Factor Rings with Algebraic Identities via Generalized Derivations

Abstract

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

1. Introduction

Consider the symbol ℜ, which denotes an associative ring with center $Z(\Re )$. A ring ℜ is defined as a prime ring if for any $\omega ,\mu \in \Re$, $\omega \Re \mu =\left\{0\right\}$, then $\omega =0$ or $\mu =0$. A prime ideal is referred to as a proper ideal $\Pi$ satisfying the condition $\omega \Re \mu \subseteq \Pi$ with $\omega \in \Pi$ or $\mu \in \Pi$. An integral domain refers to the concept of a commutative ring with a multiplicative identity, while a ring without nonzero divisors is known as a domain. Every integral domain is a prime ring, but the converse does not need to be true in general. Examples are available in the literature. For $\omega ,\mu \in \Re$, $[\omega ,\mu ]=\omega \mu -\mu \omega (resp.\omega \circ \mu =\omega \mu +\mu \omega )$ denotes the commutator (the anticommutator, respectively).

An additive mapping $\varphi :\Re ⟶\Re$ is a derivation if $\varphi \left(\omega \mu \right)=\varphi \left(\omega \right)\mu +\omega \varphi \left(\mu \right)$ for all $\omega ,\mu \in \Re$. For a fixed $\tau \in \Re$, a mapping ${\varphi }_{\tau }:\Re ⟶\Re$ such that $\omega \mapsto [\tau ,\omega ]$ is a derivation, which is called the inner derivation induced by $\tau$. An additive mapping $\Psi :\Re ⟶\Re$ is a generalized derivation if there exists a derivation $\varphi$ of ℜ such that $\Psi \left(\omega \mu \right)=\Psi \left(\omega \right)\mu +\omega \varphi \left(\mu \right)$ for all $\omega ,\mu \in \Re$. The familiar examples of generalized derivation are derivations and left multipliers (which is an additive mapping $\Psi :\Re ⟶\Re$ satisfies that $\Psi \left(\omega \mu \right)=\Psi \left(\omega \right)\mu$.

One of the important results that established the effect of derivation on the commutativity of prime rings is the first and second Posner theorems [1]. The first theorem states that if a prime ring with a characteristic not equal to two admits a product of two derivations, then one of them must be equal to zero. The second theorem states that a prime ring is commutative if it admits a nonzero centralizing derivation.

During the past few decades, several authors have established a relationship between derivations, generalized derivations, and the behavior of prime and semiprime rings that satisfy specific algebraic criteria on appropriate subsets of the rings. For more information, interested readers can refer to references [2,3,4].

Ashraf et al. [5] studied the effect of a generalized derivation $\Psi$ on the commutativity of a ring ℜ that satisfies any one of the following identities: $\Psi \left(\omega \mu \right)\pm \omega \mu \in Z(\Re )$, $\Psi \left(\omega \mu \right)\pm \mu \omega \in Z(\Re )$ and $\Psi \left(\omega \right)\Psi \left(\mu \right)\pm \omega \mu \in Z(\Re )$ for all $\omega ,\mu \in I$, where I is an ideal of ℜ. Dhara et al. [6] continued this study on prime rings that admit specific identities such as $\Psi \left(\omega \right)\Psi \left(\mu \right)\pm \mu \omega \in Z(\Re )$ for all $\omega ,\mu \in L$, where L is a Lie ideal.

Tiwari et al. [7] expanded on previous studies involving two derivations. They studied the behavior of a prime ring ℜ that admits two generalized derivations $\Psi$ and $\Omega$ that satisfy one of the following identities: $\Omega \left(\omega \mu \right)\pm \Psi \left(\omega \right)\Psi \left(\mu \right)\pm \omega \mu \in Z(\Re )$, $\Omega \left(\omega \mu \right)\pm \Psi \left(\omega \right)\Psi \left(\mu \right)\pm \mu \omega \in Z(\Re )$, $\Omega \left(\omega \mu \right)\pm \Psi \left(\mu \right)\Psi \left(\omega \right)\pm \omega \mu \in Z(\Re )$, $\Omega \left(\omega \mu \right)\pm \Psi \left(\mu \right)\Psi \left(\omega \right)\pm \mu \omega \in Z(\Re )$, $\Omega \left(\omega \mu \right)\pm \Psi \left(\mu \right)\Psi \left(\omega \right)\pm [\omega ,\mu ]\in Z(\Re )$ and $\Omega \left(\omega \mu \right)\pm \Psi \left(\omega \right)\Psi \left(\mu \right)\pm \left[\alpha \right(\omega ),\mu ]\in Z(\Re )$ for all $\omega ,\mu \in I$, where $\alpha$ is any mapping on ℜ, and I is an ideal of ℜ.

Garg et al. [8] and Bera et al. [9] further investigated closed Lie ideals and left ideals of prime and semiprime rings that admit generalized $(\alpha ,\beta )$ derivations satisfying certain identities such as $\Omega \left(\omega \mu \right)+\varphi \left(\omega \right)\Psi \left(\mu \right)=0$, $\Omega \left(\omega \mu \right)+\varphi \left(\omega \right)\Psi \left(\mu \right)+\alpha \left(\mu \omega \right)=0$, $\Omega \left(\omega \mu \right)+\varphi \left(\mu \right)\Psi \left(\omega \right)=0$, $\Omega \left(\omega \mu \right)+\varphi \left(\mu \right)\Psi \left(\omega \right)+\alpha \left(\mu \omega \right)=0$, $\Omega \left(\omega \mu \right)+\Psi \left(\omega \right)\Psi \left(\mu \right)=0$ and $\Omega \left(\omega \mu \right)+\Psi \left(\mu \right)\Psi \left(\omega \right)=0$. In Creedon [10], an improvement was made to Posner’s first theorem by proving that if a ring ℜ admits the product of two derivations contained in a prime ideal $\Pi$ such that $char(\Re /\Pi )\ne 2$, then one of the derivations is contained in $\Pi$.

Almahdi et al. [11] expanded Posner’s second theorem without any restrictions on a ring ℜ. They demonstrated that if $\varphi :\Re ⟶\Re$ is a derivation such that ℜ satisfies $\left[\varphi \right(\omega ),\omega ],\mu ]\subseteq \Pi$ for all $\omega ,\mu \in \Re$, then the factor ring $\Re /\Pi$ is a commutative integral domain or $\varphi (\Re )\subseteq \Pi$, where $\Pi$ is a prime ideal of ℜ. Numerous researchers have continued their studies in this area, as evidenced by works such as [12,13,14,15,16].

Motivated by previous studies, this article further studies the behavior of a ring $\Re /\Pi$ when ℜ admits two generalized derivations that satisfy any of the following identities $\forall \omega ,\mu \in \Re$: $\left(i\right)$ $\Omega \left(\omega \mu \right)\pm \varphi \left(\omega \right)\Psi \left(\mu \right)\pm \omega \mu \in \Pi$, $\left(ii\right)$ $\Omega \left(\omega \mu \right)\pm \varphi \left(\omega \right)\Psi \left(\mu \right)\pm \mu \omega \in \Pi$, $\left(iii\right)$ $\Omega \left(\omega \mu \right)\pm \varphi \left(\mu \right)\Psi \left(\omega \right)\pm \omega \mu \in \Pi$, $\left(iv\right)$ $\Omega \left(\omega \mu \right)\pm \varphi \left(\mu \right)\Psi \left(\omega \right)\pm \mu \omega \in \Pi$, $\left(v\right)$ $\Omega \left(\omega \mu \right)\pm \Psi \left(\omega \right)\Psi \left(\mu \right)\in \Pi$, $\left(vi\right)$ $\Omega \left(\omega \mu \right)\pm \Psi \left(\mu \right)\Psi \left(\omega \right)\in \Pi$, and $\left(vii\right)$ $\Psi \left(\omega \mu \right)\pm \Omega \left(\omega \right)\mu \pm \mu \omega \in \Pi$. Additionally, several consequences are derived. Finally, examples are provided to illustrate the importance of the primeness of $\Pi$ in our theorems.

