Basic subgroups in abelian group rings

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.