At the heart of ring theory lies the concept of derivations, over the years, several researchers have extended the notion of derivations in various directions such as generalized derivations, $(\alpha ,\beta )$-derivations, bi-derivations, higher derivations, symmetric n-derivations, etc. and have studied the structure of rings as well as the structure of additive mappings (refer to [3,4,5,6,12,13,22,23]). In this research article, we present a comprehensive investigation of symmetric generalized n-derivations, seeking to establish a theoretical connection between symmetric generalized n-derivations and other fundamental algebraic concepts. “Throughout the discussion, we will consider $\mathfrak{S}$ to be an associative ring with $Z\left(\mathfrak{S}\right)$ being its center. A ring $\mathfrak{S}$ is said to be prime if, $\varrho \mathfrak{S}\xi =\left\{0\right\}$ implies that either $\varrho =0$ or $\xi =0$, and semiprime if, $\varrho \mathfrak{S}\varrho =\left\{0\right\}$ implies that $\varrho =0$, where $\varrho ,\xi \in \mathfrak{S}$. The symbols $[\varrho ,\xi ]$ and $\varrho \circ \xi$ denote the commutator, $\varrho \xi -\xi \varrho$ and the anti-commutator, $\varrho \xi +\xi \varrho$, respectively, for all $\varrho ,\xi \in \mathfrak{S}$. A ring $\mathfrak{S}$ is said to be n-torsion free if $n\varrho =0$ implies that $\varrho =0$ for all $\varrho \in \mathfrak{S}$. If $\mathfrak{S}$ is $n!$-torsion free, then it is d-torsion free for every divisor d of $n!$. Recall that an ideal $\mathfrak{P}$ of $\mathfrak{S}$ is said to be prime if, $\mathfrak{P}\ne \mathfrak{S}$ and for $\varrho ,\xi \in \mathfrak{S}$, $\varrho \mathfrak{S}\xi \subseteq \mathfrak{P}$ implies that $\varrho \in \mathfrak{P}$ or $\xi \in \mathfrak{P}$. An additive mapping $d:\mathfrak{S}\to \mathfrak{S}$ is called a derivation if $d\left(\varrho \xi \right)=d\left(\varrho \right)\xi +\varrho d\left(\xi \right)$ holds for all $\varrho ,\xi \in \mathfrak{S}$. Following [14], an additive mapping $g:\mathfrak{S}⟶\mathfrak{S}$ is said to be a generalized derivation on $\mathfrak{S}$ if there exists a derivation $d:\mathfrak{S}⟶\mathfrak{S}$ such that $g\left(\varrho \xi \right)=g\left(\varrho \right)\xi +\varrho d\left(\xi \right)$ holds for all $\varrho ,\xi \in \mathfrak{S}$. A bi-additive map $\mathfrak{D}:\mathfrak{S}\times \mathfrak{S}\to \mathfrak{S}$ is said to be symmetric if $\mathfrak{D}(\varrho ,\xi )=\mathfrak{D}(\xi ,\varrho )$ for all $\varrho ,\xi \in \mathfrak{S}$. A symmetric bi-additive map is said to be symmetric bi-derivation if $\mathfrak{D}(\varrho \xi ,z)=\varrho \mathfrak{D}(\xi ,z)+\mathfrak{D}(\varrho ,z)\xi$ for all $\varrho ,\xi ,z\in \mathfrak{S}$. The concept of symmetric bi-derivation in rings was introduced by G. Maksa [16]. Suppose n is a fixed positive integer and ${\mathfrak{S}}^n=\mathfrak{S}\times \mathfrak{S}\times \cdots \times \mathfrak{S}$. A map $\mathfrak{D}:{\mathfrak{S}}^n\to \mathfrak{S}$ is said to be symmetric(permuting) if the relation $\mathfrak{D}\left({\varrho }_1,{\varrho }_2,\dots ,{\varrho }_n\right)=\mathfrak{D}\left({\varrho }_{\pi \left(1\right)},{\varrho }_{\pi \left(2\right)},\dots ,{\varrho }_{\pi \left(n\right)}\right)$ holds for all ${\varrho }_i\in \mathfrak{S}$ and for every permutation $\left\{\pi \right(1),\pi (2),\dots ,\pi (n\left)\right\}$. The concept of derivation and symmetric bi-derivation was generalized by Park [18] as follows: a permuting map $\mathfrak{D}:{\mathfrak{S}}^n\to \mathfrak{S}$ is said to be a permuting n-derivation if $\mathfrak{D}$ is n-additive (i.e.; additive in each coordinate) and $\mathfrak{D}\left({\varrho }_1,{\varrho }_2,\cdots ,{\varrho }_i{\varrho }_i^{\prime },\cdots ,{\varrho }_n\right)={\varrho }_i\mathfrak{D}\left({\varrho }_1,{\varrho }_2,\cdots ,{\varrho }_i^{\prime },\cdots ,{\varrho }_n\right)+$$\mathfrak{D}\left({\varrho }_1,{\varrho }_2,\cdots ,{\varrho }_i,\cdots ,{\varrho }_n\right){\varrho }_i^{\prime }$ holds for all ${\varrho }_i,{\varrho }_i^{\prime }\in \mathfrak{S}$. A 1-derivation is a derivation and a 2-derivation is a symmetric bi-derivation while a 3-derivation is known as permuting tri-derivation (viz., [2,7,12,17,23,24,25]). Let $n\ge 2$ be a fixed integer and a map $\delta :\mathfrak{S}\to \mathfrak{S}$ defined by $\delta \left(\varrho \right)=\mathfrak{D}(\varrho ,\varrho ,\dots ,\varrho )$ for all $\varrho \in \mathfrak{S}$, where $\mathfrak{D}:{\mathfrak{S}}^n\to \mathfrak{S}$ is a permuting map, be the trace of $\mathfrak{D}$. If $\mathfrak{D}$ is symmetric and n-additive, then the trace d of $\mathfrak{D}$ satisfies the relation $d(\varrho +\xi )=d\left(\varrho \right)+d\left(\xi \right)+{\sum }_{t=1}^{n-1}{}^n{C}_t\mathfrak{D}(\underset{\underset{(n-t)-\mathrm{times}}{︸}}{\varrho ,\dots ,\varrho },\underset{\underset{t-\mathrm{times}}{︸}}{\xi ,\dots ,\xi })$ for all $\varrho ,\xi \in \mathfrak{S}$."

