On a Quotient Ring That Satisfies Certain Identities via Generalized Reverse Derivations

Abstract

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

1. Introduction

The study of derivations on rings plays an important role and has many applications in other areas of mathematics, such as analysis, algebraic geometry, and the properties of algebraic systems. These applications are outside the scope of current study.

In this article, ℜ is an associative ring and $Z(\Re )$ is its center. A proper ideal $\rho$ of ℜ is prime if, for each pair of elements $\xi$ and $\eta$ in ℜ, the condition $\xi \Re \eta \subseteq \rho$ implies that either $\xi$ belongs to $\rho$ or $\eta$ belongs to $\rho$. A ring ℜ is prime if and only if the set $\left\{0\right\}$ is a prime ideal of ℜ. A domain is a ring that does not have any non-zero divisors.

An additive mapping $\partial :\Re \to \Re$ is called a derivation if it satisfies the equation $\partial \left(\xi \eta \right)=\partial \left(\xi \right)\eta +\xi \partial \left(\eta \right)$ for all $\xi ,\eta \in \Re$. An additive mapping $\vartheta :\Re \to \Re$ is a generalized derivation associated with the derivation ∂ if the equation $\vartheta \left(\xi \eta \right)=\vartheta \left(\xi \right)\eta +\xi \partial \left(\eta \right)$ is satisfied for all $\xi ,\eta \in \Re$. For a fixed $r\in \Re$, a mapping ${\partial }_r:\Re \to \Re$ such that ${\partial }_r\left(\xi \right)=[r,\xi ]$ for any $\xi \in \Re$ is a derivation, which is called the inner derivation induced by r. For a non-trivial example of a derivation on a non-commutative ring, the reader can refer to [1] (Example 2.2).

The concept of a reverse derivation was initially defined by Herstein in [2] when he proved that the prime ring ℜ is a commutative integral domain whenever the imposed derivation is a Jordan derivation. It was defined to be an additive mapping $\partial :\Re \to \Re$ that satisfies the equation $\partial \left(\xi \eta \right)=\partial \left(\eta \right)\xi +\eta \partial \left(\xi \right)$ for any $\xi ,\eta \in \Re$. It can be noted that, in the case of Lie algebras, the concept of a reverse derivation is analogy to the concept of the antiderivation. According to this fact, several authors have studied the reverse derivation on algebra and subalgebra (see, for example [3,4,5]). In ref. [6], a study was conducted by Samman et al. on the reverse derivation of the semiprime ring.

In [7], Aboubakr et al. discussed the correlation between a generalized reverse derivation and a generalized derivation on a semiprime ring. A generalized reverse derivation is defined as an additive map $\vartheta$ that satisfies the equation $\vartheta \left(\xi \eta \right)=\vartheta \left(\eta \right)\xi +\eta \partial \left(\xi \right)$ for all $\xi$ and $\eta$ in ℜ, where ∂ is a reverse derivation of ℜ. In the previous literature, there are numerous non-trivial examples of generalized reverse derivations on non-commutative rings. For example, please see reference [8]. Furthermore, we will provide several concrete examples of generalized reverse derivations on non-commutative rings at the end of this article. It is known that every generalized reverse derivation is a reverse derivation. However, it is important to note that the converse is not always true. The concepts of generalized reverse derivations are related to generalizations of generalized derivations. It is clear that if ℜ is commutative, then both generalized reverse derivations and generalized derivations are the same. However, the converse may not be true in general, as shown in [9] (Example 1).

In a study by Ibraheem [10], it was proven that a prime ring is commutative, if $\left[\vartheta \right(\xi ),\xi ]\in Z(\Re )$ for all $\xi$ belonging to a right ideal ℵ of a ring, given that the right ideal $\aleph \cap Z(\Re )\ne 0$. Here, $\vartheta$ represents a generalized reverse derivation associated with a nonzero reverse derivation ∂. In a related study by Bulak et al. [11], further exploration of generalized reverse derivations was conducted. The first part of the study focused on the commutativity of prime rings under the influence of differential identities provided by two generalized reverse derivations. The second part examined the relationships between r-generalized reverse derivations and l-generalized derivations, as well as l-generalized reverse derivations and r-generalized derivations, in a non-central square closed Lie ideal in a semiprime ring

Building upon prior findings, many researchers have achieved multiple outcomes regarding commutativity across diverse algebraic structures, including prime and semiprime rings. These outcomes have been attained through the utilization of suitable mappings, such as derivations, generalized derivations, and generalized reverse derivations, which adhere to specific identities when operating on suitable subsets of ℜ. The interested readers can be referred to [1,8,9,12].

Recently, in continuation of the above studies, several authors have discussed the situation of a quotient ring $\Re /\rho$ and the way it behaves under derivation or generalized derivation that satisfies certain identities involving a prime ideal (for more details, refer to [13,14,15,16,17,18,19,20]).

In [21], the concept of the generalized derivation $\vartheta$ was replaced by a generalized reverse derivation, and the commutativity of $\Re /\rho$ was studied whenever the proposed algebraic identities contained in a prime ideal were concerned with $\vartheta$.

The main aim of this article is to study further in this direction. More precisely, assuming that ℜ is an arbitrary ring that admits a generalized reverse derivation $\vartheta$ associated with a reverse derivation ∂, we prove that if $\vartheta$ satisfies certain identities involving a prime ideal $\rho$, then the quotient ring $\Re /\rho$ is a commutative integral domain. In some cases, it comes out that the range of the generalized reverse derivation $\vartheta$ is in a prime ideal $\rho$, i.e., $\vartheta (\Re )\subseteq \rho$. Moreover, some consequences as well as special cases are obtained. Examples that illustrate the necessity of the assumptions stated in our theorems are provided.

2. Preliminary Results

We begin this section by recalling the following basic concepts: Let $\xi ,\eta \in \Re$. We may define the commutator $[\xi ,\eta ]$ as the difference between $\xi \eta$ and $\eta \xi$, and the anticommutator $\xi \circ \eta$ as the sum of $\xi \eta$ and $\eta \xi$. The following identities will be used extensively throughout this article to facilitate access to the proofs of our theorems, which are satisfied for all $\xi ,\eta ,\zeta \in \Re$:

$$[\xi \eta ,\zeta ]=\xi [\eta ,\zeta ]+[\xi ,\zeta ]\eta ,$$

$$[\xi ,\eta \zeta ]=\eta [\xi ,\zeta ]+[\xi ,\eta ]\zeta ,$$

$$\xi \circ \left(\eta \zeta \right)=(\xi \circ \eta )\zeta -\eta [\xi ,\zeta ]=\eta (\xi \circ \zeta )+[\xi ,\eta ]\zeta ,$$

$$\left(\xi \eta \right)\circ \zeta =\xi (\eta \circ \zeta )-[\xi ,\zeta ]\eta =(\xi \circ \zeta )\eta +\xi [\eta ,\zeta ].$$

For the purpose of developing our proofs, we will present the following important remark and lemmas: The proof of Remark 1 is based on the fact that a group cannot be written as the set-theoretic union of its two proper subsets, and the proof of Lemma 1 can be found in [20].