2. Preliminaries

In this section, we provide some important identities and lemmas that are used throughout this current article to make it easier to access the proofs of our theorems. To begin, the following identities hold for all $\omega ,\mu ,\tau \in \Re$:

$$\begin{gathered} [\omega \mu ,\tau ]=\omega [\mu ,\tau ]+[\omega ,\tau ]\mu , \\ [\omega ,\mu \tau ]=\mu [\omega ,\tau ]+[\omega ,\mu ]\tau , \\ \omega \circ \left(\mu \tau \right)=(\omega \circ \mu )\tau -\mu [\omega ,\tau ]=\mu (\omega \circ \tau )+[\omega ,\mu ]\tau , \\ \left(\omega \mu \right)\circ \tau =\omega (\mu \circ \tau )-[\omega ,\tau ]\mu =(\omega \circ \tau )\mu +\omega [\mu ,\tau ]. \end{gathered}$$

The proof of the following two lemmas can be found in [17].

Lemma 1

([17] (Lemma 1.2)). If Π is a prime ideal of a ring ℜ, then $\Re /\Pi$ is a commutative integral domain if $[\omega ,\mu ]\in \Pi$ for every two elements $\omega ,\mu \in \Re$.

Lemma 2

([17] (Proposition 1.3)). Let Π be a prime ideal of a ring ℜ. Then, $\Re /\Pi$ is a commutative integral domain, or ϕ maps ℜ to Π, if $[\Psi (\omega ),\omega ]\in \Pi$ for every $\omega \in \Re$.

The following special case arises when $\Psi =\varphi$ in the previous lemma:

Corollary 1

([11] (Lemma 2.1)). Let Π be a prime ideal of a ring ℜ. Then, $\Re /\Pi$ is a commutative integral domain, or ϕ maps ℜ to Π, if $\left[\varphi \right(\omega ),\omega ]\in \Pi$ for every $\omega \in \Re$.

Additionally, the following fact is used frequently:

Remark 1.

If Ω is a generalized derivation on ℜ, then $\Omega \pm I{d}_{\Re }$ is also a generalized derivation.

3. Main Results

In the following, let $\Pi$ denote a prime ideal of a ring ℜ. Additionally, let $(\Psi ,\varphi )$ and $(\Omega ,\delta )$ represent the generalized derivations $\Psi$ and G associated with derivations $\varphi$ and $\delta$, respectively. Furthermore, the map $I{d}_{\Re }:\Re ⟶\Re$, defined by the relation $I{d}_{\Re }\left(\omega \right)=\omega$$\forall \omega \in \Re ,$ denotes the identity mapping on ℜ.

Theorem 1.

Let Π be a prime ideal of any ring ℜ. If $(\Psi ,\varphi )$ and $(\Omega ,\delta )$ are generalized derivations of ℜ such that $\Omega \left(\omega \mu \right)\pm \varphi \left(\omega \right)\Psi \left(\mu \right)\pm \omega \mu \in \Pi$ for all $\omega ,\mu \in \Re$, then $\varphi (\Re )\subseteq \Pi$, $\delta (\Re )\subseteq \Pi ,$ and $(\Omega +I{d}_{\Re })(\Re )\subseteq \Pi$.

Proof. 

We prove this in the following three cases:

Case (i): If $\Psi =0$ and $\Omega \ne 0$, then we have

$$\Omega \left(\omega \mu \right)\pm \omega \mu \in \Pi \forall \omega ,\mu \in \Re .$$

(1)

Replacing $\mu$ by $\mu \tau$ in the previous equation gives

$$\Omega \left(\omega \mu \right)\tau +\omega \mu \delta \left(\tau \right)\pm \omega \mu \tau \in \Pi \forall \omega ,\mu ,\tau \in \Re .$$

(2)

Subtracting Equation (1) from Equation (2) gives $\omega \Re \delta \left(\tau \right)\subseteq \Pi$ for all $\omega ,\tau \in \Re$. The primeness of $\Pi$ implies that $\omega \in \Re$ or $\delta \left(\tau \right)\in \Pi$ for all $\omega ,\tau \in \Re$. The first case is contradictory, while the second gives $\delta (\Re )\subseteq \Pi$. Applying this in Equation (2), we obtain $(\Omega (\omega )\pm \omega )\Re \tau \subseteq \Pi$ for all $\omega ,\tau \in \Re$. Therefore, the primeness of $\Pi$ forces $(\Omega \pm I{d}_{\Re })(\Re )\subseteq \Pi$.

Case (ii): If $\Psi \ne 0$ and $\Omega =0$, then we have $\varphi \left(\omega \right)\Psi \left(\mu \right)+\omega \mu \in \Pi$ for all $\omega ,\mu \in \Re$. Replacing $\mu$ by $\mu \tau$ in the previous equation and utilizing it, we obtain $\varphi \left(\omega \right)\Re \varphi \left(\tau \right)\subseteq \Pi$ for all $\omega ,\tau \in \Re$. The primeness of $\Pi$ leads to $\varphi (\Re )\subseteq \Pi$. Therefore, our hypothesis becomes $\omega \mu \in \Pi$ for all $\omega ,\mu \in \Re$, which means that $\omega \Re \mu \subseteq \Pi$ for all $\omega ,\mu \in \Re$. The primeness of $\Pi$ implies that $\omega \in \Re$ or $\mu \in \Re$, which contradicts $\Pi \ne \Re$.

Case (iii): If $\Psi \ne 0$ and $\Omega \ne 0$, then we have

$$\Omega \left(\omega \mu \right)\pm \varphi \left(\omega \right)\Psi \left(\mu \right)+\omega \mu \in \Pi \forall \omega ,\mu \in \Re .$$

(3)

Replace $\mu$ with $\mu \tau$ in Equation (3) and use it to find

$$\omega \mu \delta \left(\tau \right)\pm \varphi \left(\omega \right)\mu \varphi \left(\tau \right)\in \Pi \forall \omega ,\mu ,\tau \in \Re .$$

(4)

Setting $\mu =\omega \mu$ in Equation (4), we have ${\omega }^2\mu \delta \left(\tau \right)\pm \varphi \left(\omega \right)\omega \mu \varphi \left(\tau \right)\in \Pi$. Multiplying the left of Equation (4) with $\omega$ and comparing it with the previous relation, we obtain

$$\left[\varphi \right(\omega ),\omega ]\mu \varphi \left(\tau \right)\in \Pi \forall \omega ,\mu ,\tau \in \Re .$$

(5)

By replacing $\tau$ with $\tau \omega$ in Equation (5), we obtain $\left[\varphi \right(\omega ),\omega ]\mu \tau \varphi \left(\omega \right)\in \Pi \forall \omega ,\mu ,\tau \in \Re$. Multiplying the right-hand side of Equation (5) by $\tau$ and comparing it with the previous equation, we obtain $\left[\varphi \right(\omega ),\omega ]\mu \left[\varphi \right(\omega ),\tau ]\in \Pi \forall \omega ,\mu ,\tau \in \Re$. In particular, $\left[\varphi \right(\omega ),\omega ]\Re \left[\varphi \right(\omega ),\omega ]\subseteq \Pi \forall \omega \in \Re .$ Since $\Pi$ is prime, we obtain $\left[\varphi \right(\omega ),\omega ]\subseteq \Pi \forall \omega \in \Re$. By using Corollary 1, we find that $\Re /\Pi$ is a commutative integral domain, or $\varphi (\Re )\subseteq \Pi$. In the first case, Equation (4) reduces to

$$\omega \delta \left(\tau \right)\pm \varphi \left(\omega \right)\varphi \left(\tau \right)\in \Pi \forall \omega ,\tau \in \Re .$$

(6)

Replace $\omega$ with $\kappa \omega$ in Equation (6) to obtain $\kappa \omega \delta \left(\tau \right)\pm \varphi \left(\kappa \right)\omega \varphi \left(\tau \right)\pm \kappa \varphi \left(\omega \right)\varphi \left(\tau \right)\in \Pi \forall \omega ,\tau ,\kappa \in \Re$. Multiplying the left of Equation (6) by $\kappa$ and comparing it with the last relation, we have $\varphi \left(\kappa \right)\Re \varphi \left(\tau \right)\subseteq \Pi \forall \tau ,\kappa \in \Re$. The primeness of $\Pi$ leads to $\varphi (\Re )\subseteq \Pi$. Therefore, Equation (6) becomes $\omega \Re \delta \left(\tau \right)\subseteq \Pi$ for all $\omega ,\tau \in \Re$. Again, the primeness of $\Pi$ gives $\delta (\Re )\subseteq \Pi$. Therefore, Equation (3) reduces to $\Omega \left(\omega \right)\mu \pm \omega \mu \in \Pi \forall \omega ,\mu \in \Re$. Hence, we find that $(\Omega \pm I{d}_{\Re })(\Re )\subseteq \Pi$. □

Now, we are able to explore the following corollaries:

Corollary 2.

