Let $R$ be an associative ring with identity and let $J(R)$ denote the Jacobson radical of $R$. $R$ is said to be semilocal if $R/J(R)$ is Artinian. In this paper we give necessary and sufficient conditions for the group ring $RG$, where $G$ is an abelian group, to be semilocal.
The structure of the unit group of the group algebra of the group $A_4$ over any finite field of characteristic 2 is established in terms of split extensions of cyclic groups.
Let $FG$ be a group algebra of a group $G$ over a field $F$ and ${\mathcal U}(FG)$ the unit group of $FG$. It is a classical question to determine the structure of the unit group of the group algebra of a finite group over a finite field. In this article, the structure of the unit group of the group algebra of the non-abelian group $G$ with order $21$ over any finite field of characteristic $3$ is established. We also characterize the structure of the unit group of $FA_4$ over any finite field of characteristic $3$ and the structure of the unit group of $FQ_{12}$ over any finite field of characteristic $2$, where $Q_{12}=\langle x, y; x^6=1, y^2=x^3, x^y=x^{-1} \rangle $.