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Basic subgroups in Abelian group rings. (English) Zbl 1003.16026
Summary: Suppose $$R$$ is a commutative ring with identity of prime characteristic $$p$$ and $$G$$ is an arbitrary Abelian $$p$$-group. In the present paper, a basic subgroup and a lower basic subgroup of the $$p$$-component $$U_p(RG)$$ and of the factor-group $$U_p(RG)/G$$ of the unit group $$U(RG)$$ in the modular group algebra $$RG$$ are established, in the case when $$R$$ is weakly perfect. Moreover, a lower basic subgroup and a basic subgroup of the normed $$p$$-component $$S(RG)$$ and of the quotient group $$S(RG)/G_p$$ are given when $$R$$ is perfect and $$G$$ is arbitrary with $$G/G_p$$ $$p$$-divisible. These results extend and generalize a result due to N. A. Nachev [Houston J. Math. 22, No. 2, 225-232 (1996; Zbl 0859.16025)], when the ring $$R$$ is perfect and $$G$$ is $$p$$-primary. Some other applications in this direction are also obtained for the direct factor problem and for a kind of an arbitrary basic subgroup.

##### MSC:
 16U60 Units, groups of units (associative rings and algebras) 16S34 Group rings 20C07 Group rings of infinite groups and their modules (group-theoretic aspects) 20E07 Subgroup theorems; subgroup growth 20K10 Torsion groups, primary groups and generalized primary groups
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##### References:
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