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.

Let $\mathcal{A}$ be a Banach algebra and $n>1$, a fixed integer. The main objective of this paper is to talk about the commutativity of Banach algebras via its projections. Precisely, we prove that if $\mathcal{A}$ is a prime Banach algebra admitting a continuous projection $\mathcal{P}$ with image in $\mathcal{Z}(\mathcal{A})$ such that $\mathcal{P}(a^n)=a^n\text{for all}a\in \mathcal{G}$, the nonvoid open subset of $\mathcal{A}$, then $\mathcal{A}$ is commutative and $\mathcal{P}$ is the identity mapping on $\mathcal{A}$. Apart from proving some other results, as an application we prove that, a normed algebra is commutative iff the interior of its center is non-empty. Furthermore, we provide some examples to show that the assumed restrictions cannot be relaxed. Finally, we conclude our paper with a direction for further research.

Let $n\ge 1$ be a fixed integer. Within this study, we present a novel approach for discovering reversible codes over rings, leveraging the concept of $r$-glifted polynomials. This technique allows us to achieve optimal reversible codes. As we extend our methodology to the domain of DNA codes, we establish a correspondence between $4t$-bases of DNA and elements within the ring $R_{2t}=F_{4^{2t}}[u]/(u^2-1)$. By employing a variant of $r$-glifted polynomials, we successfully address the challenges of reversibility and complementarity in DNA codes over this specific ring. Moreover, we are able to generate reversible and reversible-complement DNA codes that transcend the limitations of being linear cyclic codes generated by a factor of $x^n-1$.