Action of $n$-derivations and $n$-multipliers on ideals of (semi)-prime rings

The present paper aims to investigate the containment of nonzero central ideal in a ring $\mathcal{R}$ when the trace of symmetric $n$-derivations satisfies some differential identities. Lastly, we prove that in a prime ring $\mathcal{R}$ of suitable torsion restriction, if $\mathfrak{D},\mathcal{G}:{\mathcal{R}}^n\to \mathcal{R}$ are two nonzero symmetric $n$-derivations such that $\mathcal{f}(\vartheta )\vartheta +\vartheta \mathcal{g}(\vartheta )=0$ holds $\mathrm{\forall }\vartheta \in \mathcal{W}$, a nonzero left ideal of $\mathcal{R}$ where $\mathcal{f}$ and $\mathcal{g}$ are the traces of $\mathfrak{D}$ and $\mathcal{G}$, respectively, then either $\mathcal{R}$ is commutative or $\mathcal{G}$ acts as a left $n$-multiplier. Finally, we characterize symmetric $n$-derivations in terms of left $n$-multipliers.