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If there is an exactly λ-free abelian group then there is an exactly λ-separable one in λ

Published online by Cambridge University Press:  12 March 2014

Saharon Shelah
Saharon Shelah*
Affiliation:
Institute of Mathematics, The Hebrew University, Jerusalem, Israel Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903, USA
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Abstract

We give a solution stated in the title to problem 3 of part 1 of the problems listed in the book of Eklof and Mekler [2], p. 453. There, in pp. 241-242, this is discussed and proved in some cases. The existence of strongly λ-free ones was proved earlier by the criteria in [5] and [3]. We can apply a similar proof to a large class of other varieties in particular to the variety of (non-commutative) groups.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 61 , Issue 4 , December 1996 , pp. 1261 - 1278
Copyright
Copyright © Association for Symbolic Logic 1996

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References

REFERENCES

[1]Eklof, P. and Mekler, A., Categoricity results for L∞,κ-free algebras, Annals of Pure and Applied Logic, vol. 37, pp. 81–99.CrossRefGoogle Scholar
[2]Eklof, P. and Mekler, A., Almost free modules; Set theoretic methods, North Holland Library, 1990.Google Scholar
[3]Mekler, Alan H. and Shelah, Saharon, When κ-free implies strongly κ-free, Abelian group theory (Oberwolfach 1985), Gordon and Breach, New York, 1987, pp. 137–148.Google Scholar
[4]Mekler, Alan H. and Shelah, Saharon, Almost free algebras, Israel Journal of Mathematics, vol. 89 (1995), pp. 237–259.CrossRefGoogle Scholar
[5]Shelah, Saharon, Incompactness in regular cardinals, Notre Dame Journal of Formal Logic, vol. 26 (1985), pp. 195–228.CrossRefGoogle Scholar