Basic subgroups in abelian group rings

Peter Vassilev Danchev

Czechoslovak Mathematical Journal (2002)

  • Volume: 52, Issue: 1, page 129-140
  • ISSN: 0011-4642

Abstract

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Suppose R is a commutative ring with identity of prime characteristic p and G is an arbitrary abelian p -group. In the present paper, a basic subgroup and a lower basic subgroup of the p -component U p ( R G ) and of the factor-group U p ( R G ) / G of the unit group U ( R G ) in the modular group algebra R G are established, in the case when R is weakly perfect. Moreover, a lower basic subgroup and a basic subgroup of the normed p -component S ( R G ) and of the quotient group S ( R G ) / G p are given when R is perfect and G is arbitrary whose G / G p is p -divisible. These results extend and generalize a result due to Nachev (1996) published in Houston J. Math., when the ring R is perfect and G is p -primary. Some other applications in this direction are also obtained for the direct factor problem and for a kind of an arbitrary basic subgroup.

How to cite

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Danchev, Peter Vassilev. "Basic subgroups in abelian group rings." Czechoslovak Mathematical Journal 52.1 (2002): 129-140. <http://eudml.org/doc/30690>.

@article{Danchev2002,
abstract = {Suppose $R$ is a commutative ring with identity of prime characteristic $p$ and $G$ is an arbitrary abelian $p$-group. In the present paper, a basic subgroup and a lower basic subgroup of the $p$-component $U_p(RG)$ and of the factor-group $U_p(RG)/G$ of the unit group $U(RG)$ in the modular group algebra $RG$ are established, in the case when $R$ is weakly perfect. Moreover, a lower basic subgroup and a basic subgroup of the normed $p$-component $S(RG)$ and of the quotient group $S(RG)/G_p$ are given when $R$ is perfect and $G$ is arbitrary whose $G/G_p$ is $p$-divisible. These results extend and generalize a result due to Nachev (1996) published in Houston J. Math., when the ring $R$ is perfect and $G$ is $p$-primary. Some other applications in this direction are also obtained for the direct factor problem and for a kind of an arbitrary basic subgroup.},
author = {Danchev, Peter Vassilev},
journal = {Czechoslovak Mathematical Journal},
keywords = {basic and lower basic subgroups; units; modular abelian group rings; lower basic subgroups; groups of units; modular Abelian group rings},
language = {eng},
number = {1},
pages = {129-140},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Basic subgroups in abelian group rings},
url = {http://eudml.org/doc/30690},
volume = {52},
year = {2002},
}

TY - JOUR
AU - Danchev, Peter Vassilev
TI - Basic subgroups in abelian group rings
JO - Czechoslovak Mathematical Journal
PY - 2002
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 52
IS - 1
SP - 129
EP - 140
AB - Suppose $R$ is a commutative ring with identity of prime characteristic $p$ and $G$ is an arbitrary abelian $p$-group. In the present paper, a basic subgroup and a lower basic subgroup of the $p$-component $U_p(RG)$ and of the factor-group $U_p(RG)/G$ of the unit group $U(RG)$ in the modular group algebra $RG$ are established, in the case when $R$ is weakly perfect. Moreover, a lower basic subgroup and a basic subgroup of the normed $p$-component $S(RG)$ and of the quotient group $S(RG)/G_p$ are given when $R$ is perfect and $G$ is arbitrary whose $G/G_p$ is $p$-divisible. These results extend and generalize a result due to Nachev (1996) published in Houston J. Math., when the ring $R$ is perfect and $G$ is $p$-primary. Some other applications in this direction are also obtained for the direct factor problem and for a kind of an arbitrary basic subgroup.
LA - eng
KW - basic and lower basic subgroups; units; modular abelian group rings; lower basic subgroups; groups of units; modular Abelian group rings
UR - http://eudml.org/doc/30690
ER -

References

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  10. 10.1007/BF01897031, Acta Math. Hungar. 15 (1964), 153–155. (1964) MR0162849DOI10.1007/BF01897031
  11. Basic subgroups of the group of normalized units in modular group rings, Houston J.  Math. 22 (1996), 225–232. (1996) MR1402745
  12. Invariants of the Sylow p -subgroup of the unit group of commutative group ring of characteristic p , Compt. Rend. Acad. Bulg. Sci. 47 (1994), 9–12. (1994) MR1319683
  13. 10.1080/00927879508825355, Commun. in Algebra 23 (1995), 2469–2489. (1995) Zbl0828.16037MR1330795DOI10.1080/00927879508825355

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