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

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 \(\mathbb{Z}\)?


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


Zbl 1463.03017
Full Text: DOI arXiv


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