\partial \left(\kappa \right)\in \wp ,$$\mathrm{for}\mathrm{all}\varsigma ,\kappa \in \Im$. Now, we put $\ell \varsigma$ instead of $\varsigma$ in the last equation and use it to have $\partial (\ell )\alpha \left(\varsigma \right)\partial \left(\kappa \right)\in \wp$ for all $\varsigma ,\ell ,\kappa \in \Im$. Since $\alpha$ is an automorphism, the previous equation becomes $\partial (\ell )\Im \partial \left(\kappa \right)\in \wp$ for all $\ell ,\kappa \in \Im$. Thus, either $\partial (\ell )\in \wp$ or $\partial \left(\kappa \right)\in \wp$. Both cases yield $\partial (\Im )\subseteq \wp$. Therefore, as above, we conclude that $(\vartheta +\alpha )(\Im )\subseteq \wp$ as required. □

In the case that either $\alpha =\beta =I$ or ℑ is prime, we derive the following two corollaries, respectively:

Corollary 3. 

Assume that $(\vartheta ,\partial )$ is a generalized derivation on an arbitrary ring ℑ such that $\vartheta \left(\varsigma \iota \right)+\partial \left(\varsigma \right)\vartheta \left(\iota \right)+\varsigma \iota \in \wp$ for every $\varsigma ,\iota \in \Im$, where ℘ is a prime ideal of ℑ. Then, ∂ maps ℑ to ℘ and $(\vartheta +I)(\Im )\subseteq \wp$.

Corollary 4. 

Assume that $(\vartheta ,\partial )$ is a generalized $(\alpha$, β)-derivation on a prime ring ℑ such that $\vartheta \left(\varsigma \iota \right)+\partial \left(\varsigma \right)\vartheta \left(\iota \right)+\alpha \left(\varsigma \iota \right)=0$ for every $\varsigma ,\iota \in \Im$. Then, the associated $(\alpha ,\beta )$-derivation ∂ is zero and $\vartheta =-\alpha .$

Theorem 9. 

Assume that $(\vartheta ,\partial )$ is a generalized $(\alpha$, β)-derivation on an arbitrary ring ℑ such that $\vartheta \left(\varsigma \iota \right)+\partial \left(\iota \right)\vartheta \left(\varsigma \right)+\alpha \left(\iota \varsigma \right)\in \wp$ for every $\varsigma ,\iota \in \wp$, where ℘ is a prime ideal of ℑ. Then, $\Im /\wp$ is a commutative integral domain.

Proof. 

According to the given hypothesis, we have

$$\vartheta \left(\varsigma \iota \right)+\partial \left(\iota \right)\vartheta \left(\varsigma \right)+\alpha \left(\iota \varsigma \right)\in \wp ,\mathrm{for}\mathrm{all}\varsigma ,\iota \in \Im .$$

(14)

Setting $\varsigma \iota$ instead of $\varsigma$ in Equation (14) gives

$$\vartheta \left(\varsigma \iota \right)\alpha \left(\iota \right)+\beta \left(\varsigma \iota \right)\partial \left(\iota \right)+\partial \left(\iota \right)\left(\vartheta \right(\varsigma \left)\alpha \right(\iota )+\beta (\varsigma )\partial (\iota \left)\right)+\alpha \left(\iota \varsigma \right)\alpha \left(\iota \right)\in \wp .$$

(15)

Multiplying Equation (14) from the right by $\alpha \left(\iota \right)$ and then comparing with Equation (15) yields

$$\beta \left(\varsigma \iota \right)\partial \left(\iota \right)+\partial \left(\iota \right)\beta \left(\varsigma \right)\partial \left(\iota \right)\in \wp ,\mathrm{for}\mathrm{all}\varsigma ,\iota \in \Im .$$

(16)

If we set $\iota \varsigma$ instead of $\varsigma$ in Equation (16), we obtain

$$\beta \left(\iota \right)\beta \left(\varsigma \iota \right)\partial \left(\iota \right)+\partial \left(\iota \right)\beta \left(\iota \right)\beta \left(\varsigma \right)\partial \left(\iota \right)\in \wp ,\mathrm{for}\mathrm{all}\varsigma ,\iota \in \Im .$$

(17)