Let Π be a prime ideal of any ring ℜ. If $(\Psi ,\varphi )$ and $(\Omega ,\delta )$ are generalized derivations of ℜ such that $\Omega \left(\omega \mu \right)\pm \varphi \left(\omega \right)\Psi \left(\mu \right)\in \Pi$ for all $\omega ,\mu \in \Re$, then $\varphi (\Re )\subseteq \Pi$, $\delta (\Re )\subseteq \Pi ,$ and $\Omega (\Re )\subseteq \Pi$.

Proof. 

Apply Remark 1 to the identity stated in Theorem 1 and use similar techniques to reach the desired conclusion. □

Corollary 3.

Let Π be a prime ideal of any ring ℜ. If $(\Psi ,\varphi )$ is a generalized derivation of ℜ such that $\Psi \left(\omega \mu \right)\pm \varphi \left(\omega \right)\Psi \left(\mu \right)\pm \omega \mu \in \Pi$ for all $\omega ,\mu \in \Re$, then $\varphi (\Re )\subseteq \Pi$, and $\Psi (\Re )\subseteq \Pi$.

Proof. 

By letting $\Omega =\Psi$ in Theorem 1 and following similar arguments, we can easily obtain the desired result. □

Theorem 2.

Let Π be a prime ideal of any ring ℜ. If $(\Psi ,\varphi )$ and $(\Omega ,\delta )$ are generalized derivations of ℜ such that $\Omega \left(\omega \mu \right)\pm \varphi \left(\omega \right)\Psi \left(\mu \right)\pm \mu \omega \in \Pi$ for all $\omega ,\mu \in \Re$, then $\delta (\Re )\subseteq \Pi$ and $(\Omega \pm I{d}_{\Re })(\Re )\subseteq \Pi$.

Proof. 

In cases where $\Omega =0$ and $\Psi \ne 0$, or $\Omega \ne 0$ and $\Psi =0$, the proof parallels the proof om the first paragraph of Theorem 1.

Therefore, we assume $\Omega \ne 0$ and $\Psi \ne 0$. We have

$$\Omega \left(\omega \mu \right)+\varphi \left(\omega \right)\Psi \left(\mu \right)+\mu \omega \in \Pi \forall \omega ,\mu \in \Re .$$

(7)

Replacing $\mu$ by $\mu \tau$ in Equation (7) and using it, we get

$$\omega \mu \delta \left(\tau \right)+\varphi \left(\omega \right)\mu \varphi \left(\tau \right)+\mu [\tau ,\omega ]\in \Pi \forall \omega ,\mu ,\tau \in \Re .$$

(8)

Replacing $\omega$ with $\tau \omega$ in Equation (8) and using it, we obtain

$$\varphi \left(\tau \right)\omega \mu \varphi \left(\tau \right)+\mu \tau [\tau ,\omega ]-\tau \mu [\tau ,\omega ]\in \Pi \forall \omega ,\mu ,\tau \in \Re .$$

(9)

By putting $\omega =\tau$ in Equation (9), we have $\varphi \left(\omega \right)\omega \Re \varphi \left(\omega \right)\subseteq \Pi \forall \omega \in \Re$. The primeness of $\Pi$ leads to $\varphi \left(\omega \right)\omega \in \Pi$ or $\varphi \left(\omega \right)\in \Pi \mathrm{for}\mathrm{all}\omega \in \Re$. In both cases, we have $\varphi \left(\omega \right)\omega \in \Pi \mathrm{for}\mathrm{all}\omega \in \Re$. Linearizing the last relation, we obtain $\varphi \left(\omega \right)\mu +\varphi \left(\mu \right)\omega \in \Pi \mathrm{for}\mathrm{all}\omega ,\mu \in \Re$. Replacing $\omega$ with $\omega \kappa$ in the last equation and using it, we have $\varphi \left(\omega \right)\kappa \mu -\varphi \left(\omega \right)\mu \kappa +\omega \varphi \left(\kappa \right)\mu \in \Pi \forall \omega ,\mu ,\kappa \in \Re$. Again, by replacing $\mu$ by $\mu \tau$ in the last equation and using it, we obtain $\varphi \left(\omega \right)\mu [\kappa ,\tau ]\in \Pi \forall \omega ,\mu ,\kappa ,\tau \in \Re$, i.e., $\varphi \left(\omega \right)\Re [\kappa ,\tau ]\subseteq \Pi \forall \omega ,\kappa ,\tau \in \Re$. The primeness of $\Pi$ implies $\varphi \left(\omega \right)\in \Pi$ or $[\kappa ,\tau ]\in \Pi \forall \omega ,\kappa ,\tau \in \Re$. If $\varphi \left(\omega \right)\in \Pi$ for all $\omega \in \Re$, then Equation (9) reduces to $\mu \tau [\tau ,\omega ]-\tau \mu [\tau ,\omega ]\in \Pi \forall \omega ,\mu ,\tau \in \Re$. By replacing $\mu$ with $\omega \mu$ in the previous expression and using it with some simple calculations, we arrive at $[\omega ,\tau ]\Re [\tau ,\omega ]\subseteq \Pi$ for all $\omega ,\tau \in \Re$. Again, the primeness of $\Pi$ yields $[\tau ,\omega ]\in \Pi$ for all $\omega ,\tau \in \Re$. Hence, $\Re /\Pi$ is a commutative integral domain, as shown in Lemma 1. By using this in Equation (8), we obtain $\omega \mu \delta \left(\tau \right)\in \Pi$ for all $\omega ,\mu ,\tau \in \Re$. Therefore, as previously discussed, we find that $\delta \left(\tau \right)\in \Pi$ for all $\tau \in \Re$. From this, Equation (7) becomes $\Omega \left(\omega \right)\mu +\omega \mu \in \Pi \forall \omega ,\mu \in \Re$. As discussed previously, we can deduce that $(\Omega +I{d}_{\Re })(\Re )\subseteq \Pi$.

If $[\kappa ,\tau ]\in \Pi \forall \kappa ,\tau \in \Re$, using Lemma 1, we conclude that $\Re /\Pi$ is a commutative integral domain. Hence, Equation (9) becomes $\varphi \left(\tau \right)\omega \mu \varphi \left(\tau \right)\in \Pi$ for all $\omega ,\mu ,\tau \in \Re$. Therefore, as discussed above, we find that $\varphi \left(\tau \right)\in \Pi$ for all $\tau \in \Re$. By repeating the same discussion as before, we can conclude that $(\Omega +I{d}_{\Re })(\Re )\subseteq \Pi$.

Similarly, the theorem can be proven for the identity $\Omega \left(\omega \mu \right)\pm \varphi \left(\omega \right)\Psi \left(\mu \right)-\mu \omega \in \Pi \forall \omega ,\mu \in \Re$. □

By setting $\Omega =\Omega -I{d}_{\Re }$ in the identity $\Omega \left(\omega \mu \right)\pm \varphi \left(\omega \right)\Psi \left(\mu \right)+\mu \omega \in \Pi \forall \omega ,\mu \in \Re$, we can easily deduce the following corollary:

Corollary 4.

