Some Equations in Rings Involving Semiprime Ideals and Multiplicative Generalized Semiderivations

Abstract

This paper examines the commutativity of the quotient ring $\mathcal{F}/\mathrm{Y}$ by utilizing specific differential identities in a general ring $\mathcal{F}$ that contains a semiprime ideal $\mathrm{Y}$. This study particularly focuses on the role of a multiplicative generalized semiderivation $\psi$, which is associated with a map $\theta$, in determining the commutative nature of the quotient ring.

1. Introduction

In the study of associative rings, the structure and properties of ideals play a crucial role. Prime and semiprime ideals, in particular, are significant due to their ability to characterize the ring’s behavior.

Let $\mathcal{F}$ be an associative ring with center Z. A prime ideal in an associative ring $\mathcal{F}$ is a proper ideal $\mathrm{Y}$ such that for any ${\upsilon }_1,{\upsilon }_2\in \mathcal{F}$, if ${\upsilon }_1\mathcal{F}{\upsilon }_2\subseteq \mathrm{Y}$, then either ${\upsilon }_1\in \mathrm{Y}$ or ${\upsilon }_2\in \mathrm{Y}$. A ring $\mathcal{F}$ is called a prime ring if the zero ideal $\left(0\right)$ is a prime ideal, meaning that for any ${\upsilon }_1,{\upsilon }_2\in \mathcal{F}$, ${\upsilon }_1\mathcal{F}{\upsilon }_2=\left(0\right)$ implies either ${\upsilon }_1=0$ or ${\upsilon }_2=0$.

A semiprime ideal is defined as a proper ideal $\mathrm{Y}$ where for any ${\upsilon }_1\in \mathcal{F}$, if ${\upsilon }_1\mathcal{F}{\upsilon }_1\subseteq \mathrm{Y}$, then ${\upsilon }_1\in \mathrm{Y}$. A ring $\mathcal{F}$ is termed a semiprime ring if the zero ideal $\left(0\right)$ is a semiprime ideal, which indicates that the ideal generated by any element squared lies in the ideal if the element itself does.

An additive mapping $\theta$ on $\mathcal{F}$ is a derivation if $\theta \left({\upsilon }_1{\upsilon }_2\right)=\theta \left({\upsilon }_1\right){\upsilon }_2+{\upsilon }_1\theta \left({\upsilon }_2\right)$ for all ${\upsilon }_1,{\upsilon }_2\in \mathcal{F}$. The first study on derivations of prime rings was published in 1957 by Posner [1]. Several mathematicians have since generalized the notion of derivation in various ways, leading to the introduction of different kinds of derivations in the literature.

One such generalization is the concept of semiderivations, introduced by Bergen [2]. An additive mapping $\theta :\mathcal{F}\to \mathcal{F}$ is a semiderivation if there exists a function $\gamma :\mathcal{F}\to \mathcal{F}$, such that

(i)

$\theta \left({\upsilon }_1{\upsilon }_2\right)=\theta \left({\upsilon }_1\right)\gamma \left({\upsilon }_2\right)+{\upsilon }_1\theta \left({\upsilon }_2\right)=\theta \left({\upsilon }_1\right){\upsilon }_2+\gamma \left({\upsilon }_1\right)\theta \left({\upsilon }_2\right)$ and

(ii)

$\theta \left(\gamma \left({\upsilon }_1\right)\right)=\gamma \left(\theta \left({\upsilon }_1\right)\right)$

for all ${\upsilon }_1,{\upsilon }_2\in \mathcal{F}$. If $\gamma$ is the identity map, semiderivations reduce to derivations, making semiderivations a broader concept.

In 1991, Bresar [3] introduced the notion of generalized derivations: an additive mapping $\psi$ on $\mathcal{F}$ is called a generalized derivation associated with a derivation $\theta$ if $\psi \left({\upsilon }_1{\upsilon }_2\right)=\psi \left({\upsilon }_1\right){\upsilon }_2+{\upsilon }_1\theta \left({\upsilon }_2\right)$ for all ${\upsilon }_1,{\upsilon }_2\in \mathcal{F}$. Similarly, Daif [4] introduced multiplicative derivations by removing the additive condition, defining $\theta$ as a multiplicative derivation if $\theta \left({\upsilon }_1{\upsilon }_2\right)=\theta \left({\upsilon }_1\right){\upsilon }_2+{\upsilon }_1\theta \left({\upsilon }_2\right)$ for all ${\upsilon }_1,{\upsilon }_2\in \mathcal{F}$. Daif and Tammam El-Sayiad [5] further extended this to multiplicative generalized derivations.

Let S be a nonempty subset of $\mathcal{F}$. A mapping $\psi$ from $\mathcal{F}$ to $\mathcal{F}$ is called centralizing on S if $[\psi \left({\upsilon }_1\right),{\upsilon }_1]\in Z$ for all ${\upsilon }_1\in S$, and commuting on S if $[\psi \left({\upsilon }_1\right),{\upsilon }_1]=0$ for all ${\upsilon }_1\in S$. This concept has been generalized to $\theta$-commuting maps: $\psi :\mathcal{F}\to \mathcal{F}$ is a $\theta$-commuting map on S if $[\psi \left({\upsilon }_1\right),{\upsilon }_1]\in \theta$ for all ${\upsilon }_1\in S$ and some $\theta \subseteq \mathcal{F}$.

Daif and Bell in 1992 [6] showed that an ideal I of a semiprime ring $\mathcal{F}$ is contained in the center of $\mathcal{F}$ if any of the following conditions are satisfied for all ${\upsilon }_1,{\upsilon }_2\in I$:

$$\theta \left([{\upsilon }_1,{\upsilon }_2]\right)=[{\upsilon }_1,{\upsilon }_2]\mathrm{and}\theta \left([{\upsilon }_1,{\upsilon }_2]\right)=-[{\upsilon }_1,{\upsilon }_2].$$

In particular, if $I=\mathcal{F}$, then $\mathcal{F}$ is commutative. This result has been extended by many authors, who have also obtained commutativity results for prime or semiprime rings with derivations satisfying certain polynomial identities (see, e.g., [7,8,9,10,11,12]).

The commutativity of prime and semiprime rings admitting derivations remains an active area of research. Recent approaches involve examining commutativity conditions in quotient rings $\mathcal{F}/\mathrm{Y}$ rather than assuming the ring is prime (see, e.g., [13,14,15,16]).

The principal aim of this study is to explore the definition of semiderivation given by Bergen. Inspired by the notion of multiplicative generalized derivation [17], we introduce the concept of a multiplicative generalized semiderivation. A mapping $\psi$ on $\mathcal{F}$ is a multiplicative generalized semiderivation if there exists a multiplicative semiderivation $\theta$ associated with a map $\gamma$ on $\mathcal{F}$, such that

(i)

$\psi \left({\upsilon }_1{\upsilon }_2\right)=\psi \left({\upsilon }_1\right){\upsilon }_2+\gamma \left({\upsilon }_1\right)\theta \left({\upsilon }_2\right)=\theta \left({\upsilon }_1\right)\gamma \left({\upsilon }_2\right)+{\upsilon }_1\psi \left({\upsilon }_2\right)$ and

(ii)

$\psi \left(\gamma \left({\upsilon }_1\right)\right)=\gamma \left(\psi \left({\upsilon }_1\right)\right)$

for all ${\upsilon }_1,{\upsilon }_2\in \mathcal{F}$.

