Commutative Group Algebras of Σ-summable Abelian Groups

In this note we study the commutative modular and semisimple group rings of σ-summable abelian p-groups, which group class was introduced by R. Linton and Ch. Megibben. It is proved that S(RG) is σ-summable if and only if Gp is σ-summable, provided G is an abelian group and R is a commutative ring with 1 of prime characteristic p, having a trivial nilradical. If Gp is a σ-summable p-group and the group algebras RG and RH over a field R of characteristic p are R-isomorphic, then Hp is a σ-summable p-group, too. In particular Gp ∼= Hp provided Gp is totally projective of a countable length. Moreover, when K is a first kind field with respect to p and G is p-torsion, S(KG) is σ-summable if and only if G is a direct sum of cyclic groups. Let p be a fixed prime. Throughout this paper G is an abelian group, G ω = ⋂ n<ω G p is its first p-Ulm subgroup (a first Ulm subgroup with respect to p) and G[p] is the socle of the p-primary component Gp = ⋃ n<ω G[p ]. In this note we investigate the commutative group algebras of σ-summable abelian p-groups. We obtain some characterizations of σ-summability for normed p-torsion components of abelian group rings. For this we need certain definitions (cf. [5]) and observations. If λ is a limit ordinal, we call a p-group G a Cλ-group provided G/G p is totally projective for all α < λ. Thus every abelian p-group is a Cω-group. Recall that the length of a reduced p-group G is just the smallest ordinal λ such that G λ = 1. If λ = ω, we call such an abelian p-groupG separable, i.e. the p-primaryG is said to be separable if G ω = 1. We shall say that a reduced abelian p-group G is σ-summable if its socle G[p] is the ascending union of a sequence of subgroups {Sn}n<ω, where for each n there is an ordinal αn less than the length of G such that Sn∩Gpαn = 1. Thus the reduced abelian p-group G is σ-summable (by Linton-Megibben [5]) if and only if G[p] = ⋃ n<ω Sn, where Sn ⊆ Sn+1 and Sn ∩ G αn = 1 for every natural n and some ordinal αn < lengthG. This definition for σ-summability, as a generalization of the classical Kulikov criterion for direct sums of cyclic p-groups, possesses the following properties: Some subgroups and direct sums of σ-summable groups are σ-summable; the separable abelian p-group G is σ-summable if and only Received by the editors March 3, 1995 and, in revised form, April 12, 1996. 1991 Mathematics Subject Classification. Primary 20C07; Secondary 20K10, 20K21.