Basic subgroups in abelian group rings

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.