This study aims to investigate identities involving multiplicative generalized semiderivations in semiprime ideals. Since every prime ideal is semiprime, but the converse is not true, examining identities involving multiplicative generalized semiderivations in semiprime ideals is appropriate. We will prove Daif’s theorem and discuss some functional identities, extending and unifying several known results.

2. Main Results

Throughout this paper, $\mathcal{F}$ will be an arbitrary ring, $\mathrm{Y}$ will be a semiprime ideal of $\mathcal{F}$, $\psi$ will be a multiplicative generalized semiderivation associated with a map $\theta$ of $\mathcal{F}$, and $\gamma$ is an epimorphism on $\mathcal{F}$. For any ${\upsilon }_1,{\upsilon }_2\in \mathcal{F},$ as usual, $[{\upsilon }_1,{\upsilon }_2]={\upsilon }_1{\upsilon }_2-{\upsilon }_2{\upsilon }_1$ and ${\upsilon }_{1}o{\upsilon }_2={\upsilon }_1{\upsilon }_2+{\upsilon }_2{\upsilon }_1$ will show the associated Lie and Jordan product, respectively. The following expressions will make our work easier for us:

$$\begin{gathered} [{\upsilon }_1,{\upsilon }_2{\upsilon }_3]={\upsilon }_2[{\upsilon }_1,{\upsilon }_3]+[{\upsilon }_1,{\upsilon }_2]{\upsilon }_3 \\ [{\upsilon }_1{\upsilon }_2,{\upsilon }_3]=[{\upsilon }_1,{\upsilon }_3]{\upsilon }_2+{\upsilon }_1[{\upsilon }_2,{\upsilon }_3] \\ {\upsilon }_{1}o\left({\upsilon }_2{\upsilon }_3\right)=\left({\upsilon }_{1}o{\upsilon }_2\right){\upsilon }_3-{\upsilon }_2[{\upsilon }_1,{\upsilon }_3]={\upsilon }_2\left({\upsilon }_{1}o{\upsilon }_3\right)+[{\upsilon }_1,{\upsilon }_2]{\upsilon }_3 \\ \left({\upsilon }_1{\upsilon }_2\right)o{\upsilon }_3={\upsilon }_1\left({\upsilon }_{2}o{\upsilon }_3\right)-[{\upsilon }_1,{\upsilon }_3]{\upsilon }_2=\left({\upsilon }_{1}o{\upsilon }_3\right){\upsilon }_2+{\upsilon }_1[{\upsilon }_2,{\upsilon }_3]. \end{gathered}$$

Theorem 1. 

Let $\mathcal{F}$ be a ring admitting a semiprime ideal Y and a multiplicative generalized semiderivation ψ associated with a map θ, and γ be an epimorphism on $\mathcal{F}$. If any one of the following conditions is satisfied for all ${\upsilon }_1,{\upsilon }_2\in \mathcal{F}$, then $[\psi \left({\upsilon }_1\right),{\upsilon }_1]\in \mathrm{Y},$ for all ${\upsilon }_1\in \mathcal{F}.$

(i) 

$\psi \left([{\upsilon }_1,{\upsilon }_2]\right)\in \mathrm{Y}$;

(ii) 

$\psi \left({\upsilon }_{1}o{\upsilon }_2\right)\in \mathrm{Y}$;

(iii) 

$\psi \left([{\upsilon }_1,{\upsilon }_2]\right)\pm {\upsilon }_1{\upsilon }_2\in \mathrm{Y}$;

(iv) 

$\psi \left([{\upsilon }_1,{\upsilon }_2]\right)\pm {\upsilon }_2{\upsilon }_1\in \mathrm{Y}$;

(v) 

$\psi \left({\upsilon }_{1}o{\upsilon }_2\right)\pm {\upsilon }_1{\upsilon }_2\in \mathrm{Y}$;

(vi) 

$\psi \left({\upsilon }_{1}o{\upsilon }_2\right)\pm {\upsilon }_2{\upsilon }_1\in \mathrm{Y}$.

Proof. 

$\left(i\right)$ We are assuming that

$$\psi \left([{\upsilon }_1,{\upsilon }_2]\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

(1)

Substituting ${\upsilon }_2{\upsilon }_1$ in place of ${\upsilon }_2$ in (1), we obtain

$$\psi \left([{\upsilon }_1,{\upsilon }_2]{\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

This implies that

$$\psi \left([{\upsilon }_1,{\upsilon }_2]\right){\upsilon }_1+\gamma \left(\left[{\upsilon }_1,{\upsilon }_2\right]\right)\theta \left({\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

Using our hypothesis, we have

$$\gamma \left([{\upsilon }_1,{\upsilon }_2]\right)\theta \left({\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

(2)

Replacing ${\upsilon }_2$ by ${\upsilon }_2{\upsilon }_3,$${\upsilon }_3\in \mathcal{F}$ in the last expression, we obtain

$$\gamma \left([{\upsilon }_1,{\upsilon }_2{\upsilon }_3]\right)\theta \left({\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

Since $\gamma$ is an epimorphism of $\mathcal{F}$ and using (2), we can write this equation as

$$\gamma \left(\left[{\upsilon }_1,{\upsilon }_2\right]\right)\gamma \left({\upsilon }_3\right)\theta \left({\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2,{\upsilon }_3\in \mathcal{F}.$$

(3)

Again replacing ${\upsilon }_3$ with ${\upsilon }_3{\upsilon }_1$ in (3), we obtain

$$\gamma \left(\left[{\upsilon }_1,{\upsilon }_2\right]\right)\gamma \left({\upsilon }_3\right)\gamma \left({\upsilon }_1\right)\theta \left({\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2,{\upsilon }_3\in \mathcal{F}.$$

(4)

By right-multiplying (3) by $\gamma \left({\upsilon }_1\right),$ we obtain

$$\gamma \left(\left[{\upsilon }_1,{\upsilon }_2\right]\right)\gamma \left({\upsilon }_3\right)\theta \left({\upsilon }_1\right)\gamma \left({\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2,{\upsilon }_3\in \mathcal{F}.$$

(5)

Subtracting (4) from (5), we arrive at

$$\gamma \left(\left[{\upsilon }_1,{\upsilon }_2\right]\right)\gamma \left({\upsilon }_3\right)\left[\gamma \left({\upsilon }_1\right),\theta \left({\upsilon }_1\right)\right]\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2,{\upsilon }_3\in \mathcal{F}.$$

Since $\gamma$ is an epimorphism of $\mathcal{F},$ we have

$$\left[\gamma \left({\upsilon }_1\right),t\right]\mathcal{F}\left[\gamma \left({\upsilon }_1\right),\theta \left({\upsilon }_1\right)\right]\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,t\in \mathcal{F}.$$

(6)

Writing $\theta \left({\upsilon }_1\right)$ instead of t in the last equation, we obtain

$$\left[\gamma \left({\upsilon }_1\right),\theta \left({\upsilon }_1\right)\right]\mathcal{F}\left[\gamma \left({\upsilon }_1\right),\theta \left({\upsilon }_1\right)\right]\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1\in \mathcal{F}.$$

