MATHEMATICA BOHEMICA, Vol. 129, No. 1, pp. 79-90 (2004)

Basic subgroups in commutative modular group rings

Peter V. Danchev

Peter V. Danchev, Mathematical Department, Plovdiv University, 4000 Plovdiv, Bulgaria; Insurance Supervision Directorate, Ministry of Finance, 1000 Sofia, Bulgaria, e-mail: pvdanchev@yahoo.com

Abstract: 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. \endgraf 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. \endgraf 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).

Classification (MSC2000): 16S34, 16U60, 20K10, 20K20, 20K21

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