Symmetric Reverse n-Derivations on Ideals of Semiprime Rings

Abstract

This paper focuses on examining a new type of n-additive map called the symmetric reverse n-derivation. As implied by its name, it combines the ideas of n-additive maps and reverse derivations, with a 1-reverse derivation being the ordinary reverse derivation. We explore several findings that expand our knowledge of these maps, particularly their presence in semiprime rings and the way rings respond to specific functional identities involving elements of ideals. Also, we provide examples to help clarify the concept of symmetric reverse n-derivations. This study aims to deepen our understanding of these symmetric maps and their properties within mathematical structures.

1. Introduction

In the present paper, T denotes an associative ring, and $\mathcal{Z}\left(T\right)$ its center. This paper focuses on the study of prime and semiprime rings only. By a prime ring, we mean whenever $\vartheta T\ell =\left\{0\right\}$, it implies either $\vartheta =0$ or $\ell =0$, and similarly, in case of a semiprime, if $\vartheta T\vartheta =\left\{0\right\}$, then $\vartheta =0$ with $\vartheta ,\ell \in T$. For any $\vartheta ,\ell \in T$, the symbols $[\vartheta ,\ell ]$ and $\vartheta o\ell$ represent the commutator $\vartheta \ell -\ell \vartheta$ and the anti-commutator $\vartheta \ell +\ell \vartheta$, respectively. By an n-torsion free ring, we mean whenever $n\vartheta =0$ for some $\vartheta \in T$, then the only choice left for $\vartheta$ is 0. An additive map $\alpha :T\to T$ is called a derivation if $\alpha (\vartheta \ell )=\alpha \left(\vartheta \right)\ell +\vartheta \alpha (\ell )$ holds for all $\vartheta ,\ell \in T$. The idea of derivation has been translated in many directions, and one among these is the definition of a derivation on the Cartesian product of rings. In this direction, Maksa [1] defined what is called a bi-derivation. Yes, the concept is expanded from bi to tri and then to n-derivations. This definition comes very naturally. Let us give the formal definition of these maps. As derivation is additive, when translating this idea to a Cartesian product, the additivity part refers to the additivity in both components, called a bi-additive map. A bi-additive map $\mathfrak{D}:T\times T\to T$ is known as a bi-derivation if it is a derivation in both of its components, i.e.,

$$\mathfrak{D}(\vartheta {\vartheta }^{\prime },\ell )=\mathfrak{D}(\vartheta ,\ell ){\vartheta }^{\prime }+\vartheta \mathfrak{D}({\vartheta }^{\prime },\ell ),$$

and

$$\mathfrak{D}(\vartheta ,\ell {\ell }^{\prime })=\mathfrak{D}(\vartheta ,\ell ){\ell }^{\prime }+\ell \mathfrak{D}(\vartheta ,{\ell }^{\prime }),$$

for all $\vartheta ,{\vartheta }^{\prime },\ell ,{\ell }^{\prime }\in T$. These two conditions can be clubbed into one if the map $\mathfrak{D}$ is also symmetric, i.e., $\mathfrak{D}(\vartheta ,\ell )=\mathfrak{D}(\ell ,\vartheta )$ for all $\vartheta ,\ell \in T$. The idea of bi-derivation was studied extensively by Vukman in [2,3]. Thus, a bi-additive, symmetric $\mathfrak{D}$ which is a derivation in any of its two components, is termed a symmetric bi-derivation. Several authors have studied symmetric bi-derivations on rings and fuzzification on rings (see [4,5,6,7,8] and references therein) and produced highly useful outcomes. Along similar lines, $\ddot{\mathrm{O}}$zt$\ddot{\mathrm{u}}$rk [9] initiated the study of tri-derivations. Taking this forward, Park [10] introduced the notion of permuting n-derivations. The definition is as follows:

Definition 1. 

Let $n\ge 2$ be a fixed integer. A map $\mathfrak{D}:T^n=\underset{n-times}{\underbrace{T\times T\times \dots \times T}}\to T$, is said to form a symmetric (permuting) n-derivation if $\mathfrak{D}$ is symmetric and an n-additive and, in addition to this, it is an n-derivation, that is

$$\begin{array}{c}\hfill \mathfrak{D}\left({\vartheta }_1,{\vartheta }_2,\dots ,{\vartheta }_{i-1},\vartheta \ell ,{\vartheta }_{i+1},\dots ,{\vartheta }_n\right)=\mathfrak{D}\left({\vartheta }_1,{\vartheta }_2,\dots ,{\vartheta }_{i-1},\vartheta ,{\vartheta }_{i+1},\dots ,{\vartheta }_n\right)\ell \\ \hfill +\vartheta \mathfrak{D}\left({\vartheta }_1,{\vartheta }_2,\dots ,{\vartheta }_{i-1},\ell ,{\vartheta }_{i+1},\dots ,{\vartheta }_n\right)\end{array}$$

holds for all ${\vartheta }_i,\ell \in T,1\le i\le n$.

In 1957, Herstein [11] introduced the concept of a reverse derivation, defining it as an additive map $\alpha$ that satisfies $\alpha \left(\omega \theta \right)=\alpha \left(\theta \right)\omega +\theta \alpha \left(\omega \right)$ for all $\omega ,\theta \in T$. He demonstrated that reverse derivations generally do not exist in the case of prime rings. Later, Brešar and Vukman [12] studied reverse derivations in rings with involution. In this vein, Barros et al. [13] examined the additivity of the multiplicative of *-reverse derivations over alternative algebras and provided a decomposition of Jordan *-reverse derivations as the sum of a *-reverse derivation and a singular Jordan *-reverse derivation. In 2015, Aboubakr and Gonzalez [14] explored the relationship between generalized reverse derivations and generalized derivations on an ideal in semiprime rings. More recently, Sögütcü [15] investigated multiplicative (generalized) reverse derivations in semiprime rings, established some important results and discussed continuous reverse derivations with applications in Banach algebras.

Inspired by these studies, we introduce a new concept called the reverse n-derivation. As the name implies, a reverse n-derivation is essentially a reverse derivation in each of its components. We give the formal definition of reverse n- derivations below:

Definition 2. 

Let $n\ge 2$ be a fixed integer. A map $\mathfrak{D}:T^n\to T$ is said to be a symmetric (permuting) reverse n-derivation if $\mathfrak{D}$ is symmetric and an n-additive, and if

$$\begin{array}{c}\hfill \mathfrak{D}\left({\vartheta }_1,{\vartheta }_2,\dots ,{\vartheta }_{i-1},\vartheta \ell ,{\vartheta }_{i+1},\dots ,{\vartheta }_n\right)=\ell \mathfrak{D}\left({\vartheta }_1,{\vartheta }_2,\dots ,{\vartheta }_{i-1},\vartheta ,{\vartheta }_{i+1},\dots ,{\vartheta }_n\right)\\ \hfill +\mathfrak{D}\left({\vartheta }_1,{\vartheta }_2,\dots ,{\vartheta }_{i-1},\ell ,{\vartheta }_{i+1},\dots ,{\vartheta }_n\right)\vartheta \end{array}$$

holds for all ${\vartheta }_i,\ell \in T,1\le i\le n$.

