Symmetric $n$-derivations on prime ideals with applications

Let $\mathfrak{S}$ be a ring. The main objective of this paper is to analyze the structure of quotient rings, which are represented as $\mathfrak{S}/\mathfrak{P}$, where $\mathfrak{S}$ is an arbitrary ring and $\mathfrak{P}$ is a prime ideal of $\mathfrak{S}$. The paper aims to establish a link between the structure of these rings and the behaviour of traces of symmetric $n$-derivations satisfying some algebraic identities involving prime ideals of an arbitrary ring $\mathfrak{S}$. Moreover, as an application of the main result, we investigate the structure of the quotient ring $\mathfrak{S}/\mathfrak{P}$ and traces of symmetric $n$-derivations.