Let Π be a prime ideal of any ring ℜ. If $(\Psi ,\varphi )$ and $(\Omega ,\delta )$ are generalized derivations of ℜ such that $\Omega \left(\omega \mu \right)\pm \varphi \left(\omega \right)\Psi \left(\mu \right)-[\omega ,\mu ]\in \Pi$ for all $\omega ,\mu \in \Re$, then $\delta (\Re )\subseteq \Pi$ and $\Omega (\Re )\subseteq \Pi$.

Furthermore, if we let $\Omega =\Psi$ in Theorem 2, we can obtain the following corollary:

Corollary 5.

Let Π be a prime ideal of any ring ℜ. If $(\Psi ,\varphi )$ is a generalized derivation of ℜ such that $\Psi \left(\omega \mu \right)\pm \varphi \left(\omega \right)\Psi \left(\mu \right)\pm \mu \omega \in \Pi$ for all $\omega ,\mu \in \Re$, then $\varphi (\Re )\subseteq \Pi$ and $(\Psi \pm I{d}_{\Re })(\Re )\subseteq \Pi$.

Theorem 3.

Let Π be a prime ideal of any ring ℜ. If $(\Psi ,\varphi )$ and $(\Omega ,\delta )$ are generalized derivations of ℜ such that $\Omega \left(\omega \mu \right)\pm \varphi \left(\mu \right)\Psi \left(\omega \right)\pm \omega \mu \in \Pi$ for all $\omega ,\mu \in \Re$, then $\varphi (\Re )\subseteq \Pi$, $\delta (\Re )\subseteq \Pi$ and $(\Omega \pm I{d}_{\Re })(\Re )\subseteq \Pi$.

Proof. 

In the cases where $\Omega =0$ and $\Psi \ne 0$ or $\Omega \ne 0$ and $\Psi =0$, the proof parallels the proof of the first paragraph of Theorem 1. Therefore, we assume $\Omega \ne 0$ and $\Psi \ne 0.$ Then, we have

$$\Omega \left(\omega \mu \right)+\varphi \left(\mu \right)\Psi \left(x\right)+\omega \mu \in \Pi \forall \omega ,\mu \in \Re .$$

(10)

Setting $\omega =\omega \mu$ in Equation (10) and applying it, we obtain

$$\omega \mu \delta \left(\mu \right)\pm \varphi \left(\mu \right)\omega \varphi \left(\mu \right)\in \Pi \forall \omega ,\mu \in \Re .$$

(11)

Again, replacing $\omega$ by $\mu \omega$ in Equation (11), we have $\mu \omega \mu \delta \left(\mu \right)\pm \varphi \left(\mu \right)\mu \omega \varphi \left(\mu \right)\in \Pi$ for all $\omega ,\mu \in \Pi$. Multiplying the left of Equation (11) by $\mu$ and then comparing it with the last equation, we deduce that $\left[\varphi \right(\mu ),\mu ]\omega \varphi \left(\mu \right)\in \Pi$ for all $\mu \in \Re$. The primeness of $\Pi$ together and using the methods after Equation (5) give $\Re /\Pi$ a commutative integral domain or $\varphi (\Re )\subseteq \Pi .$ We start with the first case. Using the commutativity of $\Re /\Pi$ in our initial hypothesis, we obtain $\Omega \left(\mu \omega \right)+\varphi \left(\mu \right)\Psi \left(\omega \right)+\mu \omega \in \Pi \forall \omega ,\mu \in \Re$. This is the same as in Theorem 1, so we obtain the desired conclusion immediately.

On the other hand, assuming that $\varphi (\Re )\subseteq \Pi$, then Equation (11) reduces to $\omega \Re \delta \left(\mu \right)\subseteq \Pi \forall \omega ,\mu \in \Re$. Hence, the primeness of $\Pi$ together with $\Pi \ne \Re$ gives $\delta (\Re )\subseteq \Pi$. Utilizing these in our initial hypothesis, we have $\Omega \left(\omega \right)\mu \pm \omega \mu \in \Pi \forall \omega ,\mu \in \Re$, which implies that $(\Omega \pm I{d}_{\Re })\left(\omega \right)\Re \subseteq \Pi$ for all $\omega \in \Re$. Thus, the primeness of $\Pi$ gives $(\Omega \pm I{d}_{\Re })(\Re )\subseteq \Pi$. □

From Theorem 3, we can provide the following corollaries:

Corollary 6.

Let Π be a prime ideal of any ring ℜ. If $(\Psi ,\varphi )$ and $(\Omega ,\delta )$ are generalized derivations of ℜ such that $\Omega \left(\omega \mu \right)\pm \varphi \left(\mu \right)\Psi \left(\omega \right)\in \Pi$ for all $\omega ,\mu \in \Re$, then $\varphi (\Re )\subseteq \Pi$, $\delta (\Re )\subseteq \Pi$ and $\Omega (\Re )\subseteq \Pi$.

Proof. 

It is easy to verify that by substituting $\Omega \mp I{d}_{\Re }$ in Theorem 3 and following similar arguments, we can obtain the desired result. □

Corollary 7.

Let Π be a prime ideal of any ring ℜ. If $(\Psi ,\varphi )$ is a generalized derivation of ℜ such that $\Psi \left(\omega \mu \right)\pm \varphi \left(\mu \right)\Psi \left(\omega \right)\pm \omega \mu \in \Pi$ for all $\omega ,\mu \in \Re$, then $\varphi (\Re )\subseteq \Pi$ and $\Psi (\Re )\subseteq \Pi$.

Theorem 4.

Let Π be a prime ideal of any ring ℜ. If $(\Psi ,\varphi )$ and $(\Omega ,\delta )$ are generalized derivations of ℜ such that $\Omega \left(\omega \mu \right)\pm \varphi \left(\mu \right)\Psi \left(\omega \right)\pm \mu \omega \in \Pi$ for all $\omega ,\mu \in \Re$, then $\varphi (\Re )\subseteq \Pi$, $\delta (\Re )\subseteq \Pi$, $\Re /\Pi$ is a commutative integral domain, and $(\Omega \pm I{d}_{\Re })(\Re )\subseteq \Pi$.

Proof. 

In cases where $\Omega =0$ and $\Psi \ne 0$, or $\Omega \ne 0$ and $\Psi =0$, the proof follows the structure of the first paragraph in Theorem 1. Therefore, we assume $\Omega \ne 0$ and $\Psi \ne 0.$ Then, we have

$$\Omega \left(\omega \mu \right)\pm \varphi \left(\mu \right)\Psi \left(\omega \right)+\mu \omega \in \Pi \forall \omega ,\mu \in \Re .$$

(12)

This can be rewritten as

$$\Omega \left(\omega \right)\mu +\omega \delta \left(\mu \right)\pm \varphi \left(\mu \right)\Psi \left(\omega \right)+\mu \omega \in \Pi \forall \omega ,\mu \in \Re .$$

(13)

Substituting $\omega$ with $\omega \tau$ in Equation (13), then using it, we have

$$\Omega \left(\omega \right)[\tau ,\mu ]+\omega \delta \left(\tau \right)\mu -\omega \delta \left(\mu \right)\tau +\omega \tau \delta \left(\mu \right)\pm \varphi \left(\mu \right)\omega \varphi \left(\tau \right)\in \Pi \forall \omega ,\mu ,\tau \in \Re .$$

(14)

Replacing $\tau$ by $\tau \mu$ in Equation (14) and using it, we have

$$\omega \tau \mu \delta \left(\mu \right)\pm \varphi \left(\mu \right)\omega \tau \varphi \left(\mu \right)\in \Pi \forall \omega ,\mu ,\tau \in \Re .$$

(15)