Based on the semiprimeness of $\mathrm{Y},$ we arrive at

$$[\gamma \left({\upsilon }_1\right),\theta \left({\upsilon }_1\right)]\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1\in \mathcal{F}.$$

(7)

On the other hand, by using $\psi$ as a multiplicative generalized semiderivation of $\mathcal{F}$ and this equation, we obtain

$$\begin{array}{ccc}\hfill \psi \left({\upsilon }_1\right)& =& \psi \left({\upsilon }_1\right){\upsilon }_1+\gamma \left({\upsilon }_1\right)\theta \left({\upsilon }_1\right)=\theta \left({\upsilon }_1\right)\gamma \left({\upsilon }_1\right)+{\upsilon }_1\psi \left({\upsilon }_1\right)\hfill \\ & =& \psi \left({\upsilon }_1\right){\upsilon }_1-{\upsilon }_1\psi \left({\upsilon }_1\right)=\theta \left({\upsilon }_1\right)\gamma \left({\upsilon }_1\right)-\gamma \left({\upsilon }_1\right)\theta \left({\upsilon }_1\right)\in \mathrm{Y},\hfill \end{array}$$

and so

$$[\psi \left({\upsilon }_1\right),{\upsilon }_1]\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1\in \mathcal{F}.$$

This completes the proof.

$\left(ii\right)$ Assume that

$$\psi \left({\upsilon }_{1}o{\upsilon }_2\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

(8)

Writing ${\upsilon }_2{\upsilon }_1$ instead of ${\upsilon }_2$ in (8), we have

$$\psi \left(\left({\upsilon }_{1}o{\upsilon }_2\right){\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

Since $\psi$ is a multiplicative generalized semiderivation of $\mathcal{F},$ we obtain

$$\psi \left({\upsilon }_{1}o{\upsilon }_2\right){\upsilon }_1+\gamma \left({\upsilon }_{1}o{\upsilon }_2\right)\theta \left({\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

Using (8), we obtain

$$\gamma \left({\upsilon }_{1}o{\upsilon }_2\right)\theta \left({\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

(9)

If we put ${\upsilon }_2{\upsilon }_3$ instead of ${\upsilon }_2$ in the last equation, we have

$$\gamma \left({\upsilon }_{1}o{\upsilon }_2{\upsilon }_3\right)\theta \left({\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

Expanding this equation and using (9), we arrive at

$$\gamma \left(\left[{\upsilon }_1,{\upsilon }_2\right]\right)\gamma \left({\upsilon }_3\right)\theta \left({\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

Furthermore, the proof follows directly from Theorem 1, after Equation (3). By applying the same technique, we obtain that $[\psi \left({\upsilon }_1\right),{\upsilon }_1]\in \mathrm{Y},$ for all ${\upsilon }_1\in \mathcal{F}.$

$\left(iii\right)$ Assume that $\psi =0$. According to the hypothesis, ${\upsilon }_1{\upsilon }_2\in \mathrm{Y},$ for all ${\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$ We can easily obtain ${\upsilon }_1\mathcal{F}{\upsilon }_1\in \mathrm{Y}$. Based on the semiprimeness of $\mathcal{F},$ we arrive at ${\upsilon }_1\in \mathrm{Y},$ for all ${\upsilon }_1\in \mathcal{F},$ and so $\mathrm{Y}=\mathcal{F},$ a contradiction.

Now, we assume that $\psi \ne 0$. We also assume that

$$\psi \left([{\upsilon }_1,{\upsilon }_2]\right)\pm {\upsilon }_1{\upsilon }_2\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

(10)

Replacing ${\upsilon }_2$ with ${\upsilon }_2{\upsilon }_1$ in (10), we obtain

$$\psi \left([{\upsilon }_1,{\upsilon }_2]{\upsilon }_1\right)\pm {\upsilon }_1{\upsilon }_2{\upsilon }_1\in \mathrm{Y},$$

which can be expanded as

$$\psi \left([{\upsilon }_1,{\upsilon }_2]\right){\upsilon }_1\pm \gamma \left(\left[{\upsilon }_1,{\upsilon }_2\right]\right)\theta \left({\upsilon }_1\right)\pm {\upsilon }_1{\upsilon }_2{\upsilon }_1\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

This can be written as

$$\left(\psi \left([{\upsilon }_1,{\upsilon }_2]\right)\pm {\upsilon }_1{\upsilon }_2\right){\upsilon }_1\pm \gamma \left(\left[{\upsilon }_1,{\upsilon }_2\right]\right)\theta \left({\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

Using the hypothesis, this equation reduces to

$$\gamma \left(\left[{\upsilon }_1,{\upsilon }_2\right]\right)\theta \left({\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

This expression is identical to Equation (2) in the proof of Theorem 1. By employing similar expressions within the proof of Theorem 1, we can obtain the desired result.

$\left(iv\right)$ Assume that $\psi =0$. If we apply similar operations to those applied in Theorem 1 (iii), a contradiction is obtained. Alternatively, assume that $\psi \ne 0$. Based on the hypothesis, we then have

$$\psi \left([{\upsilon }_1,{\upsilon }_2]\right)\pm {\upsilon }_2{\upsilon }_1\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

(11)

Replacing ${\upsilon }_2$ with ${\upsilon }_2{\upsilon }_1$ in the above equation and using it accordingly, we can easily obtain

$$\gamma \left(\left[{\upsilon }_1,{\upsilon }_2\right]\right)\theta \left({\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

By applying the same techniques used after Equation (2) in the proof of Theorem 1, we obtain the required result.

$\left(v\right)$ Assume that $\psi =0$. If we perform the same operations as in Theorem 1 (iii), a contradiction arises. Thus, assume now that $\psi \ne 0$. Based on the hypothesis, we then have

$$\psi \left({\upsilon }_{1}o{\upsilon }_2\right)\pm {\upsilon }_1{\upsilon }_2\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

(12)

Replacing ${\upsilon }_2$ with ${\upsilon }_2{\upsilon }_1$ in (12) and using this equation accordingly, we find that

$$\gamma \left({\upsilon }_{1}o{\upsilon }_2\right)\theta \left({\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

This expression is identical to Equation (9) in the proof of Theorem 1. By using expressions similar to those used in the proof of Theorem 1, we can obtain the desired result.

$\left(vi\right)$ Assume that $\psi =0$. If similar operations are performed in Theorem 1 (iii), a contradiction is reached. Now assume that $\psi \ne 0$. Based the hypothesis, we then have

$$\psi \left({\upsilon }_{1}o{\upsilon }_2\right)\pm {\upsilon }_2{\upsilon }_1\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

(13)

Replacing ${\upsilon }_2$ with ${\upsilon }_2{\upsilon }_1$ in (13) and using this equation, we arrive at the following conclusion

$$\gamma \left({\upsilon }_{1}o{\upsilon }_2\right)\theta \left({\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

Again using the same lines as those used in the proof of Theorem 1, we conclude that $[\psi \left({\upsilon }_1\right),{\upsilon }_1]\in \mathrm{Y},$ for all ${\upsilon }_1\in \mathcal{F}.$ □

Theorem 2. 

