Basic subgroups in Abelian group rings.

*(English)*Zbl 1003.16026Summary: 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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\textit{P. V. Danchev}, Czech. Math. J. 52, No. 1, 129--140 (2002; Zbl 1003.16026)

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##### References:

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