A reverse 1-derivation is simply a reverse derivation and a reverse 2-derivation is a reverse bi-derivation. The most essential component of a symmetric n-derivation is that of its trace. The trace of a symmetric n-derivation plays an important role as it helps to bridge the gap between an n-derivation and that of an ordinary derivation. It becomes useful while generalizing the results already proven for derivations or bi-derivations to those of n-derivations. Regarding the trace of a reverse n-derivation, we now formally define it as follows:

Definition 3. 

The trace of a symmetric map $\mathfrak{D}:T^n\to T$ is defined on T as $f\left(\vartheta \right)=\mathfrak{D}(\vartheta ,\vartheta ,\dots ,\vartheta )$ for all $\vartheta \in T$.

For a symmetric reverse n-derivation $\mathfrak{D}$, the trace f satisfies the following relation:

$$f(\vartheta +\ell )=f\left(\vartheta \right)+f(\ell )+\sum _{i=1}^{n-1}{}^{n}C_t\mathfrak{D}(\underset{(n-t)-\mathrm{times}}{\underbrace{\vartheta ,\dots ,\vartheta }},\underset{t-\mathrm{times}}{\underbrace{\ell ,\dots ,\ell }})$$

for all $\vartheta ,\ell \in T$, where ⁿ$C_t=\left({\displaystyle \genfrac{}{}{0pt}{}{n}{t}}\right)$.

The following examples help us to understand symmetric reverse n-derivations clearly and see the obvious difference between symmetric reverse n-derivations and symmetric n-derivations.

Example 1. 

Consider the ring $T=\left\{\left[\begin{array}{cc}\vartheta & \ell \\ 0& \vartheta \end{array}\right]|a,b\in \mathbb{Z}\right\}$. Denote $T_i=\left[\begin{array}{cc}{\vartheta }_i& {\ell }_i\\ 0& {\vartheta }_i\end{array}\right]\in T$, ${\vartheta }_i,{\ell }_i\in \mathbb{Z}$, $1\le i\le n$, and let us define $\mathfrak{D}:T^n\to T$ by $\mathfrak{D}(T_1,T_2,\dots ,T_n)=\left[\begin{array}{cc}0& {\ell }_1{\ell }_2\dots {\ell }_n\\ 0& 0\end{array}\right]$ with the trace $f:T\to T$ defined by $f\left(\left[\begin{array}{cc}\vartheta & \ell \\ 0& \vartheta \end{array}\right]\right)=\left[\begin{array}{cc}0& {\ell }^n\\ 0& 0\end{array}\right]$. It is easy to verify that the above-mentioned $\mathfrak{D}$ is a symmetric reverse n-derivation.

Example 2. 

Let $\mathfrak{T}$ be any commutative ring, and define another ring T as $T=\left\{\left[\begin{array}{cc}\vartheta & \ell \\ 0& \theta \end{array}\right]:\vartheta ,\ell ,\theta \in \mathfrak{T}\right\}$. Let $\mathfrak{D}:T^n\to T$ be a map given by $\mathfrak{D}(A_1,A_2,\dots ,A_n)=\left[\begin{array}{cc}0& {\ell }_1{\ell }_2\dots {\ell }_n\\ 0& 0\end{array}\right]$, where $A_i=\left[\begin{array}{cc}{\vartheta }_i& {\ell }_i\\ 0& {\theta }_i\end{array}\right]\in T$, ${\vartheta }_i,{\ell }_i\in \mathfrak{T}$, $1\le i\le n$. $\mathfrak{D}$ so defined forms a symmetric n-derivation but is not a symmetric reverse n-derivation.

In the present study, we prove some results involving symmetric reverse n-derivations and investigate their behavior when the trace is zero on the ideals of semiprime rings. Previous research has demonstrated the crucial role played by the trace function in the study of n-derivations and their related maps. Therefore, our focus is directed towards understanding the trace and some associated maps in the setting of semiprime rings and their well-behaved subsets. One of the primary contributions of this paper lies in analyzing how the trace influences the structure of symmetric reverse n-derivations. Additionally, we highlight new properties of trace maps when restricted to ideals. This provides a deeper insight into how reverse n-derivations behave under specific constraints. The results obtained extend a few existing findings on symmetric n-derivations to those on symmetric reverse n-derivations. In the following section, an example is provided that illustrates that the requirement of semiprimeness for T in Theorem 5 is indispensable and cannot be overlooked.

2. Preliminaries

This section comprises some existing results that prove to be building blocks for the construction of our main results.

Lemma 1 

([10], Lemma 2.2). Let n be a fixed positive integer and T an $n!$-torsion-free ring. Suppose that ${\vartheta }_1,{\vartheta }_2,\dots ,{\vartheta }_n\in T$ satisfies $\lambda {\vartheta }_1+{\lambda }^2{\vartheta }_2+\dots +{\lambda }^n{\vartheta }_n=0(or\in \mathcal{Z}\left(T\right))$ for $\lambda =1,2,\dots ,n$. Then, ${\vartheta }_i=0(or\in \mathcal{Z}\left(T\right))$ for $i=1,2,\dots ,n.$

Lemma 2 

([16], Lemma 2(b)). If T is a semiprime ring, then the center of a nonzero ideal of T is contained in the center of T.

Lemma 3 

([17], Lemma 2.1). Let T be a semiprime ring and I be a nonzero two-sided ideal of T and $a\in T$ such that $a\vartheta a=0$ for all $\vartheta \in I$. Then, $a=0$.

Lemma 4 

([8]). Let T be a two-torsion-free semiprime ring and J be a nonzero ideal of T. If $[J,J]\subseteq \mathcal{Z}\left(T\right)$, then T contains a nonzero central ideal.

Lemma 5 

([18], Lemma 1.4). Let T be a semiprime ring. If a nonzero ideal I of T is in the center of T, then T is a commutative ring.

Proposition 1. 

Let $n\ge 2$ be a fixed integer and T be an $n!$-torsion-free semiprime ring, and J its non-zero ideal. If there exists a symmetric reverse n-derivation $\mathfrak{D}$ on T with the trace f such that $\left[f\left(\vartheta \right),\ell \right]\in \mathcal{Z}\left(T\right)$ for all $\vartheta ,\ell \in J$, then f is commuting on J.