Let $\mathcal{F}$ be a ring admitting a semiprime ideal Y and ψ be a multiplicative generalized semiderivation associated with a map θ, with γ being an epimorphism on $\mathcal{F}$. If any one of the following conditions is satisfied for all ${\upsilon }_1,{\upsilon }_2\in \mathcal{F}$, then $[\psi \left({\upsilon }_1\right),{\upsilon }_1]\in \mathrm{Y},$ for all ${\upsilon }_1\in \mathcal{F}.$

(i) 

$\psi \left({\upsilon }_1{\upsilon }_2\right)\pm {\upsilon }_1{\upsilon }_2\in \mathrm{Y}$;

(ii) 

$\psi \left({\upsilon }_1{\upsilon }_2\right)\pm {\upsilon }_2{\upsilon }_1\in \mathrm{Y}$;

(iii) 

$\psi \left({\upsilon }_1{\upsilon }_2\right)\pm [{\upsilon }_1,{\upsilon }_2]\in \mathrm{Y}$;

(iv) 

$\psi \left({\upsilon }_1{\upsilon }_2\right)\pm \left({\upsilon }_{1}o{\upsilon }_2\right)\in \mathrm{Y}$.

Proof. 

$\left(i\right)$ Assume that $\psi =0$. Based on the hypothesis, ${\upsilon }_1{\upsilon }_2\in \mathrm{Y},$ for all ${\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$ This expression is identical to the one in Theorem 1 $\left(iii\right)$. We now assume that $\psi \ne 0$. Using our hypothesis, we obtain

$$\psi \left({\upsilon }_1{\upsilon }_2\right)\pm {\upsilon }_1{\upsilon }_2\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

(14)

Taking ${\upsilon }_2{\upsilon }_3$ instead of ${\upsilon }_2$ within (14), we have

$$\psi \left({\upsilon }_1{\upsilon }_2{\upsilon }_3\right)\pm {\upsilon }_1{\upsilon }_2{\upsilon }_3\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2,{\upsilon }_3\in \mathcal{F}.$$

Since $\psi$ is a multiplicative generalized semiderivation of $\mathcal{F},$ we obtain

$$\psi \left({\upsilon }_1{\upsilon }_2\right){\upsilon }_3+\gamma \left({\upsilon }_1{\upsilon }_2\right)\theta \left({\upsilon }_3\right)\pm {\upsilon }_1{\upsilon }_2{\upsilon }_3\in \mathrm{Y},$$

which can be written as

$$\left(\psi \left({\upsilon }_1{\upsilon }_2\right)\pm {\upsilon }_1{\upsilon }_2\right){\upsilon }_3+\gamma \left({\upsilon }_1{\upsilon }_2\right)\theta \left({\upsilon }_3\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2,{\upsilon }_3\in \mathcal{F}.$$

Using the hypothesis, we arrive at

$$\gamma \left({\upsilon }_1{\upsilon }_2\right)\theta \left({\upsilon }_3\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2,{\upsilon }_3\in \mathcal{F}.$$

Since $\gamma$ is an epimorphism of $\mathcal{F},$ we have

$$\gamma \left({\upsilon }_1\right)\gamma \left({\upsilon }_2\right)\theta \left({\upsilon }_3\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2,{\upsilon }_3\in \mathcal{F},$$

(15)

and so

$$\gamma \left({\upsilon }_2\right)\theta \left({\upsilon }_3\right)\mathcal{F}\gamma \left({\upsilon }_2\right)\theta \left({\upsilon }_3\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_2,{\upsilon }_3\in \mathcal{F}.$$

Due to the semiprimeness of $\mathrm{Y},$ we arrive at

$$\gamma \left({\upsilon }_2\right)\theta \left({\upsilon }_3\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_2,{\upsilon }_3\in \mathcal{F}.$$

On the other hand, we know that $\psi \left({\upsilon }_1{\upsilon }_2\right)\pm {\upsilon }_1{\upsilon }_2\in \mathrm{Y}$ according to our hypothesis, and so $\psi \left({\upsilon }_1\right){\upsilon }_2+\gamma \left({\upsilon }_1\right)\theta \left({\upsilon }_2\right)\pm {\upsilon }_1{\upsilon }_2\in \mathrm{Y},$ for all ${\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$ Using the $\gamma \left({\upsilon }_1\right)\theta \left({\upsilon }_2\right)\in \mathrm{Y}$ in the last expression, we understand that

$$\left(\psi \left({\upsilon }_1\right)\pm {\upsilon }_1\right){\upsilon }_2\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F},$$

(16)

and so

$${\upsilon }_1\left(\psi \left({\upsilon }_1\right)\pm {\upsilon }_1\right)\mathcal{F}{\upsilon }_1\left(\psi \left({\upsilon }_1\right)\pm {\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1\in \mathcal{F}.$$

The semiprimeness of $\mathrm{Y}$ forces us to conclude that

$${\upsilon }_1\left(\psi \left({\upsilon }_1\right)\pm {\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1\in \mathcal{F}.$$

(17)

Taking ${\upsilon }_1$ instead of ${\upsilon }_2$ in (16) and subtracting the relations (16) and (17), we find that

$$\left[\left(\psi \left({\upsilon }_1\right)\pm {\upsilon }_1\right),{\upsilon }_1\right]\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1\in \mathcal{F},$$

and so

$$\left[\psi \left({\upsilon }_1\right),{\upsilon }_1\right]\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1\in \mathcal{F}.$$

This completes the proof.

$\left(ii\right)$ Assume that $\psi =0$. Based on the hypothesis, ${\upsilon }_1{\upsilon }_2\in \mathrm{Y},$ for all ${\upsilon }_1,{\upsilon }_2\in \mathcal{F}$. This expression is the same as the expression in Theorem 1 (iii). We now assume that $\psi \ne 0$. Using our hypothesis, we obtain

$$\psi \left({\upsilon }_1\left({\upsilon }_2{\upsilon }_3\right)\right)\pm \left({\upsilon }_2{\upsilon }_3\right){\upsilon }_1\in \mathrm{Y}$$

and

$$\psi \left(\left({\upsilon }_1{\upsilon }_2\right){\upsilon }_3\right)\pm {\upsilon }_3\left({\upsilon }_1{\upsilon }_2\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2,{\upsilon }_3\in \mathcal{F}.$$

Subtracting these two equations, we have

$${\upsilon }_2{\upsilon }_3{\upsilon }_1\mp {\upsilon }_3{\upsilon }_1{\upsilon }_2\in \mathrm{Y},$$

and so

$$\left[{\upsilon }_2,{\upsilon }_3{\upsilon }_1\right]\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2,{\upsilon }_3\in \mathcal{F}.$$

Writing ${\upsilon }_3$ instead of ${\upsilon }_2$ in this equation and using it accordingly, we find that

$${\upsilon }_3\left[{\upsilon }_1,{\upsilon }_3\right]\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_3\in \mathcal{F}.$$

(18)

Replacing ${\upsilon }_1$ with ${\upsilon }_2{\upsilon }_1$ in (18) and using (18), we obtain

$${\upsilon }_3{\upsilon }_2\left[{\upsilon }_1,{\upsilon }_3\right]\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2,{\upsilon }_3\in \mathcal{F},$$

and so

$${\upsilon }_3{\upsilon }_2{\upsilon }_3\left[{\upsilon }_1,{\upsilon }_3\right]\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2,{\upsilon }_3,{\upsilon }_3\in \mathcal{F}.$$