Remark 1. 

Let ρ be a prime ideal of an arbitrary ring $\Re ,$ and let ℵ be an additive subgroup of ℜ. Let $\ell ,\wp :\aleph \to \Re$ be additive functions such that $\ell \left(s\right)\Re \wp \left(s\right)\subseteq \rho$ for all $s\in \aleph$. Then, either $\ell \left(s\right)\in \rho$ for all $s\in \aleph$, or $\wp \left(s\right)\in \rho$ for all $s\in \aleph$.

Lemma 1. 

([20], Lemma 1.2). Let ℜ be a ring and let ρ be a prime ideal of ℜ. If $[\xi ,\eta ]\in \rho$ for all $\xi ,\eta \in \Re$, then $\Re /\rho$ is a commutative integral domain.

The following lemma is an expansion of ([21], Lemma 2.5).

Lemma 2. 

Let ρ be a prime ideal of an arbitrary ring ℜ. If ℜ admits a generalized reverse derivation ϑ associated with a reverse derivation ∂ such that $[\xi ,\vartheta (\xi \left)\right]\in \rho$ for all $\xi \in \rho$, then either $\partial (\Re )\subseteq \rho$ or $\Re /\rho$ is a commutative integral domain.

Proof. 

From the hypothesis, we have

$$[\xi ,\vartheta (\xi \left)\right]\in \rho ,\mathrm{for}\mathrm{all}\xi \in \Re .$$

(1)

By linearizing Equation $\left(1\right)$, which simply means replacing $\xi$ by $\xi +\eta$, we obtain

$$[\xi ,\vartheta (\eta \left)\right]+[\eta ,\vartheta (\xi \left)\right]\in \rho ,\mathrm{for}\mathrm{all}\xi ,\eta \in \Re .$$

(2)

By replacing $\eta$ by $\eta \xi$ in $\left(2\right)$ and utilizing $\left(1\right)$, we get

$$\vartheta \left(\xi \right)[\xi ,\eta ]+\xi [\xi ,\partial (\eta \left)\right]+[\eta ,\vartheta (\xi \left)\right]\xi \in \rho ,\mathrm{for}\mathrm{all}\xi ,\eta \in \Re .$$

(3)

Setting $\xi =\eta$ in $\left(3\right)$ and using $\left(1\right)$ again, we obtain $\xi [\xi ,\partial (\xi \left)\right]\in \rho ,\mathrm{for}\mathrm{all}\xi \in \Re$. Replacing $\partial \left(\xi \right)$ by $\partial \left(\xi \right)\tau$ in the previous equation and using it, we get $\xi \partial \left(\xi \right)[\xi ,\tau ]\in \rho$ for all $\xi ,\tau \in \Re .$ Placing $\tau \nu$ instead of $\tau$ in the previous equation and using it, we get $\xi \partial \left(\xi \right)\tau [\xi ,\nu ]\in \rho$ for all $\xi ,\tau ,\nu \in \Re .$ In other words, $\xi \partial \left(\xi \right)\Re [\xi ,\nu ]\subseteq \rho$ for all $\xi ,\nu \in \Re .$ Since $\rho$ is prime, considering Remark 1, we find that either $\xi \partial \left(\xi \right)\in \rho$ or $[\xi ,\nu ]\in \rho$ for all $\xi ,\nu \in \Re .$ If $[\xi ,\nu ]\in \rho$ for all $\xi ,\nu \in \Re$, we deduce that $\Re /\rho$ is a commutative integral domain, by Lemma 1. In the alternative scenario, we have $\xi \partial \left(\xi \right)\in \rho$ for all $\xi \in \Re$. Linearizing the previous expression, we obtain $\xi \partial \left(\eta \right)+\eta \partial \left(\xi \right)\in \rho$ for all $\xi ,\eta \in \Re .$ By replacing $\xi$ by $\nu \xi$ in the previous equation and using it we find, after appropriate treatment, that $\eta \partial \left(\xi \right)\nu -\nu \eta \partial \left(\xi \right)+\eta \xi \partial \left(\nu \right)\in \rho$ for all $\xi ,\eta ,\nu \in \Re .$ Again, placing $\tau \eta$ instead of $\eta$ in the last relation and using it, we find $[\tau ,\nu ]\eta \partial \left(\xi \right)\in \rho$ for all $\xi ,\tau ,\nu ,\eta \in \Re$. This results in $[\tau ,\nu ]\Re \partial \left(\xi \right)\subseteq \rho$ for all $\xi ,\tau ,\nu \in \Re .$ By employing the assumption that $\rho$ is prime along with Remark 1, we conclude that either $\partial \left(\xi \right)\in \rho$ or $[\tau ,\nu ]\in \rho$ for all $\xi ,\tau ,\nu \in \Re .$ Therefore, we can infer that the first case leads to $\partial (\Re )\subseteq \rho$, and for the second case, we use Lemma 1 to obtain that $\Re /\rho$ is a commutative integral domain. □

Corollary 1. 

Let ρ be a prime ideal of an arbitrary ring ℜ. If ℜ admits a reverse derivation ∂, such that $[\xi ,\partial (\xi \left)\right]\in \rho$ for all $\xi \in \Re$, then either $\partial (\Re )\subseteq \rho$ or $\Re /\rho$ is a commutative integral domain. Moreover, if $\rho =\left\{0\right\}$, then either ℜ is commutative or ∂ turns out to be zero.

Remark 2. 

In Lemma 2, if ℜ is commutative, then ϑ becomes a generalized derivation, and thus we obtain ([20], Proposition 1.3).

3. Main Results

In [14] (Theorem 2.5), Bouchannafa et al. proved that either the ring $\Re /\rho$ is a commutative integral domain or $\partial (\Re )$ is a subset of $\rho$, whenever the ring ℜ has a generalized derivation $\vartheta$ such that $\vartheta (\xi \circ \eta )-\vartheta \left(\xi \right)\circ \eta$ belongs to the center $Z(\Re /\rho )$ for all $\xi$ and $\eta$ in ℜ, where $\rho$ is a prime ideal of ℜ. In the next theorem, our objective is to achieve the same outcome by substituting the generalized derivation $\vartheta$ from the previous theorem with the notion of a generalized reverse derivation, which is associated with a reverse derivation ∂ that fulfills the condition $\vartheta \left(\xi \right)\circ \eta -\vartheta (\xi \circ \eta )\in \rho$, for all $\xi ,\eta \in \Re$.

Theorem 1. 