Proof. 

Since $\left[f\left(\vartheta \right),\ell \right]\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell \in J.$ Replacing ℓ with $\ell \theta$, where $\theta \in J$, we obtain

$$\left[f\left(\vartheta \right),\ell \theta \right]\in \mathcal{Z}\left(T\right),\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J,$$

which gives

$$\ell \left[f\left(\vartheta \right),\theta \right]+\left[f\left(\vartheta \right),\ell \right]\theta \in \mathcal{Z}\left(T\right),\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.$$

Commuting the above expression with $t\in T$, we get

$$\left[\ell \left[f\left(\vartheta \right),\theta \right]+\left[f\left(\vartheta \right),\ell \right]\theta ,t\right]=0\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.$$

On solving this further, we obtain

$$\left[\ell ,t\right]\left[f\left(\vartheta \right),\theta \right]+\ell \left[\left[f\left(\vartheta \right),\theta \right],t\right]+\left[f\left(\vartheta \right),\ell \right]\left[\theta ,t\right]+\left[\left[f\left(\vartheta \right),\ell \right],t\right]=0.$$

This implies that

$$\left[\ell ,t\right]\left[f\left(\vartheta \right),\theta \right]+\left[f\left(\vartheta \right),\ell \right]\left[\theta ,t\right]=0\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J,t\in T.$$

If we replace t with $\theta$ in the last equation, we obtain

$$\left[\ell ,\theta \right]\left[f\left(\vartheta \right),\theta \right]=0.$$

Again, if we replace ℓ with $t\ell$, $t\in T$, we have

$$\left[t,\theta \right]\ell \left[f\left(\vartheta \right),\theta \right]=0\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.$$

Upon substitution of t with $f\left(\vartheta \right)$, we obtain

$$\left[f\left(\vartheta \right),\theta \right]\ell \left[f\left(\vartheta \right),\theta \right]=0\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.$$

Therefore, applying Lemma 3 gives

$$\left[f\left(\vartheta \right),\theta \right]=0\mathrm{for}\mathrm{all}\vartheta ,\theta \in J.$$

Hence, the trace f is commuting on J. □

Using a similar approach with necessary variations, we can prove the following result.

Proposition 2. 

Let $n\ge 2$ be a fixed integer and T be an $n!$-torsion-free semiprime ring, and J its non-zero ideal. If there exists a symmetric reverse n-derivation $\mathfrak{D}$ on T with the trace f such that $f\left(\vartheta \right)o\ell \in \mathcal{Z}\left(T\right)$ for all $\vartheta ,\ell \in J$, then f is commuting on J.

3. The Results Involving Symmetric Reverse n-Derivations

Park, in [10], proved various interesting results for symmetric n-derivations, like generalization to the famous Posner’s result (cf.; [19]) and several other results. Some of his initial work on n-derivations which paved way for future developments in this area are also provided in the same paper. In 2014, Ashraf and Jamal [20] introduced certain interesting identities that help us reveal the structure of any ring, similar to the work of Daif and Bell on semiprime rings [16]. They demonstrated that a ring T is commutative if there exists a symmetric n-additive map $\mathfrak{D}:T^n\to T$ satisfying certain functional identities. Also, in the case of n-derivations, Ashraf et al. in [21] developed upon the line of inquiry provided by Park in the case of prime rings. Ashraf et al. [22] used properties of an ideal and the trace of symmetric n-derivations to achieve results for the existence of central ideals, which eventually helped them to see the commutativity of the rings under consideration. Recently, Ali et al. in [23] presented several findings regarding the containment of a nonzero central ideal in a ring T that adheres to specific functional identities involving the traces d and g of the symmetric n-derivations D and G, respectively. In addition to obtaining results about the traces of permuting n-derivations, they studied the relationship between n-derivations and n-multipliers and provided a characterization of symmetric n-derivations of prime rings in terms of left n-multipliers. It would be interesting to conduct a similar study on the characterization of symmetric reverse n-derivations in terms of n-multipliers. Motivated by the existing work on symmetric n-derivations, we wish to explore rings, specifically semiprime rings, so as to understand the behavior of symmetric reverse n-derivations.

In this paper, we examine two main aspects of symmetric reverse n-derivations. First, we investigate their behavior when the trace is zero on an ideal. Second, we explore some identities involving the trace itself. Previous research has demonstrated the crucial role played by the trace function in the study of n-derivations. Therefore, our focus is directed towards understanding the trace and some associated maps. One of the primary contributions of this paper lies in analyzing how the trace influences the structure of symmetric reverse n-derivations. Additionally, we highlight new properties of trace maps when restricted to ideals. This provides a deeper insight into how reverse n-derivations behave under specific constraints. The results obtained extend a few existing findings on symmetric n-derivations to those of symmetric reverse n-derivations. The first important result of this paper is the following:

Theorem 1. 

Let $n\ge 2$ be a fixed integer, T be an $n!$-torsion-free semiprime ring, and J be its non-zero ideal. If there exists a symmetric reverse n-derivation $\mathfrak{D}$ on T with the trace f such that $f\left(J\right)=\left\{0\right\}$, then $\mathfrak{D}=0$.

Proof. 

Based on the hypothesis, we have

$$f\left(\vartheta \right)=0\mathrm{for}\mathrm{all}\vartheta \in J.$$

Substituting $\vartheta$ with $\vartheta +m{\ell }_1$, where ${\ell }_1\in J$ for $1\le m\le n-1$, we obtain

$$f(\vartheta +m{\ell }_1)=0\mathrm{for}\mathrm{all}\vartheta ,{\ell }_1\in J,$$

which is given by

$$f\left(\vartheta \right)+f\left(m{\ell }_1\right)+\sum _{i=1}^{n-1}{}^{n}C_i\mathfrak{D}(\underset{(n-t)-\mathrm{times}}{\underbrace{\vartheta ,\dots ,\vartheta }},\underset{t-\mathrm{times}}{\underbrace{m{\ell }_1,\dots ,m{\ell }_1}})=0\mathrm{for}\mathrm{all}\vartheta ,{\ell }_1\in J.$$

Using the given hypothesis, we obtain

$$\sum _{i=1}^{n-1}{}^{n}C_i\mathfrak{D}(\underset{(n-t)-\mathrm{times}}{\underbrace{\vartheta ,\dots ,\vartheta }},\underset{t-\mathrm{times}}{\underbrace{m{\ell }_1,\dots ,m{\ell }_1}})=0\mathrm{for}\mathrm{all}\vartheta ,{\ell }_1\in J.$$