Replacing $\omega$ by $\kappa \omega$ in Equation (15), we obtain $\kappa \omega \tau \mu \delta \left(\mu \right)\pm \varphi \left(\mu \right)\kappa \omega \tau \varphi \left(\mu \right)\in \Pi$$\forall \omega ,\mu ,\tau ,\kappa \in \Re$. Multiplying the left of Equation (15) by $\kappa$ and comparing it with the last relation, we obtain $\left[\varphi \right(\mu ),\kappa ]\Re \tau \varphi \left(\mu \right)\subseteq \Pi$ for all $\mu ,\tau ,\kappa \in \Re$. The primeness of $\Pi$ forces that $\left[\varphi \right(\mu ),\kappa ]\in \Pi$ or $\tau \varphi \left(\mu \right)\in \Pi$ for all $\mu ,\tau ,\kappa \in \Re$. If we set $\Gamma =\{\mu \in \Re |\left[\varphi \right(\mu ),\kappa ]\in \Pi ,\kappa \in \Re \}$ and $\Delta =\{\mu \in \Re |\tau \varphi \left(\mu \right)\in \Pi ,\tau \in \Re \}$, $\Gamma$ and $\Delta$ are two subgroups of ℜ with $\Gamma \cup \Delta =\Re$. Brauer’s trick forces $\Gamma =\Re$ or $\Delta =\Re$. These two cases similarly follow the previous discussion regarding the technique in Equation (5), to obtain that $\Re /\Pi$ is a commutative integral domain, or $\varphi \left(\mu \right)\in \Pi$ for all $\mu \in \Re$.

Suppose that $\varphi \left(\mu \right)\in \Pi$ for all $\mu \in \Re$, then Equation (15) leads to $\mu \delta \left(\mu \right)\in \Pi$ for all $\mu \in \Re$. Linearizing the last relation, then using similar techniques as in the proof in Theorem 2, we obtain that $\Re /\Pi$ is a commutative integral domain, or $\delta \left(\mu \right)\in \Pi$ for all $\mu \in \Re$. The second case forces Equation (14) to become $\Omega \left(\omega \right)[\tau ,\mu ]\in \Pi$ for all $\omega ,\tau ,\mu \in \Re$. Replacing $\omega$ with $\omega \eta$ in the last relation and using it, we deduce that $\Omega \left(\omega \right)\Re [\tau ,\mu ]\subseteq \Pi$ for all $\omega ,\tau ,\mu \in \Re$. Applying the primeness of $\Pi$, we obtain $\Re /\Pi$ is a commutative integral domain or $\Omega \left(\mu \right)\in \Pi$ for all $\mu \in \Re$. In the case of $\Omega \left(\mu \right)\in \Pi$ for all $\mu \in \Re$, the initial hypothesis is reduced to $\mu \omega \in \Pi$ for all $\omega ,\mu \in \Re$. That is, $\mu \Re \omega \subseteq \Pi$ for all $\omega ,\mu \in \Re$. The primeness of $\Pi$ gives $\omega \in \Pi$ or $\mu \in \Pi$ for all $\omega ,\mu \in \Re$, which is contradictory.

On the other hand, assume that $\Re /\Pi$ is a commutative integral domain, then Equation (14) reduces to $\delta \left(\tau \right)\mu \pm \varphi \left(\mu \right)\varphi \left(\tau \right)\in \Pi \forall \mu ,\tau \in \Re$. Replace $\mu$ with $\mu \kappa$ in the last relation and use it to obtain $\varphi \left(\kappa \right)\Re \varphi \left(\tau \right)\subseteq \Pi \forall \kappa ,\tau \in \Re$; hence, the primeness of $\Pi$ leads to $\varphi (\Re )\subseteq \Pi$. Simply, this, together with Equation (15), yields $\delta (\Re )\subseteq \Pi$. By combining these with Equation (13), we can easily find that $\Omega \left(\omega \right)\mu +\omega \mu \in \Pi ,\forall \omega ,\mu \in \Re$. Thus, we can conclude that $(\Omega +I{d}_{\Re })(\Re )\subseteq \Pi$.

Using a similar technique, one can obtain the desired conclusion in the case of $\Omega \left(\omega \mu \right)\pm \varphi \left(\mu \right)\Psi \left(\omega \right)-\mu \omega \in \Pi \mathrm{for}\mathrm{all}\omega ,\mu \in \Re$. □

As a consequence of Theorem 4, we can present the following:

Corollary 8.

Let Π be a prime ideal of any ring ℜ. If $(\Psi ,\varphi )$ and $(\Omega ,\delta )$ are generalized derivations of ℜ such that $\Omega \left(\omega \mu \right)\pm \varphi \left(\mu \right)\Psi \left(\omega \right)-[\omega ,\mu ]\in \Pi$ for all $\omega ,\mu \in \Re$, then $\varphi (\Re )\subseteq \Pi$, $\delta (\Re )\subseteq \Pi$, and $\Re /\Pi$ are a commutative integral domain, and $\Omega (\Re )\subseteq \Pi$.

Proof. 

It is clear that by substituting $\Omega -I{d}_{\Re }$ into the identity $\Omega \left(\omega \mu \right)\pm \varphi \left(\mu \right)\Psi \left(\omega \right)+\mu \omega \in \Pi \mathrm{for}\mathrm{all}\omega ,\mu \in \Re$ and following similar arguments as those used in the proof of Theorem 4, we can obtain the desired conclusion. □

Corollary 9.

Let Π be a prime ideal of any ring ℜ. If $(\Psi ,\varphi )$ is a generalized derivation of ℜ such that $\Psi \left(\omega \mu \right)\pm \varphi \left(\mu \right)\Psi \left(\omega \right)\pm \mu \omega \in \Pi$ for all $\omega ,\mu \in \Re$, then $\varphi (\Re )\subseteq \Pi$, $\Re /\Pi$ is a commutative integral domain, and $\Psi (\Re )\subseteq \Pi$.

Proof. 

The proof follows by substituting $\Psi =\Omega$ in Theorem 4. □

Theorem 5.

Let Π be a prime ideal of any ring ℜ. If $(\Psi ,\varphi )$ and $(\Omega ,\delta )$ are generalized derivations of ℜ such that $\Omega \left(\omega \mu \right)\pm \Psi \left(\omega \right)\Psi \left(\mu \right)\in \Pi$ for all $\omega ,\mu \in \Re$, then $\Re /\Pi$ is a commutative integral domain, or $\varphi (\Re )\subseteq \Pi$, $\Psi (\Re )\subseteq \Pi$, and $\Omega (\Re )\subseteq \Pi$.

Proof. 

If $\Omega =0$ and $\Psi \ne 0$, then $\Psi \left(\omega \right)\Psi \left(\mu \right)\in \Pi \forall \omega ,\mu \in \Re$. Replacing $\mu$ with $\mu \tau$ in the last equation and using it, we have $\Psi \left(\omega \right)\Re \varphi \left(\tau \right)\subseteq \Pi \forall \omega ,\tau \in \Re$. Since $\Pi$ is prime, it follows that $\Psi (\Re )\subseteq \Pi$, or $\varphi (\Re )\subseteq \Pi$.

If $\Omega \ne 0$ and $\Psi =0$, then $\Omega \left(\omega \mu \right)\in \Pi \forall \omega ,\mu \in \Re$. Replace $\mu$ with $\mu \tau$ in the last equation and use it to obtain $\omega \Re \delta \left(\tau \right)\subseteq \Pi \forall \omega ,\tau \in \Re$. The primeness of $\Pi$, implies that $\delta (\Re )\subseteq \Pi$. Applying this in our initial hypothesis gives $\Omega \left(\omega \right)\mu \in \Pi \forall \omega ,\mu \in \Re$. Again, the primeness of $\Pi$ gives $\Omega (\Re )\subseteq \Pi$.

