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Keywords:
torsion-free abelian groups; pure subgroup; $P$-pure subgroup
torsion-free abelian groups; pure subgroup; $P$-pure subgroup
Summary:
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.
References:
[1] H. Bass: Finitistic dimension and a homological characterization of semi-primary rings. Trans. Amer. Math. Soc. 95 (1960), 466–488. DOI 10.1090/S0002-9947-1960-0157984-8 | MR 0157984
[2] L. Bican: A note on pure subgroups. (to appear). MR 1777650 | Zbl 0969.20028
[3] L. Bican, B. Torrecillas: On covers. (to appear). MR 1813494
[4] B. Eckmann, A. Schopf: Über injektive Moduln. Arch. Math. 4 (1953), 75–78. DOI 10.1007/BF01899665 | MR 0055978
[5] E. Enochs: Injective and flat covers, envelopes and resolvents. Israel J. Math. 39 (1981), 189–209. DOI 10.1007/BF02760849 | MR 0636889 | Zbl 0464.16019
[6] L. Fuchs: Infinite Abelian Groups, vol. I and II. Academic Press, New York, 1973 and 1977. MR 0255673
[7] M. L. Teply: Torsion-free covers II. Israel J. Math. 23 (1976), 132–136. MR 0417245 | Zbl 0321.16014
[8] J. Xu: Flat Covers of Modules. Lecture Notes in Mathematics 1634, Springer, Berlin, 1996. MR 1438789 | Zbl 0860.16002
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