Commutative group algebras of $\sigma$-summable abelian groups

In this note we study the commutative modular and semisimple group rings of $\sigma$-summable abelian $p$-groups, which group class was introduced by R. Linton and Ch. Megibben. It is proved that $S(RG)$ is $\sigma$-summable if and only if $G_p$ is $\sigma$-summable, provided $G$ is an abelian group and $R$ is a commutative ring with 1 of prime characteristic $p$, having a trivial nilradical. If $G_p$ is a $\sigma$-summable $p$-group and the group algebras $RG$ and $RH$ over a field $R$ of characteristic $p$ are $R$-isomorphic, then $H_p$ is a $\sigma$-summable $p$-group, too. In particular $G_p\cong H_p$ provided $G_p$ is totally projective of a countable length.

Moreover, when $K$ is a first kind field with respect to $p$ and $G$ is $p$-torsion, $S(KG)$ is $\sigma$-summable if and only if $G$ is a direct sum of cyclic groups.