Lemma 1 allows us to equate coefficients to 0; we can write

$$n\mathfrak{D}({\ell }_1,\vartheta ,\dots ,\vartheta )=0\mathrm{for}\mathrm{all}\vartheta ,{\ell }_1\in J$$

and using torsion restrictions, we have

$$\mathfrak{D}({\ell }_1,\vartheta ,\dots ,\vartheta )=0\mathrm{for}\mathrm{all}\vartheta ,{\ell }_1\in J.$$

If we substitute $\vartheta$ again with $\vartheta +m{\ell }_2$ (where ${\ell }_2\in J$ and $1\le m\le n-1$), we obtain

$$\mathfrak{D}({\ell }_1,\vartheta +m{\ell }_2,\dots ,\vartheta +m{\ell }_2)=0\mathrm{for}\mathrm{all}\vartheta ,{\ell }_1,{\ell }_2\in J.$$

Computing further, we obtain

$$\mathfrak{D}({\ell }_1,{\ell }_2,\vartheta \dots ,\vartheta )=0\mathrm{for}\mathrm{all}\vartheta ,{\ell }_1,{\ell }_2\in J.$$

This process can be continued until we obtain

$$\mathfrak{D}({\ell }_1,{\ell }_2,\dots ,{\ell }_n)=0\mathrm{for}\mathrm{all}{\ell }_1,{\ell }_2,\dots {\ell }_n\in J.$$

(1)

So, from $f\left(J\right)=0$, we arrive at $\mathfrak{D}(J,J,\dots ,J)=\left\{0\right\}$. Now, replace ${\ell }_1$ with ${\ell }_1{t}_1$ in (1) to obtain

$$\mathfrak{D}({\ell }_1{t}_1,{\ell }_2,\dots ,{\ell }_n)=0\mathrm{for}\mathrm{all}{\ell }_1,{\ell }_2,\dots {\ell }_n\in J,t_1\in T.$$

Using Equation (1), we arrive at

$$\mathfrak{D}(t_1,{\ell }_2,\dots ,{\ell }_n){\ell }_1=0\mathrm{for}\mathrm{all}{\ell }_1,{\ell }_2,\dots {\ell }_n\in J,t_1\in T.$$

Now, replace ${\ell }_2$ by ${\ell }_2{t}_2$ with $t_2\in T$, we obtain

$$\mathfrak{D}(t_1,t_2,\dots ,{\ell }_n){\ell }_2{\ell }_1=0\mathrm{for}\mathrm{all}{\ell }_1,{\ell }_2,\dots {\ell }_n\in J,t_1,t_2\in T.$$

Continuing in the same manner, we finally obtain

$$\mathfrak{D}(t_1,t_2,\dots ,t_n){\ell }_n{\ell }_{n-1}\dots {\ell }_2{\ell }_1=0\mathrm{for}\mathrm{all}{\ell }_1,{\ell }_2,\dots {\ell }_n\in J,t_1,t_2,\dots ,t_n\in T.$$

In the above equation, if we replace ${\ell }_1$ with the term ${\ell }_1\mathfrak{D}(t_1,t_2,\dots ,t_n){\ell }_n{\ell }_{n-1}\dots {\ell }_2$, we obtain

$$\mathfrak{D}(t_1,t_2,\dots ,t_n){\ell }_n{\ell }_{n-1}\dots {\ell }_{2}J\mathfrak{D}(t_1,t_2,\dots ,t_n){\ell }_n{\ell }_{n-1}\dots {\ell }_2=\left(0\right)$$

for all ${\ell }_2,\dots {\ell }_n\in J,t_1,t_2,\dots ,t_n\in T$. Since T is semiprime, by invoking Lemma 3, we obtain

$$\mathfrak{D}(t_1,t_2,\dots ,t_n){\ell }_n{\ell }_{n-1}\dots {\ell }_2=0$$

for all ${\ell }_2,\dots {\ell }_n\in J,t_1,t_2,\dots ,t_n\in T$. Continuing in a similar manner, we can keep on omitting ${\ell }_i$ one by one and obtain

$$\mathfrak{D}(t_1,t_2,\dots ,t_n){\ell }_n=0\mathrm{for}\mathrm{all}{\ell }_n\in J,t_1,t_2,\dots ,t_n\in T.$$

Lastly, replace ${\ell }_n$ with $\mathfrak{D}(t_1,t_2,\dots ,t_n)$ to arrive at

$$\mathfrak{D}(t_1,t_2,\dots ,t_n){\ell }_n\mathfrak{D}(t_1,t_2,\dots ,t_n)=0\mathrm{for}\mathrm{all}{\ell }_n\in J,t_1,t_2,\dots ,t_n\in T.$$

This can be written as

$$\mathfrak{D}(t_1,t_2,\dots ,t_n)J\mathfrak{D}(t_1,t_2,\dots ,t_n)=\left(0\right)\mathrm{for}\mathrm{all}{t}_1,t_2,\dots ,t_n\in T.$$

Therefore, using Lemma 3, we obtain

$$\mathfrak{D}(t_1,t_2,\dots ,t_n)=0\mathrm{for}\mathrm{all}{t}_1,t_2,\dots ,t_n\in T,$$

which is the required conclusion. Hence, $\mathfrak{D}=0$. □

Theorem 2. 

Let T be an $n!$-torsion-free semiprime ring, J be a non-zero ideal of T, and $\mathfrak{D}:T^n\to T$ be a symmetric reverse n-derivation on T with trace f. If any one of the following conditions holds in T, then f is commuting on J:

1. 

$f(\vartheta o\ell )\pm f\left(\vartheta \right)o\ell \in \mathcal{Z}\left(T\right)$ for all $\vartheta ,\ell \in J$,

2. 

$f\left(\right[\vartheta ,\ell \left]\right)\pm f\left(\vartheta \right)o\ell \in \mathcal{Z}\left(T\right)$, for all $\vartheta ,\ell \in J$,

3. 

$f\left(\vartheta \right)o\ell \pm \left[f\right(\ell ),\vartheta ]\in \mathcal{Z}\left(T\right)$, for all $\vartheta ,\ell \in J$.

Proof. 

