On $*$-clean group rings 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 $\mathbb F_qG$.