Basic subgroups in commutative modular group rings

Danchev, Peter Vassilev. "Basic subgroups in commutative modular group rings." Mathematica Bohemica 129.1 (2004): 79-90. <http://eudml.org/doc/249401>.

@article{Danchev2004,
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. 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).},
author = {Danchev, Peter Vassilev},
journal = {Mathematica Bohemica},
keywords = {perfect rings; Abelian $p$-groups; groups of normalized units; group rings; basic subgroups; perfect rings; Abelian -groups; groups of normalized units; group rings; basic subgroups},
language = {eng},
number = {1},
pages = {79-90},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Basic subgroups in commutative modular group rings},
url = {http://eudml.org/doc/249401},
volume = {129},
year = {2004},
}

TY - JOUR
AU - Danchev, Peter Vassilev
TI - Basic subgroups in commutative modular group rings
JO - Mathematica Bohemica
PY - 2004
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 129
IS - 1
SP - 79
EP - 90
AB - 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).
LA - eng
KW - perfect rings; Abelian $p$-groups; groups of normalized units; group rings; basic subgroups; perfect rings; Abelian -groups; groups of normalized units; group rings; basic subgroups
UR - http://eudml.org/doc/249401
ER -