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Šaroch, Jan

On the nontrivial solvability of systems of homogeneous linear equations over \(\mathbb{Z}\) in ZFC. (English) Zbl 07285998

Commentat. Math. Univ. Carol. 61, No. 2, 155-164 (2020).
Summary: Motivated by the paper by H. Herrlich and E. Tachtsis [Commentat. Math. Univ. Carol. 58, No. 2, 241–260 (2017; Zbl 1463.03017)] we investigate in ZFC the following compactness question: for which uncountable cardinals \(\kappa\), an arbitrary nonempty system \(S\) of homogeneous \(\mathbb{Z}\)-linear equations is nontrivially solvable in \(\mathbb{Z}\) provided that each of its subsystems of cardinality less than \(\kappa\) is nontrivially solvable in \(\mathbb{Z}\)?

MSC:

08A45 Equational compactness
13C10 Projective and free modules and ideals in commutative rings
20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups
03E35 Consistency and independence results
03E55 Large cardinals

Keywords:

homogeneous \(\mathbb{Z}\)-linear equation; \(\kappa\)-free group; \(\mathcal L_{\omega_1\omega}\)-compact cardinal

Citations:

Zbl 1463.03017
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\textit{J. Šaroch}, Commentat. Math. Univ. Carol. 61, No. 2, 155--164 (2020; Zbl 07285998)
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References:

[1] Bagaria J.; Magidor M., Group radicals and strongly compact cardinals, Trans. Amer. Math. Soc. 366 (2014), no. 4, 1857-1877
[2] Bagaria J.; Magidor M., On \(\omega_1\)-strongly compact cardinals, J. Symb. Log. 79 (2014), no. 1, 266-278
[3] Dugas M.; Göbel R., Every cotorsion-free ring is an endomorphism ring, Proc. London Math. Soc. (3) 45 (1982), no. 2, 319-336
[4] Eklof P. C.; Mekler A. H., Almost Free Modules, Set-theoretic methods, North-Holland Mathematical Library, 65, North-Holland Publishing, Amsterdam, 2002
[5] Göbel R.; Shelah S., \( \aleph_n\)-free modules with trivial duals, Results Math. 54 (2009), no. 1-2, 53-64
[6] Göbel R.; Trlifaj J., Approximations and Endomorphism Algebras of Modules, Volume 1., Approximations, De Gruyter Expositions in Mathematics, 41, Walter de Gruyter GmbH & Co. KG, Berlin, 2012
[7] Herrlich H.; Tachtsis E., On the solvability of systems of linear equations over the ring \(\mathbb Z\) of integers, Comment. Math. Univ. Carolin. 58 (2017), no. 2, 241-260
[8] Kanamori A., The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings, Springer Monographs in Mathematics, Springer, Berlin, 2003
[9] Shelah S., Quite free complicated abelian group, PCF and black boxes, available at ArXiv: 1404.2775v2 [math.LO] (2019), 49 pages
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