The study of linear and cyclic codes over some types of finite rings has enabled the development of highly efficient codes with optimal parameters. Due to the algebraic properties of these rings, error-correction algorithms can be well adapted to diverse noisy channels, enhancing real-time performance. Furthermore, by using ideals in polynomial rings, cyclic codes ensure that hardware implementation remains remarkably straightforward and fast. For these reasons, in this paper, we describe linear and cyclic codes over the rings of the form $R_{s,p}= \mathbb{Z} _{p}[u]/\left( f\left( u\right) /\left( u-s\right) \right)$, where $p$ is a prime number and $f\left( u\right) = u^{p}-u$, with $s\in \{0,1,\dots,p-1\}$.
Rege and Chhawchharia, as well as Nielsen, introduced the concept of a right McCoy ring, based on McCoy's theorem in 1942 for the annihilators in polynomial rings over commutative rings. Alhevaz et al. in 2012 investigated a natural generalization of a right McCoy ring that is called a weak $(\sigma, \delta)$-skew McCoy ring, where $\sigma$ is an endomorphism and $\delta$ is a $\sigma$-derivation of a given ring. In this paper, we continue to investigate the structural properties of these rings and examine their behavior under various extensions. Specifically, we prove that while the full matrix ring over a weak $(\sigma, \delta)$-skew McCoy ring does not generally preserve this property, the upper triangular matrix ring does under suitable conditions. Furthermore, we explore the stability of this property in polynomial extensions, direct products, classical right quotient rings, and skew power series rings.
We introduce a new class of rings close to von Neumann regular rings and V-rings. We call a ring $R$, a right atomic V-ring (or right AV-ring) if every atomic (in the sense of Zhou and Duns) right $R$-module is injective. We observe that a right semi-Artinian ring is a right AV-ring if and only if it is a right $\Sigma$-V-ring. Among other results, we show that every finite direct product of full matrix rings over Abelian regular rings is both a right and a left AV-ring. We also detect that there exist regular AV-rings (and hence $\Sigma$-V-rings) of infinite index (of nilpotency).
In this paper we show that an integral domain $D$ is a Krull domain if and only if $D$ is a Mori domain, $D$ satisfies Serre's condition $(R_1)$ and every associated prime of a principal ideal has height one.
The essential graph of an abelian group $G$, denoted by $\varepsilon(G)$, is an undirected graph with vertex set as the collection of all non-trivial proper subgroups of $G$ and any two distinct vertices $A$ and $B$ are adjacent if and only if $AB$ is an essential subgroup of $G$. We discuss connectedness, traversability and planarity of $\varepsilon(G)$. The concept of a divisor subgroup with respect to a subgroup of $G$ is introduced, whose effect is observed in some of the characterizations of $\varepsilon(G)$.
