Basic subgroups in commutative modular group rings.

*(English)*Zbl 1057.16028Summary: Let \(S(RG)\) be a normed Sylow \(p\)-subgroup in a group ring \(RG\) of an Abelian group \(G\) with \(p\)-component \(G_p\) and a \(p\)-basic subgroup \(B\) over a commutative unitary ring \(R\) with prime characteristic \(p\). The first central result is that \(1+I(RG;B_p)+I(R(p^i)G;G)\) is basic in \(S(RG)\) and \(B[1+I(RG;B_p)+I(R(p^i)G;G)]\) is \(p\)-basic in \(V(RG)\), and \([1+I(RG;B_p)+I(R(p^i)G;G)]G_p/G_p\) is basic in \(S(RG)/G_p\) and \([1+I(RG;B_p)+I(R(p^i)G;G)]G/G\) is \(p\)-basic in \(V(RG)/G\), provided in both cases \(G/G_p\) is \(p\)-divisible and \(R\) is such that its maximal perfect subring \(R^{p^i}\) has no nilpotents whenever \(i\) is natural. The second major result is that \(B(1+I(RG;B_p))\) is \(p\)-basic in \(V(RG)\) and \((1+I(RG;B_p))G/G\) is \(p\)-basic in \(V(RG)/G\), provided \(G/G_p\) is \(p\)-divisible and \(R\) is perfect.

In particular, under these circumstances, \(S(RG)\) and \(S(RG)/G_p\) are both starred or algebraically compact groups. The last results offer a new perspective on the long-standing classical conjecture which says that \(S(RG)/G_p\) is totally projective.

The present facts improve results concerning this topic due to N. A. Nachev [Houston J. Math. 22, No. 2, 225-232 (1996; Zbl 0859.16025)] and others obtained by us [in C. R. Acad. Bulg. Sci. 48, No. 9-10, 7-10 (1995; Zbl 0853.16040) and Czech. Math. J. 52, No. 1, 129-140 (2002; Zbl 1003.16026)].

In particular, under these circumstances, \(S(RG)\) and \(S(RG)/G_p\) are both starred or algebraically compact groups. The last results offer a new perspective on the long-standing classical conjecture which says that \(S(RG)/G_p\) is totally projective.

The present facts improve results concerning this topic due to N. A. Nachev [Houston J. Math. 22, No. 2, 225-232 (1996; Zbl 0859.16025)] and others obtained by us [in C. R. Acad. Bulg. Sci. 48, No. 9-10, 7-10 (1995; Zbl 0853.16040) and Czech. Math. J. 52, No. 1, 129-140 (2002; Zbl 1003.16026)].

##### 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 |

20K20 | Torsion-free groups, infinite rank |

20K21 | Mixed groups |