(19)

Similarly, (18) reveals that

$${\upsilon }_2{\upsilon }_3{\upsilon }_3\left[{\upsilon }_1,{\upsilon }_3\right]\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2,{\upsilon }_3,{\upsilon }_3\in \mathcal{F}.$$

(20)

Subtracting (19) from (20), we find that

$$\left[{\upsilon }_2,{\upsilon }_3\right]{\upsilon }_3\left[{\upsilon }_1,{\upsilon }_3\right]\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2,{\upsilon }_3,{\upsilon }_3\in \mathcal{F},$$

and so

$$\left[{\upsilon }_1,{\upsilon }_3\right]\mathcal{F}\left[{\upsilon }_1,{\upsilon }_3\right]\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2,{\upsilon }_3\in \mathcal{F}.$$

Due to the semiprimeness of $\mathrm{Y}$, we obtain

$$\left[{\upsilon }_1,{\upsilon }_3\right]\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_3\in \mathcal{F}.$$

Replacing $\psi \left({\upsilon }_1\right){\upsilon }_3$ with ${\upsilon }_3$ in this equation and using it accordingly, we obtain

$$\left[{\upsilon }_1,\psi \left({\upsilon }_1\right)\right]{\upsilon }_3\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_3\in \mathcal{F},$$

and so

$$\left[{\upsilon }_1,\psi \left({\upsilon }_1\right)\right]\mathcal{F}\left[{\upsilon }_1,\psi \left({\upsilon }_1\right)\right]\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1\in \mathcal{F}.$$

Again using the semiprimeness of $\mathrm{Y}$, we obtain the required result.

$\left(iii\right)$ Assume that

$$\psi \left({\upsilon }_1{\upsilon }_2\right)\pm [{\upsilon }_1,{\upsilon }_2]\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

Define the map $\gamma :\mathcal{F}\to \mathcal{F},\gamma \left(\mathcal{F}\right)=\psi \left(\mathcal{F}\right)\pm \mathcal{F},$ for all $\mathcal{F}\in \mathcal{F}.$$\gamma$ is a multiplicative generalized semiderivation associated with a nonzero map $\theta$ of $\mathcal{F}.$ Based on the hypothesis, we have $\gamma \left({\upsilon }_1{\upsilon }_2\right)\pm {\upsilon }_1{\upsilon }_2\in \mathrm{Y},$ for all ${\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$ Hence, the conclusion is obtained via Theorem 2 $\left(i\right)$.

$\left(iv\right)$ Using our hypothesis, we obtain

$$\psi \left({\upsilon }_1{\upsilon }_2\right)\pm \left({\upsilon }_{1}o{\upsilon }_2\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

Define the map $\gamma :\mathcal{F}\to \mathcal{F},\gamma \left(\mathcal{F}\right)=\psi \left(\mathcal{F}\right)\pm \mathcal{F},$ for all $\mathcal{F}\in \mathcal{F}.$$\gamma$ is a multiplicative generalized semiderivation associated with a nonzero map $\theta$ of $\mathcal{F}$, such that $\gamma \left({\upsilon }_1{\upsilon }_2\right)\pm {\upsilon }_2{\upsilon }_1\in \mathrm{Y},$ for all ${\upsilon }_1,{\upsilon }_2\in \mathcal{F}$. Using Theorem 2 (ii), we find that $[\psi \left({\upsilon }_1\right),{\upsilon }_1]\in \mathrm{Y},$ for all ${\upsilon }_1\in \mathcal{F}$. □

Theorem 3. 

Let $\mathcal{F}$ be a ring admitting a semiprime ideal Y and ψ be a multiplicative generalized semiderivation associated with a map θ, with γ being an epimorphism on $\mathcal{F}$. If any one of the following conditions is satisfied for all ${\upsilon }_1,{\upsilon }_2\in \mathcal{F}$, then $[\psi \left({\upsilon }_1\right),{\upsilon }_1]\in \mathrm{Y},$ for all ${\upsilon }_1\in \mathcal{F}.$

(i) 

$\psi \left(\left[{\upsilon }_1,{\upsilon }_2\right]\right)\pm \left[{\upsilon }_1,{\upsilon }_2\right]\in \mathrm{Y}$;

(ii) 

$\psi \left(\left[{\upsilon }_1,{\upsilon }_2\right]\right)\pm \left({\upsilon }_{1}o{\upsilon }_2\right)\in \mathrm{Y}$;

(iii) 

$\psi \left({\upsilon }_{1}o{\upsilon }_2\right)\pm \left({\upsilon }_{1}o{\upsilon }_2\right)\in \mathrm{Y}$;

(iv) 

$\psi \left({\upsilon }_{1}o{\upsilon }_2\right)\pm \left[{\upsilon }_1,{\upsilon }_2\right]\in \mathrm{Y}$.

Proof. 

$\left(i\right)$ We assume that

$$\psi \left(\left[{\upsilon }_1,{\upsilon }_2\right]\right)\pm \left[{\upsilon }_1,{\upsilon }_2\right]\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

(21)

Writing ${\upsilon }_2$ instead of ${\upsilon }_2{\upsilon }_1$ in (21), we obtain

$$\psi \left(\left[{\upsilon }_1,{\upsilon }_2\right]{\upsilon }_1\right)\pm \left[{\upsilon }_1,{\upsilon }_2\right]{\upsilon }_1\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

Using the definition of $\psi$, we have

$$\psi \left(\left[{\upsilon }_1,{\upsilon }_2\right]\right){\upsilon }_1+\gamma \left(\left[{\upsilon }_1,{\upsilon }_2\right]\right)\theta \left({\upsilon }_1\right)\pm \left[{\upsilon }_1,{\upsilon }_2\right]{\upsilon }_1\in \mathrm{Y}.$$

Based on the hypothesis, we find that

$$\gamma \left(\left[{\upsilon }_1,{\upsilon }_2\right]\right)\theta \left({\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

This expression is exactly the same as Equation (2) in the proof of Theorem 1. By using expressions similar to those used in the proof of Theorem 1, we obtain the desired result.

$\left(ii\right)$ By using our hypothesis, we obtain

$$\psi \left(\left[{\upsilon }_1,{\upsilon }_2\right]\right)\pm \left({\upsilon }_{1}o{\upsilon }_2\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

(22)

Replacing ${\upsilon }_2$ with ${\upsilon }_2{\upsilon }_1$ in (22) and using this equation, we find that

$$\gamma \left(\left[{\upsilon }_1,{\upsilon }_2\right]\right)\theta \left({\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

This expression is exactly the same as Equation (2) in the proof of Theorem 1. By using expressions similar to those used in that proof, we obtain the desired result.

