Maximal pure independent sets (Q2738747)

A subset \(S\) of a torsion-free Abelian group \(G\) is pure independent, if it is linearly independent and the subgroup it generates is pure in \(G\). This subset is maximal pure independent, if it has this property and it is not properly contained in any other subset with such a property. Although these ideas have been in existence for some time, it seems, they have not been closely examined. The authors do just that by noting that the subgroups these sets generate generalize the notion of basic subgroups and then proceed to obtain analogous results established already for the case of basic subgroups. Unlike the case of basic subgroups, purely independent subsets exist in every (torsion-free) Abelian group. Maximal pure independent subsets of infinite cardinalities have the same cardinality. There are finite rank groups, where there can be two maximal pure independent sets of differing finite cardinalities, or one such set of a finite cardinality, another of the countable cardinality (in fact cardinalities of the latter cases can be prescribed).NEWLINENEWLINENEWLINESome of the results are as follows: If \(G\) is a separable group with a maximal pure independent set of cardinality \(\kappa\), then it can be embedded as a subgroup of \(\mathbb{Z}^\kappa\); in particular, the cardinality of \(G\) is at most \(2^\kappa\) (it is not clear whether this embedding is pure). If \(G\) is a torsionless group with a maximal pure independent set of infinite cardinality \(\kappa\), then \(G\) has cardinality at most \(\kappa^{\aleph_0}\). If a separable group \(G\) has a maximal pure independent set of infinite cardinality \(\kappa\), then \(G\) can be embedded as a pure subgroup of \(\mathbb{Z}^{\kappa^{\aleph_0}}\).NEWLINENEWLINEFor the entire collection see [Zbl 0960.00043].