$C^\lambda$-groups and $\lambda$-basic subgroups in modular group rigns

Abstract

This paper is concerned with the investigation of two closely related questions. The first question is: What is the $\lambda$-basic subgroup of the group of normed units $V(RG)$ in an abelian group ring $RG$ with identity of prime characteristic $p$ if $B$ is a $\lambda$-basic subgroup of the $p$-primary abelian group $G$ ? In this way it is shown that if $B$ is a $\lambda$-basic subgroup of the $p$–torsion $G$ and $R$ is perfect, then $1+I(RG;B)$ is a $\lambda$-basic subgroup of $V(RG)$, where $\lambda$ is a countable limit ordinal. Moreover, B is a direct factor of $1+I(RG;B)$ provided that it is $\lambda$-basic. This generalizes results due to Nachev (1996) and to the author (1995). The second question is the following: What is the criterion illustrated $V(RG)$ to be a $C_{\lambda}$-group when $G$ is an abelian p-group and $R$ is an unitary commutative ring with prime characteristic p ? In this direction it is proved that $V(RG)$ is a p-primary $C_{\lambda}$-group if and only if $G$ is a p-primary $C_{\lambda} group, provided R is perfect and $\lambda\leq\Omega$. Besides, if $R$ is perfect and $G$ is a p-group which is a $C_{\lambda}$-group, then the same holds for $V(RG)/G$, provided $\lambda\leq\Omega$. Moreover, if $G$ is a p-torsion $C_{\lambda}$-group of countable length $\lambda$ and $R$ is perfect without nilpotents, then $V(RG)/G$ is totally projective and so $G$ is a direct factor of $V(RG)$. The last extends in some aspect a result of May (1979, 1988).