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Basic subgroups in commutative modular group rings. (English) Zbl 1057.16028
Summary: 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)].

##### 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
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