Consider a prime ideal ρ in a ring ℜ, where ℜ can be any ring. If ℜ admits a generalized reverse derivation ϑ that is associated with a reverse derivation ∂, and satisfies the condition $\vartheta \left(\xi \right)\circ \eta -\vartheta (\xi \circ \eta )\in \rho$ for all ξ and η in ℜ, then either $\partial (\Re )$ is a subset of ρ or the quotient ring $\Re /\rho$ is a commutative integral domain.

Proof. 

Suppose that

$$\vartheta \left(\xi \right)\circ \eta -\vartheta (\xi \circ \eta )\in \rho ,\mathrm{for}\mathrm{all}\xi ,\eta \in \Re .$$

(4)

Placing $\eta \xi$ instead of $\xi$ in (4) yields

$$\left(\vartheta \right(\xi )\circ \eta )\eta +(\xi \circ \eta )\partial \left(\eta \right)+\xi [\partial (\eta ),\eta ]-\vartheta (\xi \circ \eta )\eta -(\xi \circ \eta )\partial \left(\eta \right)\in \rho ,\mathrm{for}\mathrm{all}\xi ,\eta \in \Re .$$

(5)

By multiplying (4) by $\eta$ from the right and comparing it with (5), we obtain

$$\xi [\partial (\eta ),\eta ]\in \rho ,\mathrm{for}\mathrm{all}\xi ,\eta \in \Re .$$

The last equation is simplified as follows: $\xi \Re [\partial (\eta ),\eta ]\subseteq \rho$, where $\xi$ and $\eta$ are elements of ℜ. Given that $\rho \ne \Re$ and $\rho$ is a prime, the last equation implies that $[\partial (\eta ),\eta ]$ belongs to $\rho$ for every $\eta$ in ℜ. Therefore, according to Corollary 1, either $\Re /\rho$ is a commutative integral domain or $\partial (\Re )$ is a subset of $\rho$. □

When ℜ is a prime ring and $\vartheta =\partial$, respectively, the following corollaries can be immediately obtained from Theorem 1.

Corollary 2. 

Consider a ring ℜ, which is prime. If ℜ admits a generalized reverse derivation ϑ associated with a reverse derivation ∂, satisfying the equation $\vartheta \left(\xi \right)\circ \eta -\vartheta (\xi \circ \eta )=0$ for all $\xi ,\eta \in \Re$, then either ∂ is equal to zero or ℜ is a commutative ring.

Corollary 3. 

Consider a prime ideal ρ in a ring ℜ, where ℜ can be any ring. If ℜ admits a reverse derivation ∂, and satisfies the condition $\partial \left(\xi \right)\circ \eta -\partial (\xi \circ \eta )\in \rho$ for all ξ and η in ℜ, then either $\partial (\Re )$ is a subset of ρ or the quotient ring $\Re /\rho$ is a commutative integral domain.

Theorem 2. 

Consider a prime ideal ρ of a ring ℜ. If ℜ admits a generalized reverse derivation ϑ associated with a reverse derivation ∂, such that $\vartheta \left(\xi \right)\circ \eta +\vartheta (\xi \circ \eta )\in \rho ,\mathrm{for}\mathrm{all}\xi ,\eta \in \Re$, then either $\partial (\Re )\subseteq \rho$ or $\Re /\rho$ is a commutative integral domain of $char(\Re /\rho )=2.$

Proof. 

The given identity states that

$$\vartheta \left(\xi \right)\circ \eta +\vartheta (\xi \circ \eta )\in \rho ,\mathrm{for}\mathrm{all}\xi ,\eta \in \Re .$$

(6)

Replacing $\xi$ by $\eta \xi$ in (6), gives

$$\left(\vartheta \right(\xi )\circ \eta )\eta +(\xi \circ \eta )\partial \left(\eta \right)+\xi [\partial (\eta ),\eta ]+\vartheta (\xi \circ \eta )\eta +(\xi \circ \eta )\partial \left(\eta \right)\in \rho ,\mathrm{for}\mathrm{all}\xi ,\eta \in \Re .$$

(7)

By multiplying (6) by $\eta$ from the right-hand side and comparing it with (7), we obtain

$$2(\xi \circ \eta )\partial \left(\eta \right)+\xi [\partial (\eta ),\eta ]\in \rho ,\mathrm{for}\mathrm{all}\xi ,\eta \in \rho .$$

(8)

Now, we discuss the following two cases:

Case (i): If $char(\Re /\rho )=2$, then $\left(8\right)$ becomes $\xi [\partial (\eta ),\eta ]\in \rho ,\mathrm{for}\mathrm{all}\xi ,\eta \in \rho$. Following the same arguments as above, we find either $\partial (\Re )$ is a subset of $\rho$ or $\Re /\rho$ is a commutative integral domain.

Case (ii): If $char(\Re /\rho )\ne 2$, then replacing $\xi$ by $\tau \xi$ in (8) results in

$$2\tau (\xi \circ \eta )\partial \left(\eta \right)-2[\tau ,\eta ]\xi \partial \left(\eta \right)+\tau \xi [\partial (\eta ),\eta ]\in \rho ,\mathrm{for}\mathrm{all}\xi ,\eta ,\tau \in \rho .$$

(9)

Now, multiplying (8) by $\tau$ from the left-hand side and comparing with (9), yields

$2[\tau ,\eta ]\xi \partial \left(\eta \right)\in \rho$ for all $\xi ,\eta ,\tau \in \Re .$ Our assumption that $char(\Re /\rho )\ne 2$ leads to $[\tau ,\eta ]\xi \partial \left(\eta \right)\in \rho$ for all $\xi ,\eta ,\tau \in \Re$, and hence $[\tau ,\eta ]\Re \partial \left(\eta \right)\subseteq \rho$ for all $\eta ,\tau \in \Re$. Thus, the primeness of $\rho$ together with Remark 1 lead to either $[\tau ,\eta ]\in \rho$ for all $\tau ,\eta \in \Re$ or $\partial (\Re )\subseteq \rho$. If $\partial (\Re )$ is not a subset of $\rho$, then $[\tau ,\eta ]$ belongs to $\rho$ for every elements $\tau$ and $\eta$ in ℜ. Using Lemma 1 shows that the quotient ring $\Re /\rho$ is a commutative integral domain. By utilizing the commutativity of $\Re /\rho$ with the identity (8), we can easily deduce that $2(\xi \circ \eta )\partial \left(\eta \right)$ belongs to $\rho$ for all $\xi$ and $\eta$ in ℜ. The statement $4\xi \eta \partial \left(\eta \right)\in \rho$ holds for all $\xi ,\eta \in \Re$ because $char(\Re /\rho )\ne 2$. This implies that $\xi \eta \partial \left(\eta \right)\in \rho$ for all $\xi ,\eta \in \Re$, due to the commutativity of ℜ. Furthermore, the previous expression is equivalent to $\eta \xi \partial \left(\eta \right)\in \rho$ for all $\xi ,\eta \in \Re$, which may be written as $\eta \Re \partial \left(\eta \right)\subseteq \rho$ for all $\eta \in \Re .$ However, our hypothesis that $\partial (\Re )⊈\rho$ and $\rho$ is a prime ideal of ℜ forces $\xi \in \rho$, which eventually implies that $\Re =\rho$. This contradicts our basic hypothesis about $\rho$ being a proper ideal of ℜ. Therefore, we can deduce that $\partial (\Re )\subseteq \rho$. □

If the ring ℜ imposed in Theorem 2 is prime, meaning $\rho =\left\{0\right\}$, then the following corollary results immediately:

Corollary 4. 