Therefore, assuming that $\Psi \ne 0$ and $\Omega \ne 0$, we have

$$\Omega \left(\omega \mu \right)\pm \Psi \left(\omega \right)\Psi \left(\mu \right)\in \Pi \forall \omega ,\mu \in \Re .$$

(16)

The previous equation can be rewritten as

$$\Omega \left(\omega \right)\mu +\omega \delta \left(\mu \right)\pm \Psi \left(\omega \right)\Psi \left(\mu \right)\in \Pi \forall \omega ,\mu \in \Re .$$

(17)

Replacing $\mu$ by $\mu \tau$ in Equation (17) and using it, we get

$$\omega \mu \delta \left(\tau \right)\pm \Psi \left(\omega \right)\mu \varphi \left(\tau \right)\in \Pi \forall \omega ,\mu ,\tau \in \Re .$$

(18)

Again, replacing $\mu$ with $\omega \mu$ in Equation (18), we have ${\omega }^2\mu \delta \left(\tau \right)+\Psi \left(\omega \right)\omega \mu \varphi \left(\tau \right)\in \Pi$. Multiplying the left of Equation (18) by $\omega$ and comparing it with the last relation, we have

$$[\Psi (\omega ),\omega ]\Re \varphi \left(\tau \right)\subseteq \Pi \forall \omega ,\mu ,\tau \in \Re .$$

(19)

The primeness of $\Pi$ gives $[\Psi (\omega ),\omega ]\in \Pi$ or $\varphi \left(\tau \right)\in \Pi$ for all $\omega ,\tau \in \Re$. The first case, together with Lemma 2, implies that $\Re /\Pi$ is a commutative integral domain, or $\varphi (\Re )\subseteq \Pi$. The second case leads to $\varphi (\Re )\subseteq \Pi$. If $\varphi (\Re )\subseteq \Pi$, then Equation (18) gives $\delta (\Re )\subseteq \Pi$. Hence, our initial hypothesis reduces to

$$\Omega \left(\omega \right)\mu \pm \Psi \left(\omega \right)\Psi \left(\mu \right)\in \Pi \forall \omega ,\mu \in \Re .$$

(20)

Replacing $\omega$ by $\omega \mu$ in Equation (20), we have

$$\Omega \left(\omega \right){\mu }^2\pm \Psi \left(\omega \right)\mu \Psi \left(\mu \right)\in \Pi \forall \omega ,\mu \in \Re .$$

(21)

Setting $\mu ={\mu }^2$ in Equation (20), we obtain

$$\Omega \left(\omega \right)y^2\pm \Psi \left(\omega \right)\Psi \left(\mu \right)\mu \in \Pi \forall \omega ,\mu \in \Re .$$

(22)

Subtracting Equations (21) and (22), we obtain $\Psi \left(\omega \right)[\Psi (\mu ),\mu ]\in \Pi$ for all $\omega ,\mu \in \Re$. Replacing $\omega$ with $\omega \tau$ in the last relation, we obtain $\Psi \left(\omega \right)\Re [\Psi (\mu ),\mu ]\in \Pi$ for all $\omega ,\mu \in \Re$. The primeness of $\Pi$ gives $\Psi \left(\omega \right)\in \Pi$ or $[\Psi (\mu ),\mu ]\in \Pi$ for all $\omega ,\mu \in \Re$. In both cases, we can conclude that $\Psi (\Re )\subseteq \Pi$, or $\Re /\Pi$ is a commutative integral domain. By applying $\Psi (\Re )\subseteq \Pi$ in our initial hypothesis, we have $\Omega \left(\omega \right)\mu \in \Pi$ for all $\omega ,\mu \in \Re$. The primeness of $\Pi$ forces $\Omega (\Re )\subseteq \Pi$.

On the other hand, if $\Re /\Pi$ is commutative, then Equation (18) can be rewritten as

$$\omega \delta \left(\tau \right)\pm \Psi \left(\omega \right)\varphi \left(\tau \right)\in \Pi \forall \omega ,\tau \in \Re .$$

(23)

Replace $\omega$ with $\omega \kappa$ in Equation (23) to obtain $\kappa \omega \delta \left(\tau \right)\pm \kappa \Psi \left(\omega \right)\varphi \left(\tau \right)\pm \varphi \left(\kappa \right)\omega \varphi \left(\tau \right)\in \Pi$ for all $\omega ,\tau ,\kappa \in \Re$. Multiplying the left-hand side of Equation (23) by $\kappa$ and then comparing it with the last expression, we obtain $\varphi \left(\kappa \right)\Re \varphi \left(\tau \right)\subseteq \Pi$ for all $\kappa ,\tau \in \Re$. The primeness of $\Pi$ implies that $\varphi (\Re )\subseteq \Pi$. By following the same discussion as above, we can reach the desired conclusion. □

As an application of Theorem 5, we can derive the following corollary by setting $\Psi =\Omega$.

Corollary 10.

Let Π be a prime ideal of any ring ℜ. If $(\Psi ,\varphi )$ is a generalized derivation of ℜ such that $\Psi \left(\omega \mu \right)\pm \Psi \left(\omega \right)\Psi \left(\mu \right)\in \Pi$ for all $\omega ,\mu \in \Re$, then $\Re /\Pi$ is a commutative integral domain or $\varphi (\Re )\subseteq \Pi$ and $\Psi (\Re )\subseteq \Pi$.

Theorem 6.

Let Π be a prime ideal of any ring ℜ. If $(\Psi ,\varphi )$ and $(\Omega ,\delta )$ are generalized derivations of ℜ such that $\Omega \left(\omega \mu \right)\pm \Psi \left(\mu \right)\Psi \left(\omega \right)\in \Pi$ for all $\omega ,\mu \in \Re$, then $\Re /\Pi$ is a commutative integral domain or $\varphi (\Re )\subseteq \Pi$, $\Psi (\Re )\subseteq \Pi$, and $\Omega (\Re )\subseteq \Pi$.

Proof. 

In cases where $\Omega =0$ and $\Psi \ne 0$ or $\Omega \ne 0$ and $\Psi =0$, the proof parallels the first paragraph in Theorem 5. Therefore, assuming $\Omega \ne 0$ and $\Psi \ne 0$, we have

$$\Omega \left(\omega \mu \right)\pm \Psi \left(\mu \right)\Psi \left(\omega \right)\in \Pi \forall \omega ,\mu \in \Re .$$

(24)

Replacing $\omega$ by $\omega \mu$ in Equation (24) and using it, we have

$$\omega \mu \delta \left(\mu \right)+\Psi \left(\mu \right)\omega \varphi \left(\mu \right)\in \Pi \forall \omega ,\mu \in \Re .$$

(25)

Once again, replacing $\omega$ with $\mu \omega$ in Equation (25), we have $\mu \omega \mu \delta \left(\mu \right)+\Psi \left(\mu \right)\mu \omega \varphi \left(\mu \right)\in \Pi$. By multiplying the left of Equation (25) by $\mu$ and comparing it with the last relation, we obtain $[\Psi (\mu ),\mu ]\Re \varphi \left(\mu \right)\subseteq \Pi$ for all $\omega ,\mu \in \Re$. As discussed after Equation (19), $\Re /\Pi$ is a commutative integral domain, or $\varphi (\Re )\subseteq \Pi$.

