Illinois Journal of Mathematics

Generalizations of primary Abelian $C_\alpha$ groups

Patrick W. Keef and Peter V. Danchev

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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

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1391178545

Digital Object Identifier
doi:10.1215/ijm/1391178545

Mathematical Reviews number (MathSciNet)
MR3161348

Zentralblatt MATH identifier
1288.20072

Subjects
Primary: 20K10: Torsion groups, primary groups and generalized primary groups

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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References

  • D. Cutler and R. Dimitric, Valuated vector spaces, Kurepa's hypothesis and abelian $p$-groups, Rocky Mountain J. Math. 23 (1993), no. 4, 1253–1266.
  • P. Danchev and P. Keef, $n$-summable valuated $p^n$-socles and primary abelian groups, Comm. Algebra 38 (2010), 3137–3153.
  • P. Danchev and P. Keef, Nice elongations of primary abelian groups, Publ. Mat. 54 (2010), no. 2, 317–339.
  • L. Fuchs, Infinite abelian groups, vol. I, Academic Press, New York, 1970.
  • L. Fuchs, Infinite abelian groups, vol. II, Academic Press, New York, 1973.
  • L. Fuchs, Vector spaces with valuations, J. Algebra 35 (1975), 23–38.
  • P. Griffith, Infinite abelian group theory, Univ. Chicago Press, Chicago and London, 1970.
  • P. Hill, Isotype subgroups of direct sums of countable abelian groups, Illinois J. Math. 13 (1969), 281–290.
  • P. Hill, The recovery of some abelian groups from their socles, Proc. Amer. Math. Soc. 86 (1982), no. 4, 553–560.
  • P. Hill, The balance of Tor, Math. Z. 182 (1983), no. 2, 179–188.
  • T. Jech, Set theory, 3rd Millennium ed., Springer, Berlin, 2002.
  • P. Keef, Realization theorems for $n$-summable valuated $p^n$-socles, Rend. Semin. Mat. Univ. Padova 126, 151–173.
  • P. Keef, On the Tor functor and some classes of abelian groups, Pacific J. Math. 132 (1988), no. 1, 63–84.
  • P. Keef, A theorem on the closure of $\Omega$-pure subgroups of $C_\Omega$ groups in the $\Omega$-topology, J. Algebra 125 (1989), no. 1, 150–163.
  • P. Keef, On set theory and the balanced-projective dimension of $C_\Omega$-groups, Contemporary Mathematics, vol. 87, Amer. Math. Soc., Providence, RI, 1989, pp. 31–42.
  • P. Keef, On iterated torsion products of abelian $p$-groups, Rocky Mountain J. Math. 21 (1991), no. 3, 1035–1055.
  • P. Keef, On $\o_1$-${p^{{\o}+n}}$-projective primary abelian groups, J. Algebra and Number Theory Academia 1 (2010), no. 1, 53–87.
  • P. Keef and P. Danchev, On $m,n$-balanced projective and $m,n$-totally projective primary abelian groups, J. Korean Math. Soc. 50 (2013), no. 2, 307–330.
  • P. Keef and P. Danchev, On $n$-simply presented primary abelian groups, Houston J. Math. 38 (2012), no. 4, 1027–1050.
  • C. Megibben, A generalization of the classical theory of primary groups, Tohoku Math. J. 22 (1970), 347–356.
  • R. Nunke, Purity and subfunctors of the identity, Topics in abelian groups, Scott, Foresman and Co., Chicago, IL, 1962, pp. 121–171.
  • R. Nunke, Homology and direct sums of countable abelian groups, Math. Z. 101 (1967), no. 3, 182–212.
  • R. Nunke, On the structure of Tor, II, Pacific J. Math. 22 (1967), 453–464.
  • F. Richman and E. Walker, Valuated groups, J. Algebra 56 (1979), no. 1, 145–167.