1. According to the hypothesis, we have

$$f(\vartheta o\ell )\pm \left(f\right(\vartheta )o\ell )\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell \in J.$$

Taking ℓ in place of $\ell +m\theta$, $\theta \in J$, $1\le m\le n-1$, we obtain

$$f\left(\vartheta o\right(\ell +m\theta \left)\right)\pm \left(f\right(\vartheta \left)o\right(\ell +m\theta \left)\right)\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.$$

$$f(\vartheta o\ell +\vartheta om\theta )\pm \left(f\right(\vartheta )o\ell +f(\vartheta \left)om\theta \right)\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J$$

$$\begin{array}{c}f(\vartheta o\ell )+f\left(\vartheta om\theta \right)+\sum _{i=1}^{n-1}{}^{n}C_i\mathfrak{D}(\vartheta o\ell ,\dots ,\vartheta o\ell ,\vartheta om\theta ,\dots ,\vartheta om\theta )\hfill \\ \hfill \pm f\left(\vartheta \right)o\ell \pm f\left(\vartheta \right)om\theta \in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.\end{array}$$

Using the hypothesis, we obtain

$$\sum _{i=1}^{n-1}{}^{n}C_i\mathfrak{D}(\vartheta o\ell ,\dots ,\vartheta o\ell ,\vartheta om\theta ,\dots ,\vartheta om\theta )\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.$$

The application of Lemma 1 yields

$$n\mathfrak{D}(\vartheta o\ell ,\dots ,\vartheta o\ell ,\vartheta o\theta )\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.$$

Since T is $n!$-torsion-free, we obtain

$$\mathfrak{D}(\vartheta o\ell ,\dots ,\vartheta o\ell ,\vartheta o\theta )\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.$$

Replace ℓ with $\theta$ to obtain

$$f(\vartheta o\ell )\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell \in J.$$

Thus,

$$f\left(\vartheta \right)o\ell \in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell \in J.$$

Hence, Proposition 2 implies that f is commuting on J.

2. We are given

$$f\left(\right[\vartheta ,\ell \left]\right)\pm f\left(\vartheta \right)o\ell \in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell \in J.$$

If we replace ℓ with $\ell +m\theta$, $\theta \in J,$$1\le m\le n-1$, we obtain the following calculations:

$$f\left(\right[\vartheta ,\ell +m\theta \left]\right)\pm \left(f\right(\vartheta \left)o\right(\ell +m\theta \left)\right)\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J,$$

and

$$f\left(\right[\vartheta ,\ell ]+[\vartheta ,m\theta \left]\right)\pm f\left(\vartheta \right)o\ell \pm f\left(\vartheta \right)om\theta \in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.$$

Thus, we have

$$\begin{array}{c}\hfill f\left([\vartheta ,\ell ]\right)+f\left([\vartheta ,m\theta ]\right)+\sum _{i=1}^{n-1}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right){}^{n}C_i\mathfrak{D}([\vartheta ,\ell ][\vartheta ,\ell ]+[\vartheta ,m\theta ][\vartheta ,m\theta ])\\ \hfill \pm f\left(\vartheta \right)o\ell \pm f\left(\vartheta \right)om\theta \in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.\end{array}$$

The application of the hypothesis yields

$$\begin{array}{c}\hfill \sum _{i=1}^{n-1}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right){}^{n}C_i\mathfrak{D}([\vartheta ,\ell ],[\vartheta ,\ell ],\dots ,[\vartheta ,m\theta ][\vartheta ,m\theta ])\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.\end{array}$$

Using Lemma 1 and torsion restrictions, we have

$$\begin{array}{c}\hfill \mathfrak{D}\left(\right[\vartheta ,\ell ],[\vartheta ,\theta ],\dots [\vartheta ,\theta \left]\right)\in \mathcal{Z}\left(T\right),\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.\end{array}$$

Substituting $\theta$ with ℓ, we obtain

$$\begin{array}{c}\hfill f\left(\right[\vartheta ,\ell \left]\right)\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.\end{array}$$

So, the given hypothesis boils down to

$$f\left(\vartheta \right)o\ell \in \mathcal{Z}\left(T\right).$$

Based on Proposition 2, we conclude that f is commuting on J.

3. We are given

$$f\left(\vartheta \right)o\ell \pm \left[f\right(\ell ),\vartheta ]\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell \in J.$$

The replacement of ℓ with $\ell +m\theta$, $\theta \in J$, $1\le m\le n-1$ yields

$$\begin{array}{c}\hfill f\left(\vartheta \right)o(\ell +m\theta )\pm \left[f\right(\ell +m\theta ),\vartheta ]\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.\\ \hfill f\left(\vartheta \right)o\ell +f\left(\vartheta \right)om\theta \pm \left[f\right(\ell ),\vartheta ]\pm \left[f\right(m\theta ),\vartheta ]\pm \\ \hfill \left[\sum _{i=1}^{n-1}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right){}^{n}C_i\mathfrak{D}(\ell ,\dots ,\ell ,m\theta ,\dots ,m\theta ,\vartheta )\right]\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.\end{array}$$

Using the given condition, we obtain

$$\left[\sum _{i=1}^{n-1}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right){}^{n}C_i\mathfrak{D}(\ell ,\dots ,\ell ,m\theta ,\dots ,m\theta ),\vartheta \right]\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.$$

By applying Lemma 1 and the torsion restrictions, we obtain

$$\left[\mathfrak{D}\right(\ell ,\theta ,\dots ,\theta ),\vartheta ]\in \mathcal{Z}\left(T\right),\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.$$

Again, replace $\theta$ with ℓ; we then have

$$\left[f\right(\ell ),\vartheta ]\mathrm{for}\mathrm{all}\vartheta ,\ell \in J.$$

From the hypothesis, we arrive at

$$f\left(\vartheta \right)o\ell \in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell \in J.$$

In view of Proposition 2, we obtain our desired conclusion. □

Theorem 3. 

Let T be an $n!$-torsion-free semiprime ring, J be a non-zero ideal of T, and $\mathfrak{D}:T^n\to T$ be a symmetric reverse n-derivation on T with the trace f. If any one of the following conditions holds in T, then f is commuting on J:

1. 

$f\left(\right[\vartheta ,\ell \left]\right)\pm \left[f\right(\vartheta ),\ell ]\in \mathcal{Z}\left(T\right)$, for all $\vartheta ,\ell \in J$,

2. 

$f(\vartheta o\ell )\pm \left[f\right(\vartheta ),\ell ]\in \mathcal{Z}\left(T\right)$, for all $\vartheta ,\ell \in J$,

3. 

$\left[f\right(\vartheta ),\ell ]\pm \left[f\right(\ell ),\vartheta ]\in \mathcal{Z}\left(T\right)$, for all $\vartheta ,\ell \in J$.

Proof. 

