This paper deals with some identities on Banach ${}^{\ast }$-algebras that are equipped with linear generalized derivations. As an application of one of our results, we describe the structure of the underlying algebras. Precisely, we prove that for a linear generalized derivation $F$ on a Banach ${}^{\ast }$-algebra $A$, either we obtain the existence of a central idempotent element $e\in Q$, for which $F=0$ on $eQ$ and $(1-e)Q$ satisfies $s_4$, or the set of elements $u\in A$ such that the identity $[F(u{)}^n,F(u^{\ast }{)}^{n}F(u{)}^n]\in Z(A)$ holds for no positive integer $n$ turns out to be dense. In addition to this we consider an identity satisfied by a semisimple Banach ${}^{\ast }$-algebra and look for its commutativity. Moreover, some related results are also established.

Let $s\ge 1$ be a fixed integer. In this paper, we focus on generating cyclic codes over the ring $\mathcal{R}({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_s)$, where ${\alpha }_i\in {\mathbb{F}}_q\mathrm{\setminus }\{0\}$, $1\le i\le s$, by using the Gray map that is defined by the idempotents. Moreover, we describe the process to generate an idempotent by using the formula (2.1). As applications, we obtain both optimal and new quantum codes. Additionally, we solve the DNA reversibility problem by introducing ${\mathbb{F}}_q$ reversibility. The aim to introduce the ${\mathbb{F}}_q$ reversibility is to describe IUPAC nucleotide codes, and consequently, 5 IUPAC DNA bases are considered instead of 4 DNA bases $(A,T,G,C)$.