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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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