$\left(iii\right)$ Using our hypothesis, we obtain

$$\psi \left({\upsilon }_{1}o{\upsilon }_2\right)\pm \left({\upsilon }_{1}o{\upsilon }_2\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

(23)

Replacing ${\upsilon }_2$ with ${\upsilon }_2{\upsilon }_1$ in (23), we obtain

$$\psi \left(\left({\upsilon }_{1}o{\upsilon }_2\right){\upsilon }_1\right)\pm \left({\upsilon }_{1}o{\upsilon }_2\right){\upsilon }_1\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

Since $\psi$ is a multiplicative generalized semiderivation of $\mathcal{F},$ we obtain

$$\psi \left({\upsilon }_{1}o{\upsilon }_2\right){\upsilon }_1+\gamma \left({\upsilon }_{1}o{\upsilon }_2\right)\theta \left({\upsilon }_1\right)\pm \left({\upsilon }_{1}o{\upsilon }_2\right){\upsilon }_1\in \mathrm{Y}.$$

Based on the hypothesis, we find that

$$\gamma \left({\upsilon }_{1}o{\upsilon }_2\right)\theta \left({\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

This expression is exactly the same as Equation (9) in the proof of Theorem 1. By using expressions similar to those used in that proof, we obtain the desired result.

$\left(iv\right)$ Using our hypothesis, we obtain

$$\psi \left({\upsilon }_{1}o{\upsilon }_2\right)\pm \left[{\upsilon }_1,{\upsilon }_2\right]\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

(24)

Replacing ${\upsilon }_2$ with ${\upsilon }_2{\upsilon }_1$ in (24) and using (24), we obtain

$$\gamma \left({\upsilon }_{1}o{\upsilon }_2\right)\theta \left({\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

Using the same reasoning as applied after Equation (9) in the proof of Theorem 1, we arrive at the desired result. □

Theorem 4. 

Let $\mathcal{F}$ be a ring admitting a semiprime ideal Y and ψ be a multiplicative generalized semiderivation associated with a map θ, with γ being an epimorphism on $\mathcal{F}.$ If any of the following conditions are satisfied for all ${\upsilon }_1,{\upsilon }_2\in \mathcal{F},$$m,n\in \mathbb{N}$, then $[\psi \left({\upsilon }_1\right),{\upsilon }_1]\in \mathrm{Y},$ for all ${\upsilon }_1\in \mathcal{F}.$

(i) 

$\psi \left(\left[{\upsilon }_1,{\upsilon }_2\right]\right)\pm {\upsilon }_1^m[{\upsilon }_1,{\upsilon }_2]{\upsilon }_1^n$$\in \mathrm{Y}$;

(ii) 

$\psi \left(\left[{\upsilon }_1,{\upsilon }_2\right]\right)\pm {\upsilon }_1^m\left({\upsilon }_{1}o{\upsilon }_2\right){\upsilon }_1^n\in \mathrm{Y}$;

(iii) 

$\psi \left({\upsilon }_{1}o{\upsilon }_2\right)\pm {\upsilon }_1^m\left({\upsilon }_1\circ {\upsilon }_2\right){\upsilon }_1^n\in \mathrm{Y}$;

(iv) 

$\psi \left({\upsilon }_{1}o{\upsilon }_2\right)\pm {\upsilon }_1^m\left[{\upsilon }_1,{\upsilon }_2\right]{\upsilon }_1^n\in \mathrm{Y}$.

Proof. 

$\left(i\right)$ Based on our hypothesis

$$\psi \left(\left[{\upsilon }_1,{\upsilon }_2\right]\right)\pm {\upsilon }_1^m[{\upsilon }_1,{\upsilon }_2]{\upsilon }_1^n\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

(25)

Taking ${\upsilon }_2$ instead of ${\upsilon }_2{\upsilon }_1$ in (25), we obtain

$$\psi \left(\left[{\upsilon }_1,{\upsilon }_2{\upsilon }_1\right]\right)\pm {\upsilon }_1^m[{\upsilon }_1,{\upsilon }_2{\upsilon }_1]{\upsilon }_1^n\in \mathrm{Y},$$

and so

$$\psi \left(\left[{\upsilon }_1,{\upsilon }_2\right]{\upsilon }_1\right)\pm {\upsilon }_1^m[{\upsilon }_1,{\upsilon }_2]{\upsilon }_1^{n+1}\in \mathrm{Y},$$

which can be expanded as

$$\psi \left(\left[{\upsilon }_1,{\upsilon }_2\right]\right){\upsilon }_1+\gamma \left(\left[{\upsilon }_1,{\upsilon }_2\right]\right)\theta \left({\upsilon }_1\right)\pm {\upsilon }_1^m[{\upsilon }_1,{\upsilon }_2]{\upsilon }_1^{n+1}\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

Using (25), we find that

$$\gamma \left([{\upsilon }_1,{\upsilon }_2]\right)\theta \left({\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

(26)

Equation (2) yields the same expression. Similar expressions can be used in the proof of Theorem 1. Thus, the proof is complete.

$\left(ii\right)$ Let us assume that

$$\psi \left(\left[{\upsilon }_1,{\upsilon }_2\right]\right)\pm {\upsilon }_1^m\left({\upsilon }_{1}o{\upsilon }_2\right){\upsilon }_1^n\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

(27)

Writing ${\upsilon }_2{\upsilon }_1$ instead of ${\upsilon }_2$ in (27) and using (27), we reveal that

$$\gamma \left([{\upsilon }_1,{\upsilon }_2]\right)\theta \left({\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

Equation (2) yields the same expression. Similar expressions can be used in the proof of Theorem 1. Therefore, the proof is complete.

$\left(iii\right)$ We assume that

$$\psi \left({\upsilon }_{1}o{\upsilon }_2\right)\pm {\upsilon }_1^m\left({\upsilon }_1\circ {\upsilon }_2\right){\upsilon }_1^n\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

(28)

Taking ${\upsilon }_2$ instead of ${\upsilon }_2{\upsilon }_1$ in (28) and using this, we find that

$$\gamma \left({\upsilon }_{1}o{\upsilon }_2\right)\theta \left({\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

(29)

By applying the same arguments used after Equation (9) in the proof of Theorem 1, we obtain the required result.

$\left(iv\right)$ Let us assume that

$$\psi \left({\upsilon }_{1}o{\upsilon }_2\right)\pm {\upsilon }_1^m\left[{\upsilon }_1,{\upsilon }_2\right]{\upsilon }_1^n\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

(30)

Replacing ${\upsilon }_2$ with ${\upsilon }_2{\upsilon }_1$ in (30) and using (30), we find that

$$\gamma \left({\upsilon }_{1}o{\upsilon }_2\right)\theta \left({\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

By using the same arguments as those following Equation (9) in the proof of Theorem 1, we obtain the required result. □

Theorem 5. 

Let $\mathcal{F}$ be a 2-torsion-free ring with Y being a semiprime ideal of $\mathcal{F}$. Suppose that $\mathcal{F}$ admits a multiplicative generalized semiderivation ψ associated with a map $\theta .$ If any of the following conditions are satisfied for all ${\upsilon }_1,{\upsilon }_2\in \mathcal{F}$, then θ is a $\mathrm{Y}$-commuting map on $\mathcal{F}$.

(i) 

$\psi \left({\upsilon }_1\right)\psi \left({\upsilon }_2\right)\pm {\upsilon }_1{\upsilon }_2\in \mathrm{Y}$;

(ii) 

$\psi \left({\upsilon }_1\right)\psi \left({\upsilon }_2\right)\pm {\upsilon }_2{\upsilon }_1\in \mathrm{Y}$.

Proof. 

