On (n,σ)−(n,\sigma)-equivalence relation between skew constacyclic codes

In this paper we generalize the notion of $n$-equivalence relation introduced by Chen et al. in \cite{Chen2014} to classify constacyclic codes of length $n$ over a finite field $\mathbb{F}_q$, where $q=p^r$ is a prime power, to the case of skew constacyclic codes without derivation. We call this relation $(n,\sigma)$-equivalence relation, where $n$ is the length of the code and $\sigma$ is an automorphism of the finite field. We compute the number of $(n,\sigma)$-equivalence classes, and we give conditions on $\lambda$ and $\mu$ for which $(\sigma, \lambda)$-constacyclic codes and $(\sigma, \lambda)$-constacyclic codes are equivalent with respect to our $(n,\sigma)$-equivalence relation. Under some conditions on $n$ and $q$ we prove that skew constacyclic codes are equivalent to cyclic codes. We also prove that when $q$ is even and $\sigma$ is the Frobenius autmorphism, skew constacyclic codes of length $n$ are equivalent to cyclic codes when $\gcd(n,r)=1$. Finally we give some examples as applications of the theory developed here.Comment: 16 page

On some extensions of strongly unit nil-clean rings

An element $x \in R$ is considered (strongly) nil-clean if it can be expressed as the sum of an idempotent $e \in R$ and a nilpotent $b \in R$ (where $eb = be$). If for any $x \in R$, there exists a unit $u \in R$ such that $ux$ is (strongly) nil-clean, then $R$ is called a (strongly) unit nil-clean ring. It is worth noting that any unit-regular ring is strongly unit nil-clean. In this note, we provide a characterization of the unit regularity of a group ring, along with an additional condition. We also fully characterize the unit-regularity of the group ring $\mathbb{Z}_nG$ for every $n > 1$. Additionally, we discuss strongly unit nil-cleanness in the context of Morita contexts, matrix rings, and group rings.Comment: 12 Pages, to appear i

NonCommutative Rings and their Applications, IV ABSTRACTS Checkable Codes from Group Algebras to Group Rings

Abstract A code over a group ring is defined to be a submodule of that group ring. For a code C over a group ring RG, C is said to be checkable if there is v ∈ RG such that C = {x ∈ RG : xv = 0}. In [1], Jitman et al. introduced the notion of code-checkable group ring. We say that a group ring RG is code-checkable if every ideal in RG is a checkable code. In their paper, Jitman et al. gave a necessary and sufficient condition for the group ring FG, when F is a finite field and G is a finite abelian group, to be codecheckable. In this paper, we generalize this result for RG, when R is a finite commutative semisimple ring and G is any finite group. Our main result states that: Given a finite commutative semisimple ring R and a finite group G, the group ring RG is code-checkable if and only if G is π -by-cyclic π; where π is the set of noninvertible primes in R