Communications of the
Korean Mathematical Society CKMS

Let $R$ be a commutative ring, $G$ be an Abelian group, and let $RG$ be the group ring. We say that $RG$ is a U-group ring if $a$ is a unit in $RG$ if and only if $\epsilon (a)$ is a unit in $R$. We show that $RG$ is a U-group ring if and only if $G$ is a p-group and $p\in J(R)$. We give some properties of U-group rings and investigate some properties of well known rings, such as Hermite rings and rings with stable range, in the presence of U-group rings.

Keywords: group ring, Hermite ring, rings with stable range, Jacobson radical, p-group

MSC numbers: 13A99, 16S34, 20C05, 20C07