$\left(i\right)$ Assume that $\psi =0$. Based on the hypothesis, ${\upsilon }_1{\upsilon }_2\in \mathrm{Y},$ for all ${\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$ This expression is the same as the one in Theorem 1 $\left(iii\right)$. We now assume that $\psi \ne 0$. Using the hypothesis, we have

$$\psi \left({\upsilon }_1\right)\psi \left({\upsilon }_2\right)\pm {\upsilon }_1{\upsilon }_2\in \mathrm{Y}\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

Replacing ${\upsilon }_2$ with ${\upsilon }_2{\upsilon }_3$ in the hypothesis and using the hypothesis, we find that

$$\begin{array}{cc}\hfill \psi \left({\upsilon }_1\right)\psi \left({\upsilon }_2{\upsilon }_3\right)\mp {\upsilon }_1{\upsilon }_2{\upsilon }_3& =\psi \left({\upsilon }_1\right)(\psi \left({\upsilon }_2\right){\upsilon }_3+\gamma \left({\upsilon }_2\right)\theta \left({\upsilon }_3\right))\mp {\upsilon }_1{\upsilon }_2{\upsilon }_3\hfill \\ \hfill & =(\psi \left({\upsilon }_1\right)\psi \left({\upsilon }_2\right)\mp {\upsilon }_1{\upsilon }_2){\upsilon }_3+\psi \left({\upsilon }_1\right)\gamma \left({\upsilon }_2\right)\theta \left({\upsilon }_3\right)\in \mathrm{Y}.\hfill \end{array}$$

Again using the hypothesis, we have

$$\psi \left({\upsilon }_1\right)\gamma \left({\upsilon }_2\right)\theta \left({\upsilon }_3\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2,{\upsilon }_3\in \mathcal{F}.$$

(31)

Since $\gamma$ is an epimorphism of $\mathcal{F},$ we have

$$\psi \left({\upsilon }_1\right){\upsilon }_2\theta \left({\upsilon }_3\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2,{\upsilon }_3\in \mathcal{F}.$$

(32)

Replacing ${\upsilon }_1$ with $t{\upsilon }_1$, $t\in \mathcal{F}$, we have

$$\psi \left(t{\upsilon }_1\right){\upsilon }_2\theta \left({\upsilon }_3\right)=(\psi \left(t\right){\upsilon }_1+\gamma \left(t\right)\theta \left({\upsilon }_1\right)){\upsilon }_2\theta \left({\upsilon }_3\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2,{\upsilon }_3,t\in \mathcal{F}.$$

Applying (32), we have

$$\gamma \left(t\right)\theta \left({\upsilon }_1\right){\upsilon }_2\theta \left({\upsilon }_3\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2,{\upsilon }_3,t\in L.$$

Since $\gamma$ is an epimorphism of $\mathcal{F},$ we have ${\upsilon }_2\theta \left({\upsilon }_1\right){\upsilon }_3{\upsilon }_2\theta \left({\upsilon }_1\right)\in \mathrm{Y},$ for all ${\upsilon }_1,{\upsilon }_2,{\upsilon }_3\in \mathcal{F}.$ This implies that ${\upsilon }_2\theta \left({\upsilon }_1\right)\in \mathrm{Y},$ for all ${\upsilon }_1,{\upsilon }_2\in \mathcal{F},$ and so $\theta \left({\upsilon }_1\right)\in \mathrm{Y},$ for all ${\upsilon }_1\in \mathcal{F}.$ Hence, we obtain $\left[\theta \left({\upsilon }_1\right),{\upsilon }_1\right]\in \mathrm{Y}.$

$\left(ii\right)$ Assume that $\psi =0$. Based on the hypothesis, ${\upsilon }_1{\upsilon }_2\in \mathrm{Y},$ for all ${\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$ This expression is the same as the expression in Theorem 1 $\left(iii\right)$. We now assume that $\psi \ne 0$. Using the hypothesis, we have

$$\psi \left({\upsilon }_1\right)\psi \left({\upsilon }_2\right)-{\upsilon }_2{\upsilon }_1\in \mathrm{Y}\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

(33)

Replacing ${\upsilon }_2$ with ${\upsilon }_2{\upsilon }_1$ in (33), we obtain

$$\begin{array}{ccc}\hfill \psi \left({\upsilon }_1\right)\psi \left({\upsilon }_2{\upsilon }_1\right)-{\upsilon }_2{\upsilon }_1& =& \psi \left({\upsilon }_1\right)(\psi \left({\upsilon }_2\right){\upsilon }_1+\gamma \left({\upsilon }_2\right)\theta \left({\upsilon }_1\right))-{\upsilon }_2{\upsilon }_1\hfill \\ & =& (\psi \left({\upsilon }_1\right)\psi \left({\upsilon }_2\right)-{\upsilon }_2{\upsilon }_1){\upsilon }_1+\psi \left({\upsilon }_1\right)\gamma \left({\upsilon }_2\right)\theta \left({\upsilon }_1\right)\in \mathrm{Y}.\hfill \end{array}$$

Using the hypothesis, we obtain

$$\psi \left({\upsilon }_1\right)\gamma \left({\upsilon }_2\right)\theta \left({\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

Since $\gamma$ is an epimorphism of $\mathcal{F},$ we have

$$\psi \left({\upsilon }_1\right){\upsilon }_2\theta \left({\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

Left-multiplying this equation by $\psi \left({\upsilon }_3\right)$ for ${\upsilon }_3\in \mathcal{F}$ and using it accordingly, we obtain

$$\psi \left({\upsilon }_3\right)\psi \left({\upsilon }_1\right){\upsilon }_2\theta \left({\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2,{\upsilon }_3\in \mathcal{F}.$$

Based on the hypothesis, we can express the last equation as follows:

$${\upsilon }_1{\upsilon }_3{\upsilon }_2\theta \left({\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2,{\upsilon }_3\in \mathcal{F}.$$

(34)

Replacing ${\upsilon }_3$ with $\theta \left({\upsilon }_1\right),$ we have

$${\upsilon }_1\theta \left({\upsilon }_1\right){\upsilon }_2\theta \left({\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

Taking ${\upsilon }_2$ instead of ${\upsilon }_2{\upsilon }_1,$ we obtain

$${\upsilon }_1\theta \left({\upsilon }_1\right){\upsilon }_2{\upsilon }_1\theta \left({\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

Since $\mathrm{Y}$ is a semiprime ideal of $\mathcal{F},$ we conclude that

$${\upsilon }_1\theta \left({\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1\in \mathcal{F}.$$

(35)

Right-multiplying by ${\upsilon }_1$ in (34), we obtain

$${\upsilon }_1{\upsilon }_3{\upsilon }_2\theta \left({\upsilon }_1\right){\upsilon }_1\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,,{\upsilon }_2,{\upsilon }_3\in \mathcal{F}.$$

Left-multiplying this equation by $\theta \left({\upsilon }_1\right)$, we obtain

$$\theta \left({\upsilon }_1\right){\upsilon }_1{\upsilon }_3{\upsilon }_2\theta \left({\upsilon }_1\right){\upsilon }_1\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2,{\upsilon }_3\in \mathcal{F}.$$

Since $\mathrm{Y}$ is a semiprime ideal of $\mathcal{F},$ we conclude that

$$\theta \left({\upsilon }_1\right){\upsilon }_1\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1\in \mathcal{F}.$$

(36)

Subtracting (35) from (36), we find that

$$[{\upsilon }_1,\theta \left({\upsilon }_1\right)]\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1\in \mathcal{F}.$$

This completes the proof. □

Theorem 6. 