Motivated by the concept of generalized derivation in ring, Ashraf et al. [12] introduced the notion of permuting generalized n-derivation in ring. Let $n\ge 1$ be a fixed positive integer. A permuting n-additive map $\mathcal{G}:{\mathfrak{S}}^n\to \mathfrak{S}$ is known to be permuting generalized n-derivation if there exists a permuting n-derivation $\mathfrak{D}:{\mathfrak{S}}^n\to \mathfrak{S}$ such that $\mathcal{G}\left({\varrho }_1,{\varrho }_2,\dots ,{\varrho }_i{\varrho }_i^{\prime },\dots ,{\varrho }_n\right)=\mathcal{G}\left({\varrho }_1,{\varrho }_2,\dots ,{\varrho }_i,\dots ,{\varrho }_n\right){\varrho }_i^{\prime }+{\varrho }_i\mathfrak{D}({\varrho }_1,{\varrho }_2,\dots ,{\varrho }_i^{\prime },\dots ,{\varrho }_n)$ holds for all ${\varrho }_i,{\varrho }_i^{\prime }\in \mathfrak{S}$. In fact, in [12], the authors proved that “for a fixed positive integer $n\ge 2$, let $\mathfrak{S}$ be a $n!$-torsion free semiprime ring admitting a permuting generalized n-derivation $\Omega$ with associated n-derivation $\mathfrak{D}$ such that the trace $\omega$ of $\Omega$ is centralizing on $\mathfrak{S}$. Then $\omega$ is commuting on $\mathfrak{S}$". Also, in [8], Ashraf et al. have characterized the traces of permuting generalized n-derivations. In fact, their result was motivated by the result due to Hvala [15]. Basically, they proved that “for a fixed positive integer $n\ge 2$, let $\mathfrak{S}$ be a $n!$-torsion free prime ring. Suppose that ${\omega }_1$ and ${\omega }_2$ are the traces of permuting generalized n-derivations ${\Omega }_1$, ${\Omega }_2$ respectively and ${\delta }_1\ne 0$; ${\delta }_2$ are the traces of associated derivations ${\mathfrak{D}}_1$ and ${\mathfrak{D}}_2$ respectively. If ${\omega }_1\left(\varrho \right){\omega }_2\left(\xi \right)={\omega }_2\left(\varrho \right){\omega }_1\left(\xi \right)$ holds for all $\varrho ,\xi \in \mathfrak{S}$, then there exists $\gamma \in C$, the extended centroid of $\mathfrak{S}$ such that ${\delta }_2\left(\varrho \right)=\gamma {\delta }_1\left(\varrho \right)$."

Many researchers have extensively examined a wide range of identities involving traces of n-derivations, leading to the discovery of various interesting results (see, for example [1,2,8,10,12,19] and the associated references). Very recently, Ali et al. [2], explored some algebraic identities associated with the trace of symmetric n-derivations acting on prime ideal $\mathfrak{P}$ of $\mathfrak{S}$, but without imposing the assumption of primeness on the ring under consideration. In fact, apart from proving some other interesting results, they extended the famous result [20] for the trace of symmetric n-derivations which involves prime ideals. Precisely, they proved that for any fixed integer $n\ge 2$, let $\mathfrak{S}$ be any ring and $\mathfrak{P}$ be a prime ideal of $\mathfrak{S}$ such that $\mathfrak{S}/\mathfrak{P}$ is $n!$-torsion free. If there exists a non-zero symmetric n-derivation $\mathfrak{D}$ with trace $\mathcal{d}$ on $\mathfrak{S}$ such that $\left[\mathcal{d}\right(\varrho ),\varrho ]\in \mathfrak{P}$, for all $\varrho \in \mathfrak{S}$, then either $\mathfrak{S}/\mathfrak{P}$ is a commutative integral domain or $\mathcal{d}\left(\mathfrak{S}\right)\subseteq \mathfrak{P}$.

The main purpose of our current research is to delve into the structure of the quotient ring $\mathfrak{S}/\mathfrak{P}$ where $\mathfrak{S}$ is any ring and $\mathfrak{P}$ is a prime ideal of $\mathfrak{S}$ which admits symmetric generalized n-derivations satisfying certain algebraic identities acting on prime ideals $\mathfrak{P}$. In particular, we prove that if a ring $\mathfrak{S}$ admits a symmetric generalized n-derivation $\mathcal{G}:{\mathfrak{S}}^n\to \mathfrak{S}$ with trace $\mathcal{g}:\mathfrak{S}\to \mathfrak{S}$ associated with symmetric n-derivation $\mathfrak{D}:{\mathfrak{S}}^n⟶\mathfrak{S}$ with trace $\mathcal{d}:\mathfrak{S}\to \mathfrak{S}$ satisfying any one of the following functional identities:

for all $\varrho ,\xi \in \mathfrak{S}$, then one of the following holds:

The following auxiliary results are essential for proving the above mentioned results:

Our first main result establishes a link between the derivation and symmetric generalized n-derivation. In simpler terms, we demonstrate the following result: