GENERALIZATIONS OF PRIMARY ABELIAN C_α GROUPS

摘要

To the memory of Charles K. Megibben (October 22, 1936-March 2, 2010) who defined and investigated the concepts upon which it is based A valuated p~n-socle is C_a n-summable if for every ordinal β ＜ a, it has a β-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_a 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_(W1) groups are recast into statements involving valuated p~n-socles and their related groups.