Let $\mathcal{F}$ be a $2$-torsion-free ring with Y being a semiprime ideal of $\mathcal{F}$. Suppose that $\mathcal{F}$ admits a multiplicative generalized semiderivation ψ associated with a nonzero multiplicative semiderivation $\theta .$ If any of the following conditions are satisfied for all ${\upsilon }_1,{\upsilon }_2\in \mathcal{F}$, then θ is a $\mathrm{Y}$-commuting map on $\mathcal{F}.$

(i) 

$\theta \left({\upsilon }_1\right)\psi \left({\upsilon }_2\right)\pm [{\upsilon }_1,{\upsilon }_2]\in \mathrm{Y}$;

(ii) 

$\theta \left({\upsilon }_1\right)\psi \left({\upsilon }_2\right)\pm {\upsilon }_{1}o{\upsilon }_2\in \mathrm{Y}$.

Proof. 

$\left(i\right)$ We obtain

$$\theta \left({\upsilon }_1\right)\psi \left({\upsilon }_2\right)\pm [{\upsilon }_1,{\upsilon }_2]\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2\in \mathcal{F}.$$

(37)

Replacing ${\upsilon }_1$ with ${\upsilon }_1{\upsilon }_3,{\upsilon }_3\in \mathcal{F}$ in (37), we have

$$\theta \left({\upsilon }_1\right)\gamma \left({\upsilon }_3\right)\psi \left({\upsilon }_2\right)+{\upsilon }_1\theta \left({\upsilon }_3\right)\psi \left({\upsilon }_2\right)\pm {\upsilon }_1\left[{\upsilon }_3,{\upsilon }_2\right]\pm \left[{\upsilon }_1,{\upsilon }_2\right]{\upsilon }_3\in \mathrm{Y}\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2,{\upsilon }_3\in \mathcal{F}.$$

Using (37), we see that

$$\theta \left({\upsilon }_1\right)\gamma \left({\upsilon }_3\right)\psi \left({\upsilon }_2\right)\pm \left[{\upsilon }_1,{\upsilon }_2\right]{\upsilon }_3\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_2,{\upsilon }_3\in \mathcal{F}.$$

Substituting ${\upsilon }_2$ for ${\upsilon }_1$ in the last expression, we have

$$\theta \left({\upsilon }_1\right)\gamma \left({\upsilon }_3\right)\psi \left({\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_3\in \mathcal{F}.$$

Since $\gamma$ is an epimorphism of $\mathcal{F},$ we have

$$\theta \left({\upsilon }_1\right){\upsilon }_3\psi \left({\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_3\in \mathcal{F}.$$

(38)

Replacing ${\upsilon }_3$ with $\theta \left({\upsilon }_3\right),$ we obtain

$$\theta \left({\upsilon }_1\right)\theta \left({\upsilon }_3\right)\psi \left({\upsilon }_1\right)\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_3\in \mathcal{F}.$$

Using the hypothesis and $\mathrm{Y}\in \mathrm{Y}$, we have

$$\theta \left({\upsilon }_1\right)([{\upsilon }_3,{\upsilon }_1]+\mathrm{Y})\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_3\in \mathcal{F},$$

and so

$$\theta \left({\upsilon }_1\right)[{\upsilon }_3,{\upsilon }_1]\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_3\in \mathcal{F}.$$

(39)

Writing ${\upsilon }_3\theta \left({\upsilon }_1\right)$ instead of ${\upsilon }_3$ in (39) and using (39), we find that

$$\theta \left({\upsilon }_1\right){\upsilon }_3[{\upsilon }_1,\theta \left({\upsilon }_1\right)]\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_3\in \mathcal{F}.$$

(40)

Replacing ${\upsilon }_3$ with ${\upsilon }_1{\upsilon }_3$ in (40), we find that

$$\theta \left({\upsilon }_1\right){\upsilon }_1{\upsilon }_3[{\upsilon }_1,\theta \left({\upsilon }_1\right)]\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_3\in \mathcal{F}.$$

(41)

Multiplying (40) on the left by ${\upsilon }_1,$ we have

$${\upsilon }_1\theta \left({\upsilon }_1\right){\upsilon }_3[{\upsilon }_1,\theta \left({\upsilon }_1\right)]\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_3\in \mathcal{F}.$$

(42)

Subtracting (41) from (42), we determine that

$$[{\upsilon }_1,\theta \left({\upsilon }_1\right)]{\upsilon }_2[{\upsilon }_1,\theta \left({\upsilon }_1\right)]\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1,{\upsilon }_3\in \mathcal{F};$$

that is,

$$[{\upsilon }_1,\theta \left({\upsilon }_1\right)]\mathcal{F}[{\upsilon }_1,\theta \left({\upsilon }_1\right)]\subseteq \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1\in \mathcal{F}.$$

Since $\mathrm{Y}$ is a semiprime ideal of $\mathcal{F},$ we conclude that

$$[{\upsilon }_1,\theta \left({\upsilon }_1\right)]\in \mathrm{Y},\mathrm{for}\mathrm{all}{\upsilon }_1\in \mathcal{F},$$

and so $\theta$ is a $\mathrm{Y}$-commuting map on $\mathcal{F}.$

$\left(ii\right)$ We have

$$\theta \left({\upsilon }_1\right)\psi \left({\upsilon }_2\right)\pm {\upsilon }_{1}o{\upsilon }_2\in \mathrm{Y}.$$

(43)

If we write ${\upsilon }_1{\upsilon }_3,{\upsilon }_3\in \mathcal{F}$ instead of ${\upsilon }_1$ in (43), then, since $\mathcal{F}$ is 2-torsion-free, we find that

$$\theta \left({\upsilon }_1\right)\gamma \left({\upsilon }_3\right)\psi \left({\upsilon }_2\right)+{\upsilon }_1\theta \left({\upsilon }_3\right)\psi \left({\upsilon }_2\right)\pm ({\upsilon }_1\left({\upsilon }_{3}o{\upsilon }_2\right)-\left[{\upsilon }_1,{\upsilon }_2\right]{\upsilon }_3)\in \mathrm{Y}.$$

Using (43), we obtain

$$\theta \left({\upsilon }_1\right)\gamma \left({\upsilon }_3\right)\psi \left({\upsilon }_2\right)\mp \left[{\upsilon }_1,{\upsilon }_2\right]{\upsilon }_3\in \mathrm{Y}.$$

(44)

Replacing ${\upsilon }_2$ with ${\upsilon }_1$ in (44), we have

$$\theta \left({\upsilon }_1\right)\gamma \left({\upsilon }_3\right)\psi \left({\upsilon }_2\right)\in \mathrm{Y},$$

and so

$$\theta \left({\upsilon }_1\right){\upsilon }_3\psi \left({\upsilon }_2\right)\in \mathrm{Y}.$$

Since $\gamma$ is an epimorphism of $\mathcal{F}$, we can use the same arguments as those following Equation (38) to show that $\left[\theta \left({\upsilon }_1\right),{\upsilon }_1\right]\in \mathrm{Y}.$ This completes the proof. □