Co-Local Subgroups of Abelian Groups

Cellular Covers of Abelian Groups

“…In paper [7] the authors show that the kernels of cellular covering maps for some fixed torsionfree group may be arbitrarily large, but this can not happen whenever the kernel is a free abelian group. Buckner-Dugas [1] and Dugas [4] investigated the kernels of cellular covering maps (under the name of "co-local" subgroups, using a different point of view), and showed that these kernels are always torsion-free reduced, and every cotorsion-free abelian group appears as the kernel of a cellular covering map.…”

Cellular covers of totally ordered abelian groups

“…According to this result, for torsion-free groups all the kernels have to be cotorsion-free (that follows from [1] already). For o-groups, though the kernels are always torsionfree, the situation is different; see Example 3.2.…”

“…From a result of Buckner and Dugas [1] it follows that every cotorsion-free abelian group is the kernel of a suitably chosen cellular covering map of some torsion-free abelian group (taken in Ab). We are making use of this theorem in order to verify:…”

“…is an R-module, then so are G and K in (1), and the cellular covering map is an R-homomorphism. P r o o f. A subring R of Q admits a unique total ring order, and if the group A is an R-module, then the o-group A is automatically a module over the o-ring R. The rest follows from (ii) in the preceding lemma.…”

“…By Buckner and Dugas [1] there exists in Ab a cellular exact sequence 0 → K → G γ → A → 0 of torsion-free abelian groups. It remains to furnish A with any total order and G with the lexicographic order to complete the proof.…”