Pure subgroups

Ladislav Bican

Mathematica Bohemica (2001)

  • Volume: 126, Issue: 3, page 649-652
  • ISSN: 0862-7959

Abstract

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Let λ be an infinite cardinal. Set λ 0 = λ , define λ i + 1 = 2 λ i for every i = 0 , 1 , , take μ as the first cardinal with λ i < μ , i = 0 , 1 , and put κ = ( μ 0 ) + . If F is a torsion-free group of cardinality at least κ and K is its subgroup such that F / K is torsion and | F / K | λ , then K contains a non-zero subgroup pure in F . This generalizes the result from a previous paper dealing with F / K p -primary.

How to cite

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Bican, Ladislav. "Pure subgroups." Mathematica Bohemica 126.3 (2001): 649-652. <http://eudml.org/doc/248863>.

@article{Bican2001,
abstract = {Let $\lambda $ be an infinite cardinal. Set $\lambda _0=\lambda $, define $\lambda _\{i+1\}=2^\{\lambda _i\}$ for every $i=0,1,\dots $, take $\mu $ as the first cardinal with $\lambda _i<\mu $, $i=0,1,\dots $ and put $\kappa = (\mu ^\{\aleph _0\})^+$. If $F$ is a torsion-free group of cardinality at least $\kappa $ and $K$ is its subgroup such that $F/K$ is torsion and $|F/K|\le \lambda $, then $K$ contains a non-zero subgroup pure in $F$. This generalizes the result from a previous paper dealing with $F/K$$p$-primary.},
author = {Bican, Ladislav},
journal = {Mathematica Bohemica},
keywords = {torsion-free abelian groups; pure subgroup; $P$-pure subgroup; torsion-free Abelian groups; pure subgroups; -pure subgroups},
language = {eng},
number = {3},
pages = {649-652},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Pure subgroups},
url = {http://eudml.org/doc/248863},
volume = {126},
year = {2001},
}

TY - JOUR
AU - Bican, Ladislav
TI - Pure subgroups
JO - Mathematica Bohemica
PY - 2001
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 126
IS - 3
SP - 649
EP - 652
AB - Let $\lambda $ be an infinite cardinal. Set $\lambda _0=\lambda $, define $\lambda _{i+1}=2^{\lambda _i}$ for every $i=0,1,\dots $, take $\mu $ as the first cardinal with $\lambda _i<\mu $, $i=0,1,\dots $ and put $\kappa = (\mu ^{\aleph _0})^+$. If $F$ is a torsion-free group of cardinality at least $\kappa $ and $K$ is its subgroup such that $F/K$ is torsion and $|F/K|\le \lambda $, then $K$ contains a non-zero subgroup pure in $F$. This generalizes the result from a previous paper dealing with $F/K$$p$-primary.
LA - eng
KW - torsion-free abelian groups; pure subgroup; $P$-pure subgroup; torsion-free Abelian groups; pure subgroups; -pure subgroups
UR - http://eudml.org/doc/248863
ER -

References

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  1. Finitistic dimension and a homological characterization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466–488. (1960) MR0157984
  2. A note on pure subgroups, (to appear). (to appear) Zbl0969.20028MR1777650
  3. On covers, (to appear). (to appear) MR1813494
  4. Über injektive Moduln, Arch. Math. 4 (1953), 75–78. (1953) MR0055978
  5. Injective and flat covers, envelopes and resolvents, Israel J. Math. 39 (1981), 189–209. (1981) Zbl0464.16019MR0636889
  6. Infinite Abelian Groups, vol. I and II, Academic Press, New York, 1973 and 1977. (1973 and 1977) MR0255673
  7. Torsion-free covers II, Israel J. Math. 23 (1976), 132–136. (1976) Zbl0321.16014MR0417245
  8. Flat Covers of Modules, Lecture Notes in Mathematics 1634, Springer, Berlin, 1996. (1996) Zbl0860.16002MR1438789

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