## Illinois Journal of Mathematics

### 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.

#### Article information

Source
Illinois J. Math., Volume 56, Number 3 (2012), 705-729.

Dates
First available in Project Euclid: 31 January 2014

https://projecteuclid.org/euclid.ijm/1391178545

Digital Object Identifier
doi:10.1215/ijm/1391178545

Mathematical Reviews number (MathSciNet)
MR3161348

Zentralblatt MATH identifier
1288.20072

#### Citation

Keef, Patrick W.; Danchev, Peter V. Generalizations of primary Abelian $C_\alpha$ groups. Illinois J. Math. 56 (2012), no. 3, 705--729. doi:10.1215/ijm/1391178545. https://projecteuclid.org/euclid.ijm/1391178545

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