1. We have

$$f\left(\right[\vartheta ,\ell \left]\right)\pm \left[f\right(\vartheta ),\ell ]\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell \in J.$$

Now, if we substitute ℓ with $\ell +m\theta$, $1\le m\le n-1$, $\theta \in J$, we obtain

$$f\left(\right[\vartheta ,\ell +m\theta \left]\right)\pm \left[f\right(\vartheta ),\ell +m\theta ]\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J$$

$$f\left(\right[\vartheta ,\ell ]+[\vartheta ,m\theta \left]\right)\pm \left[f\right(\vartheta ),\ell ]\pm \left[f\right(\vartheta ),m\theta ]\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J$$

$$\begin{array}{c}f\left([\vartheta ,\ell ]\right)+f\left([\vartheta ,m\theta ]\right)+\sum _{i=1}^{n-1}{}^{n}C_i\mathfrak{D}([\vartheta ,\ell ],\dots ,[\vartheta ,\ell ],[\vartheta ,m\theta ],\dots ,[\vartheta ,m\theta ])\hfill \\ \hfill \pm \left[f\right(\vartheta ),\ell ]\pm \left[f\right(\vartheta ),m\theta ]\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.\end{array}$$

The application of the hypothesis gives,

$$\sum _{i=1}^{n-1}{}^{n}C_i\mathfrak{D}([\vartheta ,\ell ],\dots ,[\vartheta ,\ell ],[\vartheta ,m\theta ],\dots ,[\vartheta ,m\theta ])\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.$$

On solving this further, we obtain

$$n\mathfrak{D}\left(\right[\vartheta ,\ell ],[\vartheta ,\theta ],\dots ,[\vartheta ,\theta \left]\right)\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J$$

and

$$\mathfrak{D}\left(\right[\vartheta ,\ell ],[\vartheta ,\theta ],\dots ,[\vartheta ,\theta \left]\right)\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.$$

Replacing $\theta$ with ℓ in the above expression, we obtain

$$f\left(\right[\vartheta ,\ell \left]\right)\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell \in J.$$

Using the hypothesis, we obtain

$$\left[f\right(\vartheta ),\ell ]\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell \in J.$$

Therefore, Proposition 1 implies that f is commuting on J.

2. We are given

$$\begin{array}{ccc}\hfill f(\vartheta o\ell )& \pm & \left[f\right(\vartheta ),\ell ]\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell \in J.\hfill \end{array}$$

Substitute ℓ with $\ell +m\theta ,\theta \in J,1\le m\le n-1$ so that

$$\begin{array}{c}\hfill f\left(\vartheta o\right(\ell +m\theta \left)\right)\pm \left[f\right(\vartheta ),\ell +m\theta ]\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J\\ \hfill f(\vartheta o\ell +\vartheta om\theta )\pm \left[f\right(\vartheta ),\ell ]\pm \left[f\right(\vartheta ),m\theta ]\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J\\ \hfill f(\vartheta o\ell )+f\left(\vartheta om\theta \right)+\sum _{i=1}^{n-1}\mathfrak{D}(\vartheta o\ell ,\dots \vartheta o\ell ,\vartheta om\theta ,\dots ,\vartheta om\theta )\\ \hfill \pm \left[f\right(\vartheta ),\ell ]\pm \left[f\right(\vartheta ),m\theta ]\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.\end{array}$$

Using the given hypothesis, we obtain

$$\begin{array}{c}\hfill \sum _{i=1}^{n-1}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right)={}^{n}C_i\mathfrak{D}(\vartheta o\ell ,\dots \vartheta o\ell ,\vartheta om\theta ,\dots ,\vartheta om\theta )\\ \hfill \pm \left[f\right(\vartheta ),\ell ]\pm \left[f\right(\vartheta ),m\theta ]\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.\end{array}$$

The application of Lemma 1 and torsion restrictions yields

$$\begin{array}{c}\hfill \mathfrak{D}(\vartheta o\ell ,\vartheta o\theta ,\dots ,\vartheta o\theta )\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.\end{array}$$

Replacing $\theta$ with ℓ gives

$$f(\vartheta o\ell )\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell \in J.$$

Based on the hypothesis, we conclude that

$$\left[f\right(\vartheta ),\ell ]\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell \in J.$$

Hence, f is commuting on J.

3. We have

$$\left[f\right(\vartheta ),\ell ]\pm \left[f\right(\ell ),\vartheta ]\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell \in J.$$

If we replace ℓ with $\ell +m\theta$, $\theta \in J$, $1\le m\le n-1$, we obtain

$$\left[f\right(\vartheta ),\ell +m\theta ]\pm \left[f\right(\ell +m\theta ),\vartheta ]\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.$$

This implies that

$$\begin{array}{c}\left[f\right(\vartheta ),\ell ]+\left[f\right(\vartheta ),m\theta ]\pm \left[f\right(\ell ),\vartheta ]\pm \left[f\right(m\theta ),\vartheta ]\pm \hfill \\ \hfill \left[\sum _{i=1}^{n-1}\mathfrak{D}(\ell ,\dots ,\ell ,m\theta ,\dots ,m\theta ),\vartheta \right]\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.\end{array}$$

Using the hypothesis, we obtain

$$\left[\sum _{i=1}^{n-1}\mathfrak{D}(\ell ,\dots ,\ell ,m\theta ,\dots ,m\theta ),\vartheta \right]\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.$$

By using Lemma 1 and the torsion restrictions, we obtain

$$\left[\mathfrak{D}\right(\ell ,\theta ,\dots ,\theta ),\vartheta ]\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.$$

Replacing $\theta$ with ℓ in the above expression, we have

$$\left[f\right(\ell ),\vartheta ]\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell \in J.$$

In view of Proposition 1, we obtain the desired conclusion. □

Theorem 4. 

Let T be an $n!$-torsion-free semiprime ring, J be a non-zero ideal of T, and $\mathfrak{D}:T^n\to T$ be a symmetric reverse n-derivation on T with the trace f. If $f\left(\right[\vartheta ,\ell \left]\right)-\left(f\right(\vartheta )o\ell )-\left[f\right(\ell ),\vartheta ]\in \mathcal{Z}\left(T\right)$ for all $\vartheta ,\ell \in J$, then f is commuting on J.

Proof. 