Now, multiplying Equation (16) from the left by $\beta \left(\iota \right)$ and comparing with Equation (17) yields

$$\left[\beta \right(\iota ),\partial (\iota \left)\right]\beta \left(\varsigma \right)\partial \left(\iota \right)\in \wp ,\mathrm{for}\mathrm{all}\varsigma ,\iota \in \Im .$$

The automorphism property of $\beta$ gives

$$\left[\beta \right(\iota ),\partial (\iota \left)\right]\Im \partial \left(\iota \right)\subseteq \wp ,\mathrm{for}\mathrm{all}\iota \in \Im .$$

where ℘ is prime, and applying Brauer’s trick implies either $\left[\beta \right(\iota ),\partial (\iota \left)\right]\in \wp$ for all $\iota \in \Im$, or $\partial \left(\iota \right)\in \wp$ for all $\iota \in \Im$. We start with the first case when $\left[\beta \right(\iota ),\partial (\iota \left)\right]\in \wp$ for all $\iota \in \Im$ and apply Theorem 1, which implies that either $\Im /\wp$ is a commutative integral domain or ∂ maps ℑ to ℘. When ∂ maps ℑ to ℘, then Equation (14) can be reduced to $\vartheta \left(\varsigma \right)\alpha \left(\iota \right)+\alpha \left(\iota \varsigma \right)\in \wp$ for every $\varsigma ,\iota \in \Im$. We set $\iota \kappa$ instead of $\iota$ in the previous equation and apply some calculations to obtain $\alpha \left(\iota \right)\alpha \left(\kappa \right)\alpha \left(\varsigma \right)-\alpha \left(\iota \right)\alpha \left(\varsigma \right)\alpha \left(\kappa \right)\in \wp$ for all $\varsigma ,\iota ,\kappa \in \Im$, which is equivalent to $\alpha \left(\iota \right)\left[\alpha \right(\kappa ),\alpha (\varsigma \left)\right]\in \wp$ for any $\varsigma ,\iota ,\kappa \in \Im$. Since $\alpha$ is an automorphism on ℑ, the previous expression becomes $\iota [\kappa ,\varsigma ]\in \wp$ for any $\varsigma ,\iota ,\kappa \in \Im$, which means that $\Im [\kappa ,\varsigma ]\subseteq \wp$. As ℘ is prime and does not equal to ℑ, then the last relation becomes $[\kappa ,\varsigma ]\in \wp$ for any $\varsigma ,\kappa \in \Im$. Then, applying Lemma 1 implies that $\Im /\wp$ is a commutative integral domain which completes the proof. □

Corollary 5. 

Assume that $(\vartheta ,\partial )$ is a generalized derivation on an arbitrary ring ℑ such that $\vartheta \left(\varsigma \iota \right)+\partial \left(\iota \right)\vartheta \left(\varsigma \right)+\iota \varsigma \in \wp$ for every $\varsigma ,\iota \in \Im$, where ℘ is a prime ideal of ℑ. Then, $\Im /\wp$ is a commutative integral domain.

Corollary 6. 

Assume that $(\vartheta ,\partial )$ is a generalized derivation on a prime ring ℑ such that $\vartheta \left(\varsigma \iota \right)+\partial \left(\iota \right)\vartheta \left(\varsigma \right)+\iota \varsigma =0$ for every $\varsigma ,\iota \in \Im$. Then, ℑ is commutative.

Theorem 10. 

Assume that $(\vartheta ,\partial )$ is a generalized $(\alpha$, β)-derivation on an arbitrary ring ℑ such that $\vartheta \left(\varsigma \iota \right)+\partial \left(\varsigma \right)\vartheta \left(\iota \right)+\alpha \left(\iota \varsigma \right)\in \wp$ for every $\varsigma ,\iota \in \Im$, where ℘ is a prime ideal of ℑ. Then, $\Im /\wp$ is a commutative integral domain.

Proof. 

