Basic subgroups in abelian group rings

Danchev, Peter V.

Czechoslovak Mathematical Journal, Tome 52 (2002) no. 1, pp. 129-140

Résumé

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.

Détail

Classification : 16U60, 20C07, 20K10
Keywords: basic and lower basic subgroups; units; modular abelian group rings