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

A system \(S\) of homogeneous \(\mathbb{Z}\)-linear equations with a set \(X=\{ x_i:i\in I\}\) of variables is nontrivially solvable in \(\mathbb{Z}\) if there is \(f:X\rightarrow\mathbb{Z}\setminus\{0\}\) such that \(\Sigma_{j\in J}a_jf(x_j)=0\) for every equation \(\Sigma_{j\in J}a_jx_j=0\) in \(S\). Let \(\mathcal{S}\) denote the class of all infinite cardinals \(\kappa\) with the property that a system \(S\) of homogeneous \(\mathbb{Z}\)-linear equations is nontrivially solvable if and only if each subsystem of \(S\) of cardinality less than \(\kappa\) is nontrivially solvable. It is shown that (a) \(\omega_\alpha\notin\mathcal{S}\) for all \(\alpha <\omega_1\cdot\omega\), (b) under \(V=L\), \(\mathcal{S}=\emptyset\), and (c) if there exists a regular \(\mathcal{L}_{\omega_1\omega}\)-compact cardinal \(\lambda\), then any cardinal \(\kappa\geq\lambda\) lies in \(\mathcal{S}\).

MSC:

03E35 Consistency and independence results
13C10 Projective and free modules and ideals in commutative rings
20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups
03E55 Large cardinals
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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 · Zbl 1349.03055 · doi:10.1090/S0002-9947-2013-05871-0
[2] Bagaria J.; Magidor M., On \(\omega_1\)-strongly compact cardinals, J. Symb. Log. 79 (2014), no. 1, 266-278 · Zbl 1337.03076 · doi:10.1017/jsl.2013.12
[3] Dugas M.; Göbel R., Every cotorsion-free ring is an endomorphism ring, Proc. London Math. Soc. (3) 45 (1982), no. 2, 319-336 · Zbl 0506.16022
[4] Eklof P. C.; Mekler A. H., Almost Free Modules, Set-theoretic methods, North-Holland Mathematical Library, 65, North-Holland Publishing, Amsterdam, 2002 · Zbl 1054.20037
[5] Göbel R.; Shelah S., \( \aleph_n\)-free modules with trivial duals, Results Math. 54 (2009), no. 1-2, 53-64 · Zbl 1183.13012 · doi:10.1007/s00025-009-0382-0
[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 · Zbl 1292.16001
[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 · Zbl 1463.03017
[8] Kanamori A., The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings, Springer Monographs in Mathematics, Springer, Berlin, 2003 · Zbl 1022.03033
[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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