The Direct Factor Problem for Modular Group Algebras of Isolated Direct Sums of Torsion-Complete Abelian Groups

Abstract

Let R be an arbitrary commutative unitary ring of prime characteristic p and G an arbitrary abelian group whose p-component G _p is an isolated direct sum of torsion-complete abelian groups. Then G _p is a direct factor of S(RG). As a consequence, the same holds when G is a direct sum of groups for which their p-components are torsion-complete groups. In particular when G is p-mixed, it is a direct factor of V(RG) provided R is a field. The formulated results extend a classical theorem of May (Contemp. Math., 1989) for direct sums of cyclic groups and its generalization due to the author (Proc. Amer. Math. Soc., 1997).