Temporarily assuming that $\Re /\Pi$ is not a commutative integral domain, we then have $\varphi (\Re )\subseteq \Pi$. This reduces Equation (25) to $\delta (\Re )\subseteq \Pi$. Thus, the given hypothesis becomes

$$\Omega \left(\omega \right)\mu \pm \Psi \left(\mu \right)\Psi \left(\omega \right)\in \Pi \forall \omega ,\mu \in \Re .$$

(26)

Substituting $\omega$ with $\omega \mu$ in Equation (26), we have

$$\Omega \left(\omega \right){\mu }^2\pm \Psi \left(\mu \right)\Psi \left(\omega \right)\mu \in \Pi \forall \omega ,\mu \in \Re .$$

(27)

Setting $\mu ={\mu }^2$ in Equation (26), we have

$$\Omega \left(\omega \right){\mu }^2\pm \Psi \left(\mu \right)\mu \Psi \left(\omega \right)\in \Pi \forall \omega ,\mu \in \Re .$$

(28)

By comparing Equations (27) and (28), we find $\Psi \left(\mu \right)[\Psi (\omega ),\mu ]\in \Pi$ for all $\omega ,\mu \in \Re$. Replacing $\omega$ with $\omega \tau$ in the last relation and using it, we obtain $\Psi \left(\mu \right)\Psi \left(\omega \right)[\tau ,\mu ]\in \Pi$ for all $\omega ,\mu ,\tau \in \Re$. For any $\kappa \in \Re$, replacing $\omega$ by $\omega \kappa$, we find $\Psi \left(\mu \right)\Psi \left(\omega \right)\Re [\tau ,\mu ]\subseteq \Pi$. By using the primeness of $\Pi$, it is easy to deduce that $\Psi (\Re )\subseteq \Pi$, or $\Re /\Pi$ is a commutative integral domain, which contradicts our temporary assumption. Therefore, we assume that $\Psi (\Re )\subseteq \Pi$, then our initial hypothesis becomes $\Omega \left(\omega \right)\mu \in \Pi \forall \omega ,\mu \in \Re$, which implies that $\Omega (\Re )\subseteq \Pi .$ □

Corollary 11.

Let Π be a prime ideal of any ring ℜ. If $(\Psi ,\varphi )$ is a generalized derivation of ℜ such that $\Psi \left(\omega \mu \right)\pm \Psi \left(\mu \right)\Psi \left(\omega \right)\in \Pi$ for all $\omega ,\mu \in \Re$, then $\Re /\Pi$ is a commutative integral domain or $\varphi (\Re )\subseteq \Pi$, and $\Psi (\Re )\subseteq \Pi$.

Theorem 7.

Let Π be a prime ideal of any ring ℜ. If $(\Psi ,\varphi )$ and $(\Omega ,\delta )$ are generalized derivations of ℜ such that $\Psi \left(\omega \mu \right)\pm \Omega \left(\omega \right)\mu \pm \mu \omega \in \Pi$ for all $\omega ,\mu \in \Re$, then $\Re /\Pi$ is a commutative integral domain, $\varphi (\Re )\subseteq \Pi$, and $(\Psi \pm \Omega \pm I{d}_{\Re })(\Re )\subseteq \Pi$.

Proof. 

If $\Psi =0$ and $\Omega =0$, then we have $\omega \mu \in \Pi \forall \omega ,\mu \in \Re .$ That is, $\omega \Re \mu \subseteq \Pi \forall \omega ,\mu \in \Re$, which contradicts the primeness of $\Pi$.

If $\Psi =0$ and $\Omega \ne 0$, then

$$\Omega \left(\omega \right)\mu \pm \mu \omega \in \Pi \forall \omega ,\mu \in \Re .$$

(29)

Setting $\omega =\omega \mu$ in Equation (29) and using it, we have

$$\omega \delta \left(\mu \right)\mu \in \Pi \forall \omega ,\mu \in \Re .$$

(30)

The primeness of $\Pi$ yields $\delta \left(\mu \right)\mu \in \Pi$ for all $\omega ,\mu \in \Re$. Hence, as discussed in Theorem 2, we have that $\delta (\Re )\subseteq \Pi$, or $\Re /\Pi$ is a commutative integral domain. The commutativity of $\Re /\Pi$ with Equation (29) leads to $(\Omega \pm I{d}_{\Re })(\Re )\subseteq \Pi$. If $\delta (\Re )\subseteq \Pi$, then replace $\omega$ with $\omega \tau$ in Equation (29) and use it to obtain $\Omega \left(\omega \right)[\mu ,\tau ]\in \Pi$ for all $\omega ,\mu ,\tau \in \Re$. Letting $\omega =\omega \kappa$ in the last relation gives $\Omega \left(\omega \right)\Re [\mu ,\kappa ]\in \Pi$ for all $\omega ,\mu ,\tau \in \Re$. The primeness of $\Pi$ leads to $\Omega (\Re )\subseteq \Pi$, or $\Re /\Pi$ is a commutative integral domain. If $\Omega (\Re )\subseteq \Pi$, Equation (29) leads to a contradiction with the primeness of $\Pi$, as previously discussed. Therefore, $\Re /\Pi$ is a commutative integral domain. Again, Equation (29) gives us $(\Omega \pm I{d}_{\Re })(\Re )\subseteq \Pi$.

If $\Psi \ne 0$ and $\Omega =0$, then

$$\Psi \left(\omega \mu \right)\pm \mu \omega \in \Pi \forall \omega ,\mu \in \Re .$$

(31)

Thus, as discussed in the first paragraph in Theorem 2, we obtain $(\Psi \pm I{d}_{\Re })(\Re )\subseteq \Pi$.

Therefore, assuming $\Psi \ne 0$ and $\Omega \ne 0$, we have

$$\Psi \left(\omega \mu \right)+\Omega \left(\omega \right)\mu \pm \mu \omega \in \Pi \forall \omega ,\mu \in \Re .$$

(32)

Replacing $\mu$ by $\mu \tau$ in Equation (32) and using it, we have

$$\omega \mu \varphi \left(\tau \right)+\mu [\tau ,\omega ]\in \Pi \forall \omega ,\mu ,\tau \in \Re .$$

(33)

Substituting $\mu$ with $\mu \kappa$ in Equation (33), we have

$$\begin{array}{c}\hfill \omega \kappa \mu \varphi \left(\tau \right)+\kappa \mu [\tau ,\omega ]\in \Pi \forall \omega ,\mu ,\tau ,\kappa \in \Re .\end{array}$$

By multiplying the left of Equation (33) by $\kappa$ and comparing it with the last relation, we have

$$\omega \kappa \mu \varphi \left(\tau \right)-\kappa \omega \mu \varphi \left(\tau \right)\in \Pi \forall \omega ,\mu ,\tau ,\kappa \in \Re .$$

(34)

That is, $[\omega ,\kappa ]\Re \varphi \left(\tau \right)\subseteq \Pi$ for all $\omega ,\tau ,\kappa \in \Re$. The primeness of $\Pi$ using a similar tactic as after Equation (5) implies that $\varphi \left(\tau \right)\in \Pi$ for all $\tau \in \Re$ or $[\omega ,\kappa ]\in \Pi$ for all $\omega ,\kappa \in \Re$. Suppose $\varphi \left(\tau \right)\in \Pi$ for all $\tau \in \Re$. Thus, Equation (33) reduces to $\mu [\tau ,\omega ]\in \Pi \forall \omega ,\mu ,\tau \in \Re$. So, we conclude that $\Re /\Pi$ is a commutative integral domain by using Lemma 1. Applying this in Equation (32), we obtain $\Psi \left(\omega \right)\mu +\Omega \left(\omega \right)\mu +\omega \mu \in \Pi$ for all $\omega ,\mu \in \Re$. Therefore, we can conclude that $(\Psi +\Omega +I{d}_{\Re })(\Re )\subseteq \Pi$. On the other hand, if $[\omega ,\kappa ]\in \Pi$ for all $\omega ,\kappa \in \Re$, then $\Re /\Pi$ is a commutative integral domain according to Lemma 1. Thus, Equation (33) becomes $\omega \mu \varphi \left(\tau \right)\in \Pi$ for all $\omega ,\mu ,\tau \in \Re$. As previously discussed, we have $\varphi (\Re )\subseteq \Pi$. From this, together with our initial assumption, we obtain $(\Psi +\Omega +I{d}_{\Re })(\Re )\subseteq \Pi$.

Following similar arguments, we can easily obtain the desired conclusion in the case of $\Psi \left(\omega \mu \right)-\Omega \left(\omega \right)\mu -\mu \omega \in \Pi$ for all $\omega ,\mu \in \Re$ □

Now, we are ready to derive the following corollaries:

Corollary 12.

Let Π be a prime ideal of any ring ℜ. If $(\Psi ,\varphi )$ and $(\Omega ,\delta )$ are generalized derivations of ℜ such that $\Psi \left(\omega \mu \right)\pm \Omega \left(\omega \right)\mu -[\omega ,\mu ]\in \Pi$ for all $\omega ,\mu \in \Re$, then $\Re /\Pi$ is a commutative integral domain, $\varphi (\Re )\subseteq \Pi$, and $(\Psi \pm \Omega )(\Re )\subseteq \Pi$.