Consider a prime ring ℜ. If ℜ admits a generalized reverse derivation ϑ associated with a reverse derivation ∂ such that $\vartheta \left(\xi \right)\circ \eta +\vartheta (\xi \circ \eta )=0$ for all ξ and η in ℜ, then either $\partial \left(R\right)=0$ or ℜ is commutative of $char(\Re )=2.$

When we consider $\vartheta =\partial$ in Theorem 2, the following corollary is immediately obtained.

Corollary 5. 

Consider a prime ideal ρ of a ring ℜ. If ℜ admits a reverse derivation ∂ such that $\partial \left(\xi \right)\circ \eta +\partial (\xi \circ \eta )\in \rho \mathrm{for}\mathrm{all}\xi ,\eta \in \Re$, then either $\partial (\Re )\subseteq \rho$ or $\Re /\rho$ is a commutative integral domain of $char(\Re /\rho )=2.$

In ref. [20], Rehman et al. established a result stating that if ℜ is a ring and $\rho$ is a prime ideal of it, such that ℜ admits a generalized derivation $\vartheta$ associated with ∂ and meets the condition $\vartheta \left(\xi \right)\vartheta \left(\eta \right)-[\xi ,\eta ]\in \rho$ for all $\xi ,\eta \in \Re$, then either $\partial (\Re )\subseteq \rho$ or $\Re /\rho$ is a commutative integral domain.

This result prompts us to investigate the properties of the ring $\Re /\rho$ when we replace the assumption that $\vartheta$ is a generalized derivation by a generalized reverse derivation associated with a reverse derivation ∂. For this purpose, we introduce the following theorem:

Theorem 3. 

Consider ρ as a prime ideal in any ring ℜ. If ℜ admits a generalized reverse derivation ϑ which is associated with a reverse derivation ∂ and satisfies the condition $\vartheta \left(\xi \right)\vartheta \left(\eta \right)\pm [\xi ,\eta ]\in \rho$ for all ξ and η in ℜ, then $\partial (\Re )$ is a subset of ρ and the quotient ring $\Re /\rho$ is a commutative integral domain.

Proof. 

The given identity states that

$$\vartheta \left(\xi \right)\vartheta \left(\eta \right)-[\xi ,\eta ]\in \rho ,\mathrm{for}\mathrm{all}\xi ,\eta \in \Re .$$

(10)

By replacing $\eta$ with $\xi \eta$ in (10), we obtain

$$\vartheta \left(\xi \right)\vartheta \left(\eta \right)\xi +\vartheta \left(\xi \right)\eta \partial \left(\xi \right)-\xi [\xi ,\eta ]\in \rho ,\mathrm{for}\mathrm{all}\xi ,\eta \in \Re .$$

(11)

Multiplying (10) by $\xi$ on the right gives

$$\vartheta \left(\xi \right)\vartheta \left(\eta \right)\xi -[\xi ,\eta ]\xi \in \rho ,\mathrm{for}\mathrm{all}\xi ,\eta \in \Re .$$

(12)

By comparing Equations (11) and (12), we obtain

$$\vartheta \left(\xi \right)\eta \partial \left(\xi \right)-\xi [\xi ,\eta ]+[\xi ,\eta ]\xi \in \rho ,\mathrm{for}\mathrm{all}\xi ,\eta \in \Re .$$

(13)

Again, by replacing $\eta$ by $\xi \eta$ in the previous equation, we obtain

$$\vartheta \left(\xi \right)\xi \eta \partial \left(\xi \right)-{\xi }^2[\xi ,\eta ]+\xi [\xi ,\eta ]\xi \in \rho ,\mathrm{for}\mathrm{all}\xi ,\eta \in \Re .$$

(14)

Multiplying (13) from the left by $\xi$ and comparing it with (14) yield

$$\vartheta \left(\xi \right)\xi \eta \partial \left(\xi \right)-\xi \vartheta \left(\xi \right)\eta \partial \left(\xi \right)\in \rho ,\mathrm{for}\mathrm{all}\xi ,\eta \in \Re .$$

It follows that $\left[\vartheta \right(\xi ),\xi ]\Re \partial \left(\xi \right)\subseteq \rho \mathrm{for}\mathrm{all}\xi \in \Re .$ Hence, the primeness of $\rho$ together with Remark 1 forces that either $\left[\vartheta \right(\xi ),\xi ]\in \rho \mathrm{for}\mathrm{all}\xi \in \Re$ or $\partial \left(\xi \right)\in \rho$ for any $\xi \in \Re$. If $\left[\vartheta \right(\xi ),\xi ]\in \rho \mathrm{for}\mathrm{all}\xi \in \Re$, then, according to Lemma 2, $\Re /\rho$ is a commutative integral domain or $\partial (\Re )\subseteq \rho$.

Let $\Re /\rho$ be a commutative integral domain. Then, (10) can be reduced to $\vartheta \left(\xi \right)\Re \vartheta \left(\eta \right)\subseteq \rho$ for all $\xi ,\eta \in \Re .$ That is $\vartheta \left(\xi \right)\in \rho ,\mathrm{for}\mathrm{all}\xi \in \Re$. Now, we replace $\xi$ by $\eta \xi$ in the previous expression and use it to conclude that $\partial (\Re )\subseteq \rho$. On the other hand, if we assume $\partial (\Re )\subseteq \rho$, then (13) can be simplified to

$$[\xi ,[\xi ,\eta \left]\right]\in \rho ,\mathrm{for}\mathrm{all}\xi ,\eta \in \Re .$$

(15)

By setting $\xi =\xi +\zeta$ in the previous equation and using it, we can easily find $[\xi ,[\zeta ,\eta \left]\right]+[\zeta ,[\xi ,\eta \left]\right]\in \rho \mathrm{for}\mathrm{all}\xi ,\eta ,\zeta \in \Re .$ Replacing $\zeta$ by $\eta \zeta$ in the last relation and using it, we obtain $[\xi ,\eta ][\zeta ,\eta ]+[\eta ,[\xi ,\eta \left]\right]\zeta \in \rho \mathrm{for}\mathrm{all}\xi ,\eta ,\zeta \in \Re .$ By replacing $\zeta$ by $\zeta \xi$ in the last expression and using it, we deduce that $[\xi ,\eta ]\Re [\xi ,\eta ]\subseteq \rho$ for all $\xi ,\eta \in \Re$. The primeness of $\rho$ implies that $[\xi ,\eta ]\in \rho$ for all $\xi ,\eta \in \Re$. Therefore, once again, $\Re /\rho$ is a commutative integral domain, by using Lemma 1.

