On $*$-clean group rings over finite fields

TL;DR

This paper investigates the $*$-cleanness of group rings over finite fields with involutions, providing characterizations in various cases and linking to properties of LCD and self-orthogonal abelian codes.

Contribution

It introduces two classes of involutions on group rings and characterizes their $*$-cleanness, connecting algebraic properties to coding theory concepts.

Findings

Characterization of $*$-cleanness for group rings with specific involutions.

Connection between $*$-cleanness and properties of LCD and self-orthogonal codes.

Results applicable to finite abelian groups over finite fields.

Abstract

A ring $R$ is called clean if every element of $R$ is the sum of a unit and an idempotent. Motivated by a question proposed by Lam on the cleanness of von Neumann Algebras, Va\v{s} introduced a more natural concept of cleanness for $*$-rings, called the $*$-cleanness. More precisely, a $*$-ring $R$ is called a $*$-clean ring if every element of $R$ is the sum of a unit and a projection ($*$-invariant idempotent). Let $\mathbb F$ be a finite field and $G$ a finite abelian group. In this paper, we introduce two classes of involutions on group rings of the form $\mathbb FG$ and characterize the $*$-cleanness of these group rings in each case. When $*$ is taken as the classical involution, we also characterize the $*$-cleanness of $\mathbb F_qG$ in terms of LCD abelian codes and self-orthogonal abelian codes in $\mathbb F_qG$.