Article
MSC:
16S34,
16U60,
20C07,
20E07,
20K10,
20K20,
20K21 | MR 2048788 | Zbl 1057.16028 | DOI: 10.21136/MB.2004.134103
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Keywords:
perfect rings; Abelian $p$-groups; groups of normalized units; group rings; basic subgroups
perfect rings; Abelian $p$-groups; groups of normalized units; group rings; basic subgroups
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 the results concerning this topic due to Nachev (Houston J. Math., 1996) and others obtained by us in (C. R. Acad. Bulg. Sci., 1995) and (Czechoslovak Math. J., 2002).
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