By applying similar arguments to those shown earlier, with only slight adjustments, the same result can be obtained for the case $\vartheta \left(\xi \right)\vartheta \left(\eta \right)+[\xi ,\eta ]\in \rho$, for all $\xi$ and $\eta$ in ℜ. □

In Theorem 3, if ℜ is assumed to be prime, the following corollary can be immediately obtained.

Corollary 6. 

Consider ℜ is a prime ring admits a generalized reverse derivation ϑ, which is associated with a nonzero reverse derivation ∂ and satisfies the condition $\vartheta \left(\xi \right)\vartheta \left(\eta \right)\pm [\xi ,\eta ]=0$ for all ξ and η in ℜ, then the ring ℜ a commutative.

By setting $\vartheta =\partial$ in Theorem 3, we promptly obtain the subsequent corollary.

Corollary 7. 

Consider a prime ideal ρ in any ring ℜ. Suppose that ℜ admits a reverse derivation d such that $\partial \left(\xi \right)\partial \left(\eta \right)\pm [\xi ,\eta ]$ belongs to ρ for all $\xi ,\eta \in \Re$. In this case, $\partial (\Re )$ is a subset of ρ and $\Re /\rho$ is a commutative integral domain.

Proof. 

The proof can be directly obtained from Equation (10) in Theorem 3, by setting $\vartheta =\partial$ and following the same arguments and techniques of its proof. □

Theorem 4. 

Let ρ be a prime ideal in any ring ℜ. If ℜ admits a generalized reverse derivation ϑ that is associated with a reverse derivation ∂ and satisfies the condition $\vartheta \left(\xi \eta \right)\pm \partial \left(\xi \right)\vartheta \left(\eta \right)\in \rho$ for all $\xi ,\eta \in \Re$, then $\vartheta (\Re )\subseteq \rho .$

Proof. 

The given assumption states that

$$\vartheta \left(\xi \eta \right)+\partial \left(\xi \right)\vartheta \left(\eta \right)\in \rho \mathrm{for}\mathrm{all}\xi ,\eta \in \Re .$$

(16)

Replacing $\eta$ by $\xi \eta$ in (16) gives

$$\vartheta \left(\xi \eta \right)\xi +\xi \eta \partial \left(\xi \right)+\partial \left(\xi \right)\vartheta \left(\eta \right)\xi +\partial \left(\xi \right)\eta \partial \left(\xi \right)\in \rho ,\mathrm{for}\mathrm{all}\xi ,\eta \in \Re .$$

(17)

Multiplying Equation (16) by $\xi$ from the right and comparing it with (17) yield $\partial \left(\xi \right)\eta \partial \left(\xi \right)+\xi \eta \partial \left(\xi \right)\in \rho \mathrm{for}\mathrm{all}\xi ,\eta \in \Re .$ That is, $(\partial (\xi )+\xi )\Re \partial \left(\xi \right)\subseteq \rho \mathrm{for}\mathrm{all}\xi \in \Re .$ By using the primeness of $\rho$ together with Remark 1, we get either $\partial \left(\xi \right)\in \rho$ for all $\xi \in \Re$ or $(\partial (\xi )+\xi )\in \rho$ for all $\xi \in \Re$. If $\partial \left(\xi \right)\in \rho$ for all $\xi \in \Re$, (16) can be reduced to $\vartheta \left(\eta \right)\xi \in \rho$ for all $\xi ,\eta \in \Re$. Hence, we have $\vartheta (\Re )\subseteq \rho .$ On the other hand, for

$$\partial \left(\xi \right)+\xi \in \rho ,\mathrm{for}\mathrm{all}\xi \in \Re ,$$

(18)

we substitute $\eta \xi$ in the place of $\xi$ in (18) to obtain $\partial \left(\xi \right)\eta +\xi \partial \left(\eta \right)+\eta \xi \in \rho \mathrm{for}\mathrm{all}\xi ,\eta \in \Re$. Multiplying (18) by $\eta$ from the right and comparing it with the last relation, we get

$$\xi \partial \left(\eta \right)+[\eta ,\xi ]\in \rho ,\mathrm{for}\mathrm{all}\xi ,\eta \in \Re .$$

(19)

Now, placing $\tau \xi$ instead of $\xi$ in Equation (19), we get

$$\tau \xi \partial \left(\eta \right)+\tau [\eta ,\xi ]+[\eta ,\tau ]\xi \in \rho ,\mathrm{for}\mathrm{all}\xi ,\eta ,\tau \in \Re .$$

(20)

Left-multiplying (19) by $\tau$ and comparing it with (20) yields $[\eta ,\tau ]\xi \in \rho \mathrm{for}\mathrm{all}\xi ,\eta ,\tau \in \Re .$ Again, replacing $\tau$ by $\nu \tau$ in the last equation and using it gives $[\eta ,\nu ]\Re \xi \subseteq \rho$ for all $\xi ,\eta ,\nu \in \Re .$ Since $\rho$ is a prime ideal, either $\xi \in \rho$ for all $\xi \in \Re$ or $[\eta ,\nu ]\in \rho \mathrm{for}\mathrm{all}\eta ,\nu \in \Re$. If $\xi \in \rho$ for all $\xi \in \Re ,$ then it implies that $\rho =\Re ,$ which contradicts the fact that $\rho$ is a proper ideal. If $[\eta ,\nu ]\in \rho$ for all $\eta ,\nu \in \Re$, then, according to Lemma 1, $\Re /\rho$ is a commutative integral domain. In this case, (19) can be simplified to $\xi \partial \left(\eta \right)\in \rho \mathrm{for}\mathrm{all}\xi ,\eta \in \Re$, which is leading to $\partial (\Re )\subseteq \rho$. So, we can conclude, as above, that $\vartheta (\Re )\subseteq \rho .$

By following the exact techniques as described previously, we can prove the same conclusion in the case of the identity $\vartheta \left(\xi \eta \right)-\partial \left(\xi \right)\vartheta \left(\eta \right)\in \rho$ for all $\xi ,\eta \in \Re$. □

By equating $\vartheta$ to ∂ in the prior theorem, we can obtain the following conclusion as a similar version of ([16], Theorem 4(1)).

Corollary 8. 

