In the present paper, we denote T by an associative ring and $\mathcal{Z}\left(T\right)$ as its center. This paper focusses 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 mapping $\alpha :T\to T$ is called a derivation if $\alpha (\vartheta \ell )=\alpha \left(\vartheta \right)\ell +\vartheta \alpha (\ell )$ holds $forall$$\vartheta ,\ell \in T$. The idea derivation has been translated in many directions and one among those is it define a derivation on Cartesian product of rings. In this direction, Maksa [12] defined what is called a bi-derivation. Yes, the concept is expanded from bi to tri and then to n-derivations. The definition comes very naturally. Let’s give the formal definition of these maps. As derivation is additive, while translating this idea to Cartesian product the additivity part refers to the additivity in both the 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., and for all $\vartheta ,{\vartheta }^{\prime },\ell ,{\ell }^{\prime }\in T$. These two conditions can be clubbed in 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 [20,21]. Thus, a bi-additive, symmetric $\mathfrak{D}$ which is a derivation in any of its two components is termed as a symmetric bi-derivation. Several authors have studied symmetric bi-derivations on rings (see [3,11,19] and references therein) and produced highly useful outcomes. On similar lines $\ddot{\mathrm{O}}$zt$\ddot{\mathrm{u}}$rk [13] initiated the study of tri-derivations. Taking this forward Park [14] introduced the notion of permuting n-derivations. The definition goes like:

In 1957, Herstein [10] 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 case of prime rings. Later, Brešar and Vukman [8] studied reverse derivations in rings with involution. In this vein, Barros et al. [7] examined the additivity of 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 [1] explored the relationship between generalized reverse derivations and generalized derivations on an ideal in semiprime rings. More recently, Sögütcü [17] 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 as follows:

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 proved for derivations or bi-derivations to that of n-derivations. Talking about the trace of a reverse n-derivation, we now formally define it as follows:

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

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

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

Park in [14] proved various interesting results for symmetric n- derivations like generalization to the famous Posner’s result (cf.; [15]) 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 [5] introduced certain interesting identities that help us reveal the structure of any ring, similar to the work of Daif and Bell on semiprime rings [9]. 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 case of n-derivations Ashraf et al. in [4] developed upon the line of inquiry provided by Park in the case of prime rings. Ashraf et al. [6] used properties of an ideal and the trace of symmetric n-derivations to achieve results for the existence of central ideals which eventually helps to see the commutativity of the rings under consideration. Recently, Ali et al. in [2] 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 symmetric n-derivations D and G, respectively. In addition to proving results about the traces of permuting n-derivations, they studied the relation between n-derivations and n-multipliers and provided a characterization of symmetric n-derivations of prime rings in terms of left n-multipliers. A similar study of characterization of symmetric reverse n-derivations in terms of n-multipliers seems an interesting concept to venture out. Motivated by the existing work on symmetric n-derivations, we wish to explore rings specifically semiprime rings so that 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 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 few existing findings on symmetric n-derivations to that of symmetric reverse n-derivations. The first important result of this paper is the following: