Generalizations of primary Abelian $C_\alpha$ groups

Abstract

A valuated $p^{n}$-socle is $C_\alpha$ $n$-summable if for every ordinal $\beta <\alpha $, it has a $\beta $-high subgroup that is $n$-summable (i.e., a valuated direct sum of countable valuated groups). This generalizes both the classical concepts of a $C_\alpha$ group due to Megibben and of an $n$-summable valuated $p^n$-socle developed by the authors. The notion is first analyzed in the category of valuated $p^{n}$-socles and then applied to the category of Abelian $p$-groups. In particular, results of Nunke on the torsion product and results of Keef on the balanced projective dimension of $C_{\omega_{1}}$ groups are recast into statements involving valuated $p^{n}$-socles and their related groups.