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$$C_\lambda$$-groups and $$\lambda$$-basic subgroups in modular group rings. (English) Zbl 0989.16019
For a fixed limit ordinal $$\lambda$$, denote by $$C_\lambda$$ the class of all Abelian $$p$$-groups $$G$$ such that $$G/G^{p^\alpha}$$ is totally projective for all $$\alpha<\lambda$$. C. Megibben [Tôhoku Math. J., II. Ser. 22, 347-356 (1970; Zbl 0222.20017)] introduced the concept of a $$\lambda$$-basic subgroup of a $$C_\lambda$$-group.
Let $$V(RG)$$ be the group of normalized units in a commutative group ring $$RG$$ of characteristic $$p$$. The author investigates the questions, when $$V(RG)$$ is a $$C_\lambda$$-group and a subgroup $$B$$ of $$V(RG)$$ is a $$\lambda$$-basic subgroup. In the paper answers to these questions are obtained for the case when $$G$$ is an Abelian $$p$$-group, $$R$$ is a perfect ring and $$\lambda$$ is a countable limit ordinal. If $$V(RG)$$ is a $$C_\lambda$$-group, then $$G$$ is a direct factor of $$V(RG)$$ with a totally projective complement.

MSC:
 16U60 Units, groups of units (associative rings and algebras) 20C07 Group rings of infinite groups and their modules (group-theoretic aspects) 16S34 Group rings 20K10 Torsion groups, primary groups and generalized primary groups 20K27 Subgroups of abelian groups 20K21 Mixed groups
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