Geodesic
Basic subgroups in abelian group rings
Czechoslovak Mathematical Journal, Tome 52 (2002) no. 1, pp. 129-140 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
Classification : 16U60, 20C07, 20K10
Keywords: basic and lower basic subgroups; units; modular abelian group rings 
@article{CMJ_2002_52_1_a11,
     author = {Danchev, Peter V.},
     title = {Basic subgroups in abelian group rings},
     journal = {Czechoslovak Mathematical Journal},
     pages = {129--140},
     year = {2002},
     volume = {52},
     number = {1},
     mrnumber = {1885462},
     zbl = {1003.16026},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2002_52_1_a11/}
}
TY  - JOUR
AU  - Danchev, Peter V.
TI  - Basic subgroups in abelian group rings
JO  - Czechoslovak Mathematical Journal
PY  - 2002
SP  - 129
EP  - 140
VL  - 52
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/CMJ_2002_52_1_a11/
LA  - en
ID  - CMJ_2002_52_1_a11
ER  - 
%0 Journal Article
%A Danchev, Peter V.
%T Basic subgroups in abelian group rings
%J Czechoslovak Mathematical Journal
%D 2002
%P 129-140
%V 52
%N 1
%U http://geodesic.mathdoc.fr/item/CMJ_2002_52_1_a11/
%G en
%F CMJ_2002_52_1_a11
Danchev, Peter V. Basic subgroups in abelian group rings. Czechoslovak Mathematical Journal, Tome 52 (2002) no. 1, pp. 129-140. http://geodesic.mathdoc.fr/item/CMJ_2002_52_1_a11/

[1] M. F. Atiyah, I. G.  MacDonald: Introduction to Commutative Algebra. Mir, Moscow, 1972. (Russian) | MR

[2] P. V.  Danchev: Normed unit groups and direct factor problem for commutative modular group algebras. Math. Balkanica 10 (1996), 161–173. | MR | Zbl

[3] P. V. Danchev: Topologically pure and basis subgroups in commutative group rings. Compt. Rend. Acad. Bulg. Sci. 48 (1995), 7–10. | MR | Zbl

[4] P. V. Danchev: Subgroups of the basis subgroup in a modular group ring. (to appear). | MR

[5] P. V. Danchev: Isomorphism of commutative modular group algebras. Serdica Math.  J. 23 (1997), 211–224. | MR | Zbl

[6] L.  Fuchs: Infinite Abelian Groups I. Mir, Moscow, 1974. (Russian) | MR

[7] P.  Hill: Concerning the number of basic subgroups. Acta Math. Hungar. 17 (1966), 267–269. | DOI | MR | Zbl

[8] P. Hill: Units of commutative modular group algebras. J.  Pure and Appl. Algebra 94 (1994), 175–181. | DOI | MR | Zbl

[9] G.  Karpilovsky: Unit Groups of Group Rings. North-Holland, Amsterdam, 1989. | MR | Zbl

[10] S. A. Khabbaz, E. A.  Walker: The number of basic subgroups of primary groups. Acta Math. Hungar. 15 (1964), 153–155. | DOI | MR

[11] N. A.  Nachev: Basic subgroups of the group of normalized units in modular group rings. Houston J.  Math. 22 (1996), 225–232. | MR

[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

[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 | MR | Zbl