We are given

$$f\left(\right[\vartheta ,\ell \left]\right)-\left(f\right(\vartheta )o\ell )-\left[f\right(\ell ),\vartheta ]\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell \in J.$$

If we replace ℓ with $\ell +m\theta$, $\theta \in J$, $1\le m\le n-1$, we see that

$$f\left(\right[\vartheta ,\ell +m\theta \left]\right)-\left(f\right(\vartheta )o\ell +m\theta )-\left[f\right(\ell +m\theta ),\vartheta ]\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.$$

On solving this further, we obtain

$$\begin{array}{c}f\left(\right[\vartheta ,\ell ]+[\vartheta ,m\theta \left]\right)-f\left(\vartheta \right)o\ell -f\left(\vartheta \right)om\theta -\left[f\right(\ell ),\vartheta ]-\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left[f\right(m\theta ),\vartheta ]-\left[\sum _{i=1}^{n-1}{}^{n}C_i\mathfrak{D}(\ell ,\dots ,\ell ,m\theta ,\dots ,m\theta ),\vartheta \right]\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J\hfill \end{array}$$

and

$$\begin{array}{c}f\left([\vartheta ,\ell ]\right)+f\left([\vartheta ,m\theta ]\right)+\sum _{i=1}^{n-1}{}^{n}C_i\mathfrak{D}\left([\vartheta ,\ell ],\dots ,[\vartheta ,\ell ],[\vartheta ,m\theta ],\dots ,[\vartheta ,m\theta ]\right)-\hfill \\ f\left(\vartheta \right)o\ell -f\left(\vartheta \right)om\theta -\left[f\right(\ell ),\vartheta ]-\left[f\right(m\theta ),\vartheta ]-\\ \hfill \hspace{1em}\hspace{1em}\left[\sum _{i=1}^{n-1}{}^{n}C_i\mathfrak{D}(\ell ,\dots ,\ell ,m\theta ,\dots ,m\theta ),\vartheta \right]\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.\end{array}$$

From the given hypothesis, the above expression reduces to

$$\begin{array}{c}\sum _{i=1}^{n-1}{}^{n}C_i\mathfrak{D}([\vartheta ,\ell ],\dots ,[\vartheta ,\ell ],[\vartheta ,m\theta ],\dots ,[\vartheta ,m\theta ])-\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left[\sum _{i=1}^{n-1}{}^{n}C_i\mathfrak{D}(\ell ,\dots ,\ell ,m\theta ,\dots ,m\theta ),\vartheta \right]\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}x,\ell ,\theta \in J.\hfill \end{array}$$

The application of Lemma 1 yields

$$\mathfrak{D}\left(\right[\vartheta ,\ell ],[\vartheta ,\theta ],\dots ,[\vartheta ,\theta \left]\right)-\left[\mathfrak{D}\right(\ell ,\theta ,\dots ,\theta ),\vartheta ]\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.$$

Replacing $\theta$ with ℓ provides

$$f\left(\right[\vartheta ,\ell \left]\right)-\left[f\right(\ell ),\vartheta ]\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.$$

Therefore, we have

$$f(\ell )o\vartheta \in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell \in J$$

and hence, from Proposition 2, we conclude that f is commuting on J. This completes the proof. □

Theorem 5. 

Let T be an $n!$-torsion-free semiprime ring, J be a non-zero ideal of T, and $\mathfrak{D}:T^n\to T$ be a symmetric reverse n-derivation on T with the trace f. If any one of the following conditions hold, then T contains a non-zero central ideal.

1. 

$f(\vartheta \ell )+f\left(\vartheta \right)f(\ell )\pm \vartheta \ell \in \mathcal{Z}\left(T\right)$, for all $\vartheta ,\ell \in J$,

2. 

$f(\vartheta \ell )+f\left(\vartheta \right)f(\ell )\pm \ell \vartheta \in \mathcal{Z}\left(T\right)$, for all $\vartheta ,\ell \in J$,

3. 

$f(\vartheta \ell )-f(\ell \vartheta )\pm [\vartheta ,\ell ]\in \mathcal{Z}\left(T\right)$, for all $\vartheta ,\ell \in J$.

Proof. 

1. We have,

$$f(\vartheta \ell )+f\left(\vartheta \right)f(\ell )\pm \vartheta \ell \in \mathcal{Z}\left(T\right),\mathrm{for}\mathrm{all}\vartheta ,\ell \in J.$$

On replacing ℓ with $\ell +m\theta$, $\theta \in J$, $1\le m\le n-1$, we obtain

$$f\left(\vartheta \right(\ell +m\theta \left)\right)+f\left(\vartheta \right)f(\ell +m\theta )\pm \vartheta (\ell +m\theta )\in \mathcal{Z}\left(T\right),\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.$$

This further implies that

$$\begin{array}{c}f(\vartheta \ell )+f\left(\vartheta m\theta \right)+\sum _{i=1}^{n-1}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right)\mathfrak{D}(\vartheta \ell ,\dots ,\vartheta \ell ,\vartheta m\theta ,\dots ,\vartheta m\theta )+\hfill \\ \hfill f\left(\vartheta \right)(f(\ell )+f\left(m\theta \right))+f\left(\vartheta \right)\sum _{i=1}^{n-1}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right)\mathfrak{D}(\ell ,\dots ,\ell ,m\theta ,\dots ,m\theta ))\pm \vartheta \ell \pm \vartheta m\theta \in \mathcal{Z}\left(T\right).\end{array}$$

Using the given hypothesis, we obtain

$$\sum _{i=1}^{n-1}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right)\mathfrak{D}(\vartheta \ell ,\dots ,\vartheta \ell ,\vartheta m\theta ,\dots ,\vartheta m\theta )+f\left(\vartheta \right)\sum _{i=1}^{n-1}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right)\mathfrak{D}(\ell ,\dots ,\ell ,m\theta ,\dots ,m\theta )\in \mathcal{Z}\left(T\right)$$

for all $\vartheta ,\ell ,\theta \in J$. The application of Lemma 1 yields

$$n\mathfrak{D}(\vartheta \ell ,\dots ,\vartheta \ell ,\vartheta \theta )+nf\left(\vartheta \right)\mathfrak{D}(\ell ,\dots ,\ell ,\theta )\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.$$

Since T is $n!$-torsion-free, we obtain

$$\mathfrak{D}(\vartheta \ell ,\dots ,\vartheta \ell ,\vartheta \theta )+f\left(\vartheta \right)\mathfrak{D}(\ell ,\dots ,\ell ,\theta )\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.$$

Replace $\theta$ with ℓ in the above equation to obtain

$$\mathfrak{D}(\vartheta \ell ,\dots ,\vartheta \ell ,\vartheta \ell )+f\left(\vartheta \right)\mathfrak{D}(\ell ,\dots ,\ell ,\ell )\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell \in J.$$

Hence, we arrive at

$$f(\vartheta \ell )+f\left(\vartheta \right)f(\ell )\in \mathcal{Z}\left(T\right),\mathrm{for}\mathrm{all}\vartheta ,\ell \in J.$$

Using this in the given hypothesis, we obtain

$$\vartheta \ell \in \mathcal{Z}\left(T\right),\mathrm{for}\mathrm{all}\vartheta ,\ell \in J.$$

On commuting this with any $t\in T$, we obtain

$$[\vartheta \ell ,t]=0\mathrm{for}\mathrm{all}\vartheta ,\ell \in J;t\in T.$$

This can be written as

$$\vartheta [\ell ,t]+[\vartheta ,t]\ell =0\mathrm{for}\mathrm{all}\vartheta ,\ell \in J;t\in T$$

Replacing ℓ with $\ell \theta$, where $\theta \in J$, we see that

$$\vartheta \ell [\theta ,t]=0\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J;t\in T.$$

Now, substitute $[\theta ,t]$ in place of $\vartheta$ to obtain

$$[\theta ,t]\ell [\theta ,t]=0\mathrm{for}\mathrm{all}\ell ,\theta \in J;t\in T.$$

So we can write

$$[\theta ,t]\ell T[\theta ,t]\ell =\left\{0\right\},$$

which gives

$$[\theta ,t]\ell =0\mathrm{for}\mathrm{all}\ell ,\theta \in J;t\in T.$$

Taking ℓ to be $t[\theta ,t]$ and using the semiprimeness of T, we obtain $J\subseteq \mathcal{Z}\left(T\right)$. Hence, J is a non-zero central ideal in T.

