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.