By this hypothesis, we have

$$\vartheta \left(\varsigma \iota \right)+\partial \left(\varsigma \right)\vartheta \left(\iota \right)+\alpha \left(\iota \varsigma \right)\in \wp ,\mathrm{for}\mathrm{all}\varsigma ,\iota \in \Im ,$$

(18)

for any $\kappa \in \Im$, set $\iota \kappa$ instead of $\iota$ in Equation (18) to have

$$\vartheta \left(\varsigma \iota \right)\alpha \left(\kappa \right)+\beta \left(\varsigma \iota \right)\partial \left(\kappa \right)+\partial \left(\varsigma \right)\vartheta \left(\iota \right)\alpha \left(\kappa \right)+\partial \left(\varsigma \right)\beta \left(\iota \right)\partial \left(\kappa \right)+\alpha \left(\iota \kappa \varsigma \right)\in \wp .$$

(19)

Then, multiplying Equation (18) from the right by $\alpha \left(\kappa \right)$ and then comparing with Equation (19) yields

$$\beta \left(\varsigma \iota \right)\partial \left(\kappa \right)+\partial \left(\varsigma \right)\beta \left(\iota \right)\partial \left(\kappa \right)+\alpha \left(\iota \kappa \varsigma \right)-\alpha \left(\iota \varsigma \kappa \right)\in \wp ,\mathrm{for}\mathrm{all}\varsigma ,\iota ,\kappa \in \Im .$$

(20)

Choose $\varsigma =\kappa$ in Equation (20) to obtain

$$\beta \left(\varsigma \iota \right)\partial \left(\varsigma \right)+\partial \left(\varsigma \right)\beta \left(\iota \right)\partial \left(\varsigma \right)\in \wp ,\mathrm{for}\mathrm{all}\varsigma ,\iota \in \Im .$$

The previous equation is similar to Equation (16), so we repeat similar arguments and techniques to obtain the desired goal. □

We can derive the following two corollaries:

Corollary 7. 

Assume that $(\vartheta ,\partial )$ is a generalized derivation on an arbitrary ring ℑ such that $\vartheta \left(\varsigma \iota \right)+\partial \left(\varsigma \right)\vartheta \left(\iota \right)+\iota \varsigma \in \wp$ for every $\varsigma ,\iota \in \Im$, where ℘ is a prime ideal of ℑ. Then, $\Im /\wp$ is a commutative integral domain.

Corollary 8. 

Assume that $(\vartheta ,\partial )$ is a generalized derivation on a prime ring ℑ such that $\vartheta \left(\varsigma \iota \right)+\partial \left(\varsigma \right)\vartheta \left(\iota \right)+\iota \varsigma =0$ for every $\varsigma ,\iota \in \Im$. Then, ℑ is a commutative.

Finally, we devoted the following examples to emphasize the necessity of the various hypotheses imposed in our theorems:

Example 1. 

Let ℑ be the quaternions ring $\mathbb{H}$, that is

$$\Im =\mathbb{H}=\{\mu =\alpha +\beta i+\gamma j+\delta k|\alpha ,\beta ,\gamma ,\delta \in \mathbb{R}\}.$$

and let $\wp =\left\{0\right\}$ be a prime ideal of the quaternions ring $\mathbb{H}$. Define $(\vartheta ,\partial ),\alpha ,\beta :\Im ⟶\Im$ by

$$\vartheta \left(\mu \right)=\partial \left(\mu \right)=n\mu ,\alpha \left(\mu \right)=2n\mu ,\mathrm{and}\beta \left(\mu \right)=-2(n-1)\mu$$

where $n\in \mathbb{Z}$. It is easy to see that $(\vartheta ,\partial )$ is a generalized $(\alpha ,\beta )$-derivation associated with ∂ where α and β are automorphisms on ℑ. Furthermore, we can see that $[\partial (\varsigma ),\beta (\varsigma \left)\right]\in \wp$ and $\left[\vartheta \right(\varsigma ),\alpha (\varsigma \left)\right]\in \wp$ for every $\varsigma ,\iota \in \Im$, although $\Im /\wp$ is not commutative and $\partial (\Im )⊈\wp$.

Example 2. 