Let ρ be a prime ideal in any ring ℜ. If ℜ admits a reverse derivation ∂ such that $\partial \left(\xi \eta \right)\pm \partial \left(\xi \right)\partial \left(\eta \right)\in \rho$ for all $\xi ,\eta \in \Re$, then $\partial (\Re )\subseteq \rho .$

Corollary 9. 

Let ℜ be a prime ring that admits a generalized reverse derivation ϑ that is associated with a reverse derivation ∂ and satisfies the condition $\vartheta \left(\xi \eta \right)\pm \partial \left(\xi \right)\vartheta \left(\eta \right)=0$ for all $\xi ,\eta \in \Re$, then $\vartheta (\Re )=0$.

Theorem 5. 

Consider ρ as a prime ideal in any ring ℜ. If ℜ admits a generalized reverse derivation ϑ, which is associated with a reverse derivation ∂, and satisfies the condition $\vartheta \left(\xi \eta \right)\pm \partial \left(\eta \right)\vartheta \left(\xi \right)\in \rho$ for all $\xi ,\eta \in \Re$, then $\vartheta (\Re )\subseteq \rho$.

Proof. 

The given assumption states that

$$\vartheta \left(\xi \eta \right)-\partial \left(\eta \right)\vartheta \left(\xi \right)\in \rho ,\mathrm{for}\mathrm{all}\xi ,\eta \in \Re .$$

(21)

We replace $\xi$ by $\tau \xi$ in (21) and use it to obtain

$$\xi \eta \partial \left(\tau \right)-\partial \left(\eta \right)\xi \partial \left(\tau \right)\in \rho ,\mathrm{for}\mathrm{all}\xi ,\eta ,\tau \in \Re .$$

(22)

If we replace $\xi$ by $\nu \xi$ in the previous equation, we obtain $\nu \xi \eta \partial \left(\tau \right)-\partial \left(\eta \right)\nu \xi \partial \left(\tau \right)\in \rho$ for all $\xi ,\eta ,\tau ,\nu \in \Re .$ By left multiplying Equation (22) by $\nu$ and comparing it with the last equation, we find $\nu \partial \left(\eta \right)\xi \partial \left(\tau \right)-\partial \left(\eta \right)\nu \xi \partial \left(\tau \right)\in \rho \mathrm{for}\mathrm{all}\xi ,\eta ,\tau ,\nu \in \Re .$ This implies that $[\nu ,\partial (\eta \left)\right]\xi \partial \left(\tau \right)\in \rho \mathrm{for}\mathrm{all}\xi ,\eta ,\tau ,\nu \in \Re .$ In other words, $[\nu ,\partial (\eta \left)\right]\Re \partial \left(\tau \right)\subseteq \rho$ for all $\eta ,\tau ,\nu \in \Re$. Using the primeness of $\rho$ together with Remark 1 yield either $[\nu ,\partial (\eta \left)\right]\in \rho$ for all $\eta ,\nu \in \Re$ or $\partial (\Re )\subseteq \rho .$ In the second case, (21), becomes $\vartheta \left(\eta \right)\xi \in \rho \mathrm{for}\mathrm{all}\xi ,\eta \in \Re$ and therefore, $\vartheta (\Re )\subseteq \rho$. For the case of $[\nu ,\partial (\eta \left)\right]\in \rho$ for all $\eta ,\nu \in \Re$, we have, in particular, that $[\eta ,\partial (\eta \left)\right]\in \rho$ for all $\eta \in \Re$. We use Corollary 1 to obtain that either $\Re /\rho$ is a commutative integral domain or $\partial (\Re )\subseteq \rho$. If $\Re /\rho$ is a commutative integral domain, then (22) can be rewritten as $(\eta -\partial (\eta \left)\right)\xi \partial \left(\tau \right)\in \rho \mathrm{for}\mathrm{all}\xi ,\eta ,\tau \in \Re .$ That is, $(\eta -\partial (\eta \left)\right)\Re \partial \left(\tau \right)\subseteq \rho \mathrm{for}\mathrm{all}\eta ,\tau \in \Re .$ Again, using the primeness of $\rho$ together with Remark 1 give that either $(\eta -\partial (\eta \left)\right)\in \rho$ for all $\eta \in \Re$ or $\partial (\Re )\subseteq \rho$. When $(\eta -\partial (\eta \left)\right)\in \rho \mathrm{for}\mathrm{all}\eta \in \Re$, we replace $\eta$ by $\xi \eta$ in the last relation and use it to get $[\xi ,\eta ]-\eta \partial \left(\xi \right)\in \rho \mathrm{for}\mathrm{all}\xi ,\eta \in \Re$. For the other case, the commutativity of $\Re /\rho$ leads to $\eta \partial \left(\xi \right)\in \rho \mathrm{for}\mathrm{all}\xi ,\eta \in \Re ,$ and hence, $\partial (\Re )\subseteq \rho$. Thus, (21) becomes $\vartheta \left(\eta \right)\xi \in \rho \mathrm{for}\mathrm{all}\xi ,\eta \in \Re$. Therefore, $\vartheta (\Re )\subseteq \rho$.

By following the exact techniques as described previously, we can prove the same conclusion for the case of the identity $\vartheta \left(\xi \eta \right)+\partial \left(\eta \right)\vartheta \left(\xi \right)\in \rho$ for all $\xi ,\eta \in \Re$. □

Corollary 10. 

Consider ℜ is a prime ring that admits a generalized reverse derivation ϑ associated with a reverse derivation ∂ and satisfies the condition $\vartheta \left(\xi \eta \right)\pm \partial \left(\eta \right)\vartheta \left(\xi \right)=0$ for all $\xi ,\eta \in \Re$, then $\vartheta (\Re )=0$.

Corollary 11. 

Consider ρ as a prime ideal in any ring ℜ. If ℜ admits a reverse derivation ∂ and satisfies the condition $\partial \left(\xi \eta \right)\pm \partial \left(\eta \right)\partial \left(\xi \right)\in \rho$ for all $\xi ,\eta \in \Re$, then $\partial (\Re )\subseteq \rho$.

In [20] (Theorem 1.5(iii)), Rehman et al. showed that the quotient ring $\Re /\rho$ is a commutative integral domain, where $\rho$ is a prime ideal of ℜ, if ℜ admits a generalized derivation $\vartheta$ associated with a derivation ∂ that satisfies $\vartheta \left(\xi \right)\vartheta \left(\eta \right)\pm \xi \eta \in \rho ,\mathrm{for}\mathrm{every}\xi ,\eta \in \Re$.

The following theorem aims to generalize the above identity to $\vartheta \left(\xi \right)\vartheta \left(\eta \right)+\partial \left(\xi \right)\partial \left(\eta \right)\in \rho$ for every $\xi ,\eta \in \Re$ and prove that $\vartheta (\Re )$ is a subset of $\rho$ when $char(\Re /\rho )\ne 2$ and the imposed $\vartheta$ is a generalized reverse derivation associated with a reverse derivation ∂.

