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Keywords:
basic and lower basic subgroups; units; modular abelian group rings
basic and lower basic subgroups; units; modular abelian group rings
Summary:
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.
References:
[1] M. F. Atiyah, I. G. MacDonald: Introduction to Commutative Algebra. Mir, Moscow, 1972. (Russian) MR 0349645
[2] P. V. Danchev: Normed unit groups and direct factor problem for commutative modular group algebras. Math. Balkanica 10 (1996), 161–173. MR 1606535 | Zbl 0980.16500
[3] P. V. Danchev: Topologically pure and basis subgroups in commutative group rings. Compt. Rend. Acad. Bulg. Sci. 48 (1995), 7–10. MR 1405499 | Zbl 0853.16040
[4] P. V. Danchev: Subgroups of the basis subgroup in a modular group ring. (to appear). MR 2181782
[5] P. V. Danchev: Isomorphism of commutative modular group algebras. Serdica Math. J. 23 (1997), 211–224. MR 1661072 | Zbl 0977.20003
[6] L. Fuchs: Infinite Abelian Groups I. Mir, Moscow, 1974. (Russian) MR 0346073
[7] P. Hill: Concerning the number of basic subgroups. Acta Math. Hungar. 17 (1966), 267–269. DOI 10.1007/BF01894873 | MR 0201507 | Zbl 0139.25301
[8] P. Hill: Units of commutative modular group algebras. J. Pure and Appl. Algebra 94 (1994), 175–181. DOI 10.1016/0022-4049(94)90031-0 | MR 1282838 | Zbl 0806.16033
[9] G. Karpilovsky: Unit Groups of Group Rings. North-Holland, Amsterdam, 1989. MR 1042757 | Zbl 0687.16010
[10] S. A. Khabbaz, E. A. Walker: The number of basic subgroups of primary groups. Acta Math. Hungar. 15 (1964), 153–155. DOI 10.1007/BF01897031 | MR 0162849
[11] N. A. Nachev: Basic subgroups of the group of normalized units in modular group rings. Houston J. Math. 22 (1996), 225–232. MR 1402745
[12] N. A. Nachev: 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. MR 1319683
[13] N. A. Nachev: Invariants of the Sylow $p$-subgroup of the unit group of a commutative group ring of characteristic $p$. Commun. in Algebra 23 (1995), 2469–2489. DOI 10.1080/00927879508825355 | MR 1330795 | Zbl 0828.16037
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