On $\ast$-clean group rings over finite fields

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 $\ast$-rings, called the $\ast$-cleanness. More precisely, a $\ast$-ring $R$ is called a $\ast$-clean ring if every element of $R$ is the sum of a unit and a projection ($\ast$-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 $\mathbb{F}G$ and characterize the $\ast$-cleanness of these group rings in each case. When $\ast$ is taken as the classical involution, we also characterize the $\ast$-cleanness of ${\mathbb{F}}_{q}G$ in terms of LCD abelian codes and self-orthogonal abelian codes in ${\mathbb{F}}_{q}G$.