2. Using similar arguments to those in part 1, we obtain the desired conclusion.

3. We have

$$f(\vartheta \ell )-f(\ell \vartheta )\pm [\vartheta ,\ell ]\in \mathcal{Z}\left(T\right),\mathrm{for}\mathrm{all}\vartheta ,\ell \in J.$$

Substitute $\ell +m\theta$ in place of ℓ, $\theta \in J$, $1\le m\le n-1$ to obtain

$$f\left(\vartheta \right(\ell +m\theta \left)\right)-f\left(\right(\ell +m\theta \left)\vartheta \right)\pm [\vartheta ,\ell +m\theta ]\in \mathcal{Z}\left(T\right),\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.$$

This implies that

$$\begin{array}{c}f(\vartheta \ell )+f\left(\vartheta m\theta \right)+\sum _{i=1}^{n-1}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right)\mathfrak{D}(\vartheta \ell ,\dots ,\vartheta \ell ,\vartheta m\theta ,\dots ,\vartheta m\theta )-\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}f(\ell \vartheta )-f\left(m\theta \vartheta \right)-\sum _{i=1}^{n-1}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right)\mathfrak{D}(\ell \vartheta ,\dots ,\ell \vartheta ,m\theta \vartheta ,\dots ,m\theta \vartheta )\pm [\vartheta ,\ell ]\pm [\vartheta ,m\theta ]\in \mathcal{Z}\left(T\right).\hfill \end{array}$$

Using the given hypothesis, we arrive at

$$\sum _{i=1}^{n-1}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right)\mathfrak{D}(\vartheta \ell ,\dots ,\vartheta \ell ,\vartheta m\theta ,\dots ,\vartheta m\theta )-\sum _{i=1}^{n-1}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right)\mathfrak{D}(\ell \vartheta ,\dots ,\ell \vartheta ,m\theta \vartheta ,\dots ,m\theta \vartheta )\in \mathcal{Z}\left(T\right)$$

for all $\vartheta ,\ell ,\theta \in J$. In view of Lemma 1, we have

$$n\mathfrak{D}(\vartheta \ell ,\dots ,\vartheta \ell ,\vartheta \theta )-n\mathfrak{D}(\ell \vartheta ,\dots ,\ell \vartheta ,\theta \vartheta )\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.$$

Since T is $n!$-torsion-free, we have

$$\mathfrak{D}(\vartheta \ell ,\dots ,\vartheta \ell ,\vartheta \theta )-\mathfrak{D}(\ell \vartheta ,\dots ,\ell \vartheta ,\theta \vartheta )\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell ,\theta \in J.$$

On replacing $\theta$ with ℓ in the above equation, we obtain

$$\mathfrak{D}(\vartheta \ell ,\dots ,\vartheta \ell ,\vartheta \ell )-\mathfrak{D}(\ell \vartheta ,\dots ,\ell \vartheta ,\ell \vartheta )\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell \in J.$$

Therefore,

$$f(\vartheta \ell )-f(\ell \vartheta )\in \mathcal{Z}\left(T\right)\mathrm{for}\mathrm{all}\vartheta ,\ell \in J.$$

Thus, from the given hypothesis, we can conclude that

$$[\vartheta ,\ell ]\in \mathcal{Z}\left(T\right),\mathrm{for}\mathrm{all}\vartheta ,\ell \in T.$$

The application of Lemma 4 gives us the existence of a non-zero central ideal in T. Hence, we obtain the desired result. □

The subsequent example illustrates that the requirement of semiprimeness for T in Theorem 5 is indispensable and cannot be overlooked. The following example justifies this fact:

Example 3. 

Consider the ring $T=\left\{\left[\begin{array}{cc}a& b\\ 0& 0\end{array}\right]|a,b\in \mathbb{Z}\right\}$. Next, let $J=\left\{\left[\begin{array}{cc}0& b\\ 0& 0\end{array}\right]|b\in \mathbb{Z}\right\}$ be an ideal of T. Denote $A_i=\left[\begin{array}{cc}{a}_i& b_i\\ 0& 0\end{array}\right]\in T$ where $a_i,b_i\in \mathbb{Z}$, $1\le i\le n$, and let us define $\mathfrak{D}:T^n\to T$ as $\mathfrak{D}(A_1,A_2,\dots ,A_n)=\left[\begin{array}{cc}0& a_1{a}_2\dots a_n\\ 0& 0\end{array}\right]$ with the trace $\mathcal{f}:T\to T$ defined by $\mathcal{f}\left(\left[\begin{array}{cc}a& b\\ 0& 0\end{array}\right]\right)=\left[\begin{array}{cc}0& a^n\\ 0& 0\end{array}\right]$. One can easily check that $\mathfrak{D}$ is a symmetric reverse n-derivation such that all the conditions in Theorem 5 are satisfied. However, J is a non-central ideal. Hence, in Theorem 5, the hypotheses of semiprimeness cannot be omitted.

4. Conclusions

This work embarks on a thorough investigation of a novel class of maps known as symmetric reverse n-derivations, specifically studied on ideals of semiprime rings. The primary objective is to introduced these notions and to analyze the behavior of these maps. Throughout this comprehensive study, we uncovered various relationships between symmetric reverse n-derivations and their traces, particularly when these traces satisfied specific identities. These findings offer valuable insights into this area of research. By delving into the properties of symmetric reverse n-derivations, we contribute to a deeper understanding of how such maps interact with the underlying algebraic structures. Our results provide a foundation for further future exploration into the behavior of these maps in different contexts. The intricate connections revealed in this work not only advance the study of n-derivations, but also open up new avenues for researchers in this domain.

As we push the boundaries of this area, our findings present new perspectives on algebraic structures, suggesting a broader application of these maps. This study also raises important questions for future research, particularly in the interaction between symmetric maps and rings. Ultimately, this exploration expands the scope of modern ring theory and paves the way for continued advances in understanding the behavior of these new algebraic tools.