Corollary 13.

Let Π be a prime ideal of any ring ℜ. If $(\Psi ,\varphi )$ is a generalized derivation of ℜ such that $\Psi \left(\omega \mu \right)+\Psi \left(\omega \right)\mu \pm \mu \omega \in P$ for all $\omega ,\mu \in \Re$, then $\Re /\Pi$ is a commutative integral domain, $\varphi (\Re )\subseteq \Pi$, and $(2\Psi \pm I{d}_{\Re })(\Re )\subseteq \Pi$.

Finally, we provide illustrative examples that demonstrate the necessity of the assumptions in the given theorems.

Example 1.

Let $\Re =\{\omega e_{21}+\mu e_{31}+\tau e_{32}|\omega ,\mu ,\tau \in \mathbb{H}\}$, where $\mathbb{H}$ is a Hamilton ring, and let $\Pi =\left\{0\right\}$. Define $(\Psi ,\varphi )and(\Omega ,\delta ):\Re ⟶\Re$ by

$$\begin{array}{c}\hfill \Psi (\omega e_{21}+\mu e_{31}+\tau e_{32})=2\tau e_{21},\mathit{with}\varphi (\omega e_{21}+\mu e_{31}+\tau e_{32})=2\tau e_{31},\end{array}$$

and

$$\begin{array}{c}\hfill \Omega (\omega e_{21}+\mu e_{31}+\tau e_{32})=3\omega e_{21},\mathit{with}\delta (\omega e_{21}+\mu e_{31}+\tau e_{32})=3\omega e_{31}\end{array}$$

It is easy to verify that Ψ and Ω are generalized derivations associated with derivations ϕ and δ, respectively. We can also see that $\Omega \left(\omega \mu \right)\pm \Psi \left(\omega \right)\Psi \left(\mu \right)\in \Pi$, $\Omega \left(\omega \mu \right)\pm \Psi \left(\mu \right)\Psi \left(\omega \right)\in \Pi$, $\Omega \left(\omega \mu \right)\pm \varphi \left(\omega \right)\Psi \left(\mu \right)\in \Pi$ and $\Omega \left(\omega \mu \right)\pm \varphi \left(\mu \right)\Psi \left(\omega \right)\in \Pi$ are satisfied for all $\omega ,\mu \in \Re$. However, $\Re /\Pi$ is noncommutative, $\Psi (\Re )⊈\Pi$, $\Omega (\Re )⊈\Pi$, $\varphi (\Re )⊈\Pi$, and $\delta (\Re )⊈\Pi$. Since $\omega e_{21}\Re \mu e_{31}=\left\{0\right\}$, but neither $\omega e_{21}\in \Pi$ nor $\mu e_{31}\in \Pi$, then Π is not prime. Therefore, the hypothesis that Π is prime in Theorems 5 and 6 and Corollaries 2 and 6 is necessary.

Example 2.

Let $R=\{\left(\begin{array}{ccc}0& \omega & \mu \\ 0& 0& 2\tau \\ 0& 0& 0\end{array}\right)|\omega ,\mu ,\tau \in {\mathbb{Z}}_4\}$, $andlet$ $\Pi =\{\left(\begin{array}{ccc}0& 0& 2\mu \\ 0& 0& 0\\ 0& 0& 0\end{array}\right)\}$. $Define (\Psi ,\varphi )|;(\Omega ,\delta ):\Re \to \Re by$

$$\begin{array}{c}\hfill \Psi \left(\begin{array}{ccc}0& \omega & \mu \\ 0& 0& 2\tau \\ 0& 0& 0\end{array}\right)\end{array}=\left(\begin{array}{ccc}0& 2\omega & 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)\mathit{with}\varphi \left(\begin{array}{ccc}0& \omega & \mu \\ 0& 0& 2\tau \\ 0& 0& 0\end{array}\right)=\left(\begin{array}{ccc}0& 0& \tau \\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$$

and

$$\Omega \left(\begin{array}{ccc}0& \omega & \mu \\ 0& 0& 2\tau \\ 0& 0& 0\end{array}\right)=\left(\begin{array}{ccc}0& 2\mu & 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)\mathit{with}\delta \left(\begin{array}{ccc}0& \omega & \mu \\ 0& 0& 2\tau \\ 0& 0& 0\end{array}\right)=\left(\begin{array}{ccc}0& 0& \omega \\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$$

It is easy to verify that Ψ and Ω are generalized derivations associated with the derivations ϕ and δ, respectively. We can also see that $\Omega \left(\omega \mu \right)\pm \varphi \left(\omega \right)\Psi \left(\mu \right)\pm \omega \mu \in \Pi$, $\Omega \left(\omega \mu \right)\pm \varphi \left(\omega \right)\Psi \left(\mu \right)\pm \mu \omega \in \Pi$, $\Omega \left(\omega \mu \right)\pm \varphi \left(\mu \right)\Psi \left(\omega \right)\pm \omega \mu \in \Pi$, $\Omega \left(\omega \mu \right)\pm \varphi \left(\mu \right)\Psi \left(\omega \right)\pm \mu \omega \in \Pi$, $\Omega \left(\omega \mu \right)\pm \varphi \left(\omega \right)\Psi \left(\mu \right)\in \Pi$, $\Omega \left(\omega \mu \right)\pm \varphi \left(\mu \right)\Psi \left(\omega \right)\in \Pi$ and $\Psi \left(\omega \mu \right)\pm \Omega \left(\omega \right)\mu \pm \mu \omega \in \Pi$, also $\Omega \left(\omega \mu \right)\pm \varphi \left(\omega \right)\Psi \left(\mu \right)-[\omega ,\mu ]\in \Pi$, $\Omega \left(\omega \mu \right)\pm \varphi \left(\mu \right)\Psi \left(\omega \right)-[\omega ,\mu ]\in \Pi$ and $\Psi \left(\omega \mu \right)+\Psi \left(\omega \right)\mu \pm \mu \omega \in \Pi$ for all $\omega ,\mu \in \Re$, are satisfied. However, $\Re /\Pi$ is noncommutative, $\Psi (\Re )⊈\Pi$, $\Omega (\Re )⊈\Pi$, $\varphi (\Re )⊈\Pi$, $\delta (\Re )⊈\Pi$, $(\Omega \pm I{d}_{\Re })\left(R\right)⊈\Pi$ and $(\Psi \pm \Omega \pm I{d}_{\Re })(\Re )⊈\Pi$. Since ${\left(\begin{array}{ccc}0& \omega & 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)}^2=0$, but $\left(\begin{array}{ccc}0& \omega & 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$∉ Π, then Π is not prime. Therefore, the hypothesis that Π is prime in Theorems 1–7 and Corollaries 4, 8, 12 are necessary.

4. Conclusions

This article studied the relationship between the behavior of a ring $\Re /\Pi$ and generalized derivations $(\Psi ,\varphi )$ and $(\Omega ,\delta )$ that satisfy specific identities where $\Pi$ is a prime ideal of any ring ℜ. Without imposing restrictions on ℜ being prime or semiprime, we successfully generalized and expanded upon some of the literature findings, such as [5,6,7,8,9]. Moreover, we obtained several corollaries and special cases related to our findings. To emphasize the nontriviality and importance of imposing a primeness $\Pi$ in our theorems, we explored two examples.

5. Future Studies

Our current study was limited to examining the behavior of a factor ring $\Re /\Pi$, where ℜ is an arbitrary ring that admits generalized derivations satisfying differential identities linking the whole ring ℜ and a prime ideal $\Pi$ of it. In future studies, the above results could be expanded in several directions. This includes examining identities linking a nonzero ideal I to a prime ideal $\Pi$ or exploring generalized $(\alpha ,\beta )$ derivations or generalized P-derivations.