Theorem 6. 

Consider a prime ideal ρ in a ring ℜ with a characteristic not equal to 2. If ℜ admits a generalized reverse derivation ϑ that is associated with a reverse derivation ∂, and if $\vartheta \left(\xi \right)\vartheta \left(\eta \right)+\partial \left(\xi \right)\partial \left(\eta \right)\in \rho$ for all ξ and η in ℜ, then $\vartheta (\Re )$ is a subset of ρ.

Proof. 

The given hypothesis states that

$$\vartheta \left(\xi \right)\vartheta \left(\eta \right)+\partial \left(\xi \right)\partial \left(\eta \right)\in \rho ,\mathrm{for}\mathrm{all}\xi ,\eta \in \Re .$$

(23)

By replacing $\eta$ with $\tau \eta$ in (23), we obtain

$$\vartheta \left(\xi \right)\vartheta \left(\eta \right)\tau +\vartheta \left(\xi \right)\eta \partial \left(\tau \right)+\partial \left(\xi \right)\partial \left(\eta \right)\tau +\partial \left(\xi \right)\eta \partial \left(\tau \right)\in \rho ,\mathrm{for}\mathrm{all}\xi ,\eta ,\tau \in \Re .$$

(24)

Now, by multiplying Equation (23) by $\tau$ from the right and comparing it with (24), we obtain

$$\left(\vartheta \right(\xi )+\partial (\xi \left)\right)\eta \partial \left(\tau \right)\in \rho ,\mathrm{for}\mathrm{all}\xi ,\eta ,\tau \in \Re .$$

(25)

That is, $\left(\vartheta \right(\xi )+\partial (\xi \left)\right)\Re \partial \left(\tau \right)\subseteq \rho .$ Hence, the condition of $\rho$ being prime, together with Remark (1), forces either $\vartheta \left(\xi \right)+\partial \left(\xi \right)\in \rho \mathrm{for}\mathrm{all}\xi \in \Re$ or $\partial \left(\tau \right)\in \tau$ for all $\tau \in \Re$. Let us consider the first case and replace $\xi$ with $\eta \xi$. This yields $\vartheta \left(\xi \right)\eta +\xi \partial \left(\eta \right)+\partial \left(\xi \right)\eta +\xi \partial \left(\eta \right)\in \rho$ for all $\xi ,\eta \in \Re .$ By multiplying the first equation by $\eta$ from the right and comparing it with the second equation, we obtain $\xi \partial \left(\eta \right)+\xi \partial \left(\eta \right)\in \rho$ for all $\xi ,\eta \in \Re$, which in turn means that $2\xi \partial \left(\eta \right)\in \rho$ for every $\xi ,\eta \in \Re$. The basic assumption $char(\Re /\rho )\ne 2$ leads to $\xi \partial \left(\eta \right)\in \rho$ for every $\xi ,\eta \in \Re$. That is, $\Re \partial \left(\eta \right)\subseteq \rho$. As $\rho \ne \Re$ and $\rho$ is prime, we get $\partial \left(\eta \right)\in \rho$ for any $\eta \in \Re$. Thus, we deduce that $\partial (\Re )\subseteq \rho$. In the second case, we observe that $\partial \left(\tau \right)$ belongs to $\rho$ for every $\tau \in \Re$, indicating that $\partial (\Re )\subseteq \rho$. So both cases lead to $\partial (\Re )\subseteq \rho$, which reduces (23) to $\vartheta \left(\xi \right)\vartheta \left(\eta \right)\in \rho \mathrm{for}\mathrm{all}\xi ,\eta \in \Re .$ Replacing $\xi$ by $\xi \tau$ in the last relation, we get $\vartheta \left(\tau \right)\Re \vartheta \left(\eta \right)\subseteq \rho \mathrm{for}\mathrm{all}\tau ,\eta \in \Re .$ Therefore, we have $\vartheta (\Re )\subseteq \rho$ for both cases as required. □

As an immediate consequent of the above theorem, we have the following corollary when the imposed ring ℜ is prime.

Corollary 12. 

Consider a prime ring ℜ with a characteristic that is not equal to 2. If ℜ admits a generalized reverse derivation ϑ associated with a reverse derivation ∂, satisfying the equation $\vartheta \left(\xi \right)\vartheta \left(\eta \right)+\partial \left(\xi \right)\partial \left(\eta \right)=0$ for all ξ and η in ℜ, then ϑ turns out to be zero.

Next, we will explore some counterexamples that illustrate the necessity of assuming that $\rho$ is prime in the hypotheses of our theorems.

Example 1. 

Consider the ring of integers $\mathbb{Z}$ and let $\Re =\{\left(\begin{array}{cccc}0& \alpha & \beta & \gamma \\ 0& 0& 0& \beta \\ 0& 0& 0& -\alpha \\ 0& 0& 0& 0\end{array}\right)|\alpha ,\beta ,\gamma \in \mathbb{Z}\}$, $\rho =\{\left(\begin{array}{cccc}0& 0& 0& 2\gamma \\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)\}$. Define $\vartheta ,\partial :\Re ⟶\Re ,$ by $\vartheta \left(\begin{array}{cccc}0& \alpha & \beta & \gamma \\ 0& 0& 0& \beta \\ 0& 0& 0& -\alpha \\ 0& 0& 0& 0\end{array}\right)=\left(\begin{array}{cccc}0& 0& 0& -\gamma \\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)\mathit{and}\partial \left(\begin{array}{cccc}0& \alpha & \beta & \gamma \\ 0& 0& 0& \beta \\ 0& 0& 0& -\alpha \\ 0& 0& 0& 0\end{array}\right)=\left(\begin{array}{cccc}0& 0& 0& -\gamma \\ 0& 0& 0& \beta \\ 0& 0& 0& -\alpha \\ 0& 0& 0& 0\end{array}\right)$. Thus, it is evident that ℜ is a ring, ρ is an ideal of ℜ, and ϑ is a generalized reverse derivation associated with the reverse derivation ∂ that satisfies $\vartheta \left(\xi \right)\circ \eta \pm \vartheta (\xi \circ \eta )\in \rho$, $\vartheta \left(\xi \right)\vartheta \left(\eta \right)\pm [\xi ,\eta ]\in \rho$, and $\vartheta \left(\xi \right)\vartheta \left(\eta \right)+\partial \left(\xi \right)\partial \left(\eta \right)\in \rho$ for all $\xi ,\eta \in \rho$. However, $\Re /\rho$ is non-commutative, $\partial (\Re )⊈\rho$, and $\vartheta (\Re )⊈\rho$. Moreover, ρ is not a prime ideal of ℜ since $\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& \beta \\ 0& 0& 0& \alpha \\ 0& 0& 0& 0\end{array}\right)\left(\begin{array}{cccc}0& 0& \alpha & 0\\ 0& 0& 0& 0\\ 0& 0& 0& \alpha \\ 0& 0& 0& 0\end{array}\right)\in \rho$, but neither $\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& \beta \\ 0& 0& 0& \alpha \\ 0& 0& 0& 0\end{array}\right)\in \rho$ nor $\left(\begin{array}{cccc}0& 0& \alpha & 0\\ 0& 0& 0& 0\\ 0& 0& 0& \alpha \\ 0& 0& 0& 0\end{array}\right)\in \rho$; hence, ρ is not prime ideal of ℜ. Therefore, the assumption that ρ is prime in Theorems 1–3 and 6 cannot be omitted.