Let Λ be a ring with property ${\rho }^2=0$ and let $\Im =\left\{\left(\begin{array}{cc}\varsigma & \iota \\ 0& \varsigma \end{array}\right)\right|\varsigma ,\iota \in \mathsf{\Lambda }\}$. Since ${(\rho +\varrho )}^2=0$ and ${(\rho -\varrho )}^2=0$, then $\rho \circ \varrho =0$ and $[\rho ,\varrho ]=0$ for any $\rho ,\varrho \in \mathsf{\Lambda }$. Let $\wp =\left\{\left(\begin{array}{cc}0& \iota \\ 0& 0\end{array}\right)\right\}$. Define $(\vartheta ,\partial ),\alpha ,\beta :\Im ⟶\Im$ by $\vartheta \left(\left(\begin{array}{cc}\varsigma & \iota \\ 0& \varsigma \end{array}\right)\right)=\partial \left(\left(\begin{array}{cc}\varsigma & \iota \\ 0& \varsigma \end{array}\right)\right)=\left(\begin{array}{cc}0& \iota \\ 0& 0\end{array}\right)$, $\alpha \left(\left(\begin{array}{cc}\varsigma & \iota \\ 0& \varsigma \end{array}\right)\right)=\left(\begin{array}{cc}0& 0\\ 0& \varsigma \end{array}\right)$ and $\beta \left(\left(\begin{array}{cc}\varsigma & \iota \\ 0& \varsigma \end{array}\right)\right)=\left(\begin{array}{cc}\varsigma & 0\\ 0& 0\end{array}\right)$. It is easy to check that $(\vartheta ,\partial )$ is a generalized $(\alpha ,\beta )$-derivation of ℑ. Furthermore, $\left(i\right)[\partial (\varsigma ),\beta (\varsigma \left)\right]\in \wp$, $\left(ii\right)\left[\vartheta \right(\varsigma ),\alpha (\varsigma \left)\right]\in \wp$, $\left(iii\right)\vartheta \left(\right[\varsigma ,\iota \left]\right)\pm \alpha (\varsigma \circ \iota )\in \wp$, $\left(iv\right)\vartheta (\varsigma \circ \iota )\in \wp$, $\left(v\right)\vartheta \left({\varsigma }^2\right)\in \wp$, $\left(vi\right)\vartheta (\varsigma \circ \iota )\pm \alpha (\varsigma \circ \iota )\in \wp$, $\left(vii\right)\vartheta \left(\right[\varsigma ,\iota \left]\right)\in \wp$ for every $\varsigma ,\iota \in \Im$. Also, we note that $\Im /\wp$ is not an integral domain and ℘ is not a prime ideal as $\left(\begin{array}{cc}0& \iota \\ 0& 0\end{array}\right)\left(\begin{array}{cc}0& \varsigma \\ 0& 0\end{array}\right)\in \wp$, but neither $\left(\begin{array}{cc}0& \varsigma \\ 0& 0\end{array}\right)$$\in \wp$ nor $\left(\begin{array}{cc}0& \iota \\ 0& 0\end{array}\right)$$\in \wp$. Also, α and β are not automorphisms whenever $\partial (\Im )\subseteq \wp$.

Example 3. 

Let $\Im =\mathbb{Z}\left[\varsigma \right]$ be the ring of polynomial with integers coefficient, and let $\wp =<{\varsigma }^2>$. Define $(\vartheta ,\partial ),\alpha ,\beta :\mathbb{Z}\left[\varsigma \right]⟶\mathbb{Z}\left[\varsigma \right]$ by $\vartheta \left(f\left(\varsigma \right)\right)=\partial \left(f\left(\varsigma \right)\right)=\varsigma f^{\prime }\left(\varsigma \right)$, $\alpha \left(f\left(\varsigma \right)\right)=-f^{\prime }\left(\varsigma \right)$, and $\beta \left(f\left(\varsigma \right)\right)=2f^{\prime }\left(\varsigma \right)$. It is easy to verify that $(\vartheta ,\partial )$ is a generalized $(\alpha ,\beta )$-derivation associated with ∂. One can check that $\left(i\right)[\partial (\varsigma ),\beta (\varsigma \left)\right]\in \wp$, $\left(ii\right)\left[\vartheta \right(\varsigma ),\alpha (\varsigma \left)\right]\in \wp$, $\left(iii\right)\vartheta \left(\right[\varsigma ,\iota \left]\right)\pm \alpha [\varsigma ,\iota ]\in \wp$, $\left(iv\right)\vartheta \left(\right[\varsigma ,\iota \left]\right)\in \wp$ for every $\varsigma ,\iota \in \Im$, although neither $\mathbb{Z}\left[\varsigma \right]/<{\varsigma }^2>$ is an integral domain nor $\partial (\Im )\subseteq \wp$. Also, ℘ is not a prime ideal in $\mathbb{Z}\left[\varsigma \right]$, since $\varsigma (\varsigma +{\varsigma }^3)\in \wp$, but $\varsigma ,\varsigma +{\varsigma }^3\notin \wp$ as well as α and β are not automorphisms. So, the primeness of ℘ and the automorphism property of α and β are necessary conditions.

Example 4. 

Let $\Im =\left\{\left(\begin{array}{ccc}0& 0& 0\\ \varsigma & 0& 0\\ \iota & \kappa & 0\end{array}\right)\right|\varsigma ,\iota ,\kappa \in \mathbb{H}\}$, where $\mathbb{H}$ is the Hamilton ring as in Example 1 and let $\wp =\left\{\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)\right\}$. Define $(\vartheta ,\partial ),\alpha ,\beta :\Im ⟶\Im$ by $\vartheta \left(\begin{array}{ccc}0& 0& 0\\ \varsigma & 0& 0\\ \iota & \kappa & 0\end{array}\right)=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& \kappa & 0\end{array}\right)$, $\partial \left(\begin{array}{ccc}0& 0& 0\\ \varsigma & 0& 0\\ \iota & \kappa & 0\end{array}\right)=\left(\begin{array}{ccc}0& 0& 0\\ \varsigma & 0& 0\\ 0& 0& 0\end{array}\right)$, $\alpha \left(\begin{array}{ccc}0& 0& 0\\ \varsigma & 0& 0\\ \iota & \kappa & 0\end{array}\right)=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 2\kappa & 0& 0\end{array}\right)$, and $\beta \left(\begin{array}{ccc}0& 0& 0\\ \varsigma & 0& 0\\ \iota & \kappa & 0\end{array}\right)=\left(\begin{array}{ccc}0& 0& 0\\ 2\kappa & 0& 0\\ 0& 0& 0\end{array}\right)$. It is easy to verify that $(\vartheta ,\partial )$ is a generalized $(\alpha ,\beta )$-derivation associated with an $(\alpha ,\beta )$-derivation ∂. Also, one can check that $\left(i\right)[\partial (\varsigma ),\beta (\varsigma \left)\right]\in \wp$, $\left(ii\right)\vartheta \left(\right[\varsigma ,\iota \left]\right)\pm \alpha (\varsigma \circ \iota )\in \wp$, $\left(iii\right)\vartheta (\varsigma \circ \iota )\in \wp$, $\left(iv\right)\vartheta \left({\varsigma }^2\right)\in \wp$, $\left(v\right)\vartheta (\varsigma \circ \iota )\pm \alpha (\varsigma \circ \iota )\in \wp$, $\left(vi\right)\vartheta [\varsigma ,\iota ]\pm [\varsigma ,\iota ]\in \wp$, $\left(vii\right)\vartheta \left(\right[\varsigma ,\iota \left]\right)\in \wp$, $\left(viii\right)\vartheta \left(\varsigma \iota \right)+\partial \left(\varsigma \right)\vartheta \left(\iota \right)+\alpha \left(\varsigma \iota \right)\in \wp$, $\left(ix\right)\vartheta \left(\varsigma \iota \right)+\partial \left(\iota \right)\vartheta \left(\varsigma \right)+\alpha \left(\iota \varsigma \right)\in \wp$, for all $\varsigma ,\iota \in \Im$. However, neither $\Im /\wp$ commutative nor ∂ maps the ring ℑ to a prime ideal ℘. Note that ℘ is not a prime ideal of ℑ since ${\left(\begin{array}{ccc}0& 0& 0\\ \varsigma & 0& 0\\ 0& 0& 0\end{array}\right)}^2\in \wp$ but $\left(\begin{array}{ccc}0& 0& 0\\ \varsigma & 0& 0\\ 0& 0& 0\end{array}\right)\notin \wp$.

4. Conclusions

In the current article, we continued the study of generalized $(\alpha ,\beta )$-derivation associated with $(\alpha ,\beta )$-derivation via a contemporary approach wherein we assume the ring ℑ is without restriction and the studied identities involved in prime ideal ℘. We have reached the following results: associated derivation maps a ring ℑ to ℘, or a quotient ring of ℑ by prime ideal ℘ becomes a commutative integral domain, or a combination of generalized $(\alpha ,\beta )$-derivation with automorphism $\alpha$ maps a ring ℑ to ℘, where one or more holds, as proven in this article. We conclude with four examples clarifying the necessity of the considered assumption herein.