Example 2. 

Let $\Re =\{p\xi e_{12}+\eta e_{13}+\tau e_{23}|\xi ,\eta ,\tau \in \mathbb{C},p\mathrm{is}\mathrm{a}\mathrm{prime}\mathrm{number}\}$, where $\mathbb{C}$ is the complex number ring. Let $\rho =\left\{p\eta e_{13}\right\}$. Define $\vartheta ,\partial :\Re ⟶\Re$ as follows:

$$\vartheta (p\xi e_{12}+\eta e_{13}+\tau e_{23})=p\tau e_{23},\mathit{with}\partial (p\xi e_{12}+\eta e_{13}+\tau e_{23})=\xi e_{13}.$$

Thus, it is evident that ℜ is a ring, ρ is an ideal of ℜ, and ϑ is a generalized reverse derivation associated with the reverse derivation ∂ that satisfies $\vartheta \left(\xi \right)\circ \eta \pm \vartheta (\xi \circ \eta )\in \rho$, $\vartheta \left(\xi \right)\vartheta \left(\eta \right)\pm [\xi ,\eta ]\in \rho$, $\vartheta \left(\xi \eta \right)\pm \partial \left(\xi \right)\partial \left(\eta \right)\in \rho$, $\vartheta \left(\xi \eta \right)\pm \partial \left(\eta \right)\partial \left(\xi \right)\in \rho$, and $\vartheta \left(\xi \right)\vartheta \left(\eta \right)+\partial \left(\xi \right)\partial \left(\eta \right)\in \rho$ for all $\xi ,\eta \in \Re$. However, $\Re /\rho$ is non-commutative and $\partial (\Re )⊈\rho$. Moreover, ρ is not a prime ideal of ℜ since $\eta e_{31}\Re (\xi e_{31}+\eta e_{23})\in \rho$, but $\eta e_{31}\notin \rho$ and $\xi e_{31}+\eta e_{23}\notin \rho$. Therefore, assumption that ρ is prime in Theorems 1–6 cannot be omitted.

Example 3. 

Consider the ring of integers $\mathbb{Z}$ and let $\Re =\{\left(\begin{array}{ccc}0& 0& 0\\ \alpha & 0& 0\\ \beta & 2\gamma & 0\end{array}\right)|\alpha ,\beta ,\gamma \in \mathbb{Z}\},$$\rho =\{\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 2\beta & 0& 0\end{array}\right)|\beta \in \mathbb{Z}\}$. Define $\vartheta ,\partial :\Re ⟶\Re$ by $\vartheta \left(\begin{array}{ccc}0& 0& 0\\ \alpha & 0& 0\\ \beta & 2\gamma & 0\end{array}\right)=\left(\begin{array}{ccc}0& 0& 0\\ 2\alpha & 0& 0\\ 0& 0& 0\end{array}\right),\partial \left(\begin{array}{ccc}0& 0& 0\\ \alpha & 0& 0\\ \beta & 2\gamma & 0\end{array}\right)=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ \gamma & 0& 0\end{array}\right)$. Thus, it is evident that ℜ is a ring, ρ is an ideal of ℜ, and ϑ is a generalized reverse derivation associated with the reverse derivation ∂ that satisfies the following identities for all $\xi ,\eta ,\tau \in \Re$: $\vartheta \left(\xi \right)\circ \eta -\vartheta (\xi \circ \eta )\in \rho$, $\vartheta \left(\xi \right)\circ \eta +\vartheta (\xi \circ \eta )\in \rho$, $\vartheta \left(\xi \right)\vartheta \left(\eta \right)\pm [\xi ,\eta ]\in \rho$, $\vartheta \left(\xi \right)\vartheta \left(\eta \right)+\partial \left(\xi \right)\partial \left(\eta \right)\in \rho$ for all $\left(\begin{array}{ccc}0& 0& 0\\ \alpha & 0& 0\\ \beta & 2\gamma & 0\end{array}\right)\in \Re$. However, $\Re /\rho$ is non-commutative and $\partial (\Re )⊈\rho$. Moreover, ρ is not a prime ideal of ℜ since $\left(\begin{array}{ccc}0& 0& 0\\ \alpha & 0& 0\\ 0& 0& 0\end{array}\right)\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 2\gamma & 0\end{array}\right)\in \rho$, but $\left(\begin{array}{ccc}0& 0& 0\\ \alpha & 0& 0\\ 0& 0& 0\end{array}\right)\notin \rho$ and $\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 2\gamma & 0\end{array}\right)\notin \rho$. Therefore, the assumption that ρ is prime in Theorems 1–3 and 6 cannot be omitted.

Example 4. 

In Example 3, one can note that $\vartheta \left(\xi \eta \right)\pm \partial \left(\xi \right)\partial \left(\eta \right)\in \rho$ and $\vartheta \left(\xi \eta \right)\pm \partial \left(\eta \right)\partial \left(\xi \right)\in \rho$ hold for all $\xi ,\iota \in \Re$ though $\vartheta (\Re )⊈\rho$. This emphasizes the necessity of primeness of ρ.

4. Conclusions

In the current article, we continued the study of generalized reverse derivation associated with reverse derivation via a contemporary approach wherein we assume that ring ℜ has no restrictions and the studied identities are contained in prime ideal $\rho$. We have reached the following results: associated derivation maps a ring ℜ to $\rho$, or a quotient ring of ℜ by prime ideal $\rho$ becomes a commutative integral domain, or the generalized reverse derivation mapping the ring to the chosen prime ideal $\rho$ as proven in this article. We conclude with two examples clarifying the necessity of the considered assumption hypotheses.

In future studies of our current topic, the behavior of a quotient ring $\Re /\rho$ can be explored by replacing the generalized reverse derivation with any of the following mappings: a generalized $(\alpha ,\beta )$-derivation where $\alpha$ and $\beta$ are automorphisms on ℜ, or a multiplicative reverse derivation, or a generalized P-reverse derivation, or a generalized reverse homoderivation.