Semi-rigid classes of cotorsion-free Abelian groups
A class of groups is semi-rigid if for every pair (H,G) in it, either Hom(H,G) or \(Hom(G,H)=0\), and rigid if both are zero. The authors use combinatorics of infinite sets to prove powerful theorems on the existence of large semi-rigid families (in ZFC) and large rigid families (in ZFC \(+\) additional axioms). The main results depend on the following theorem: for every strong limit singular cardinal m of uncountable cofinality, there is a cotorsion-free group G of cardinal m such that every proper subgroup U with cotorsion- free quotient has smaller cardinality. Moreover G may be chosen to be homogeneous of any non-divisible type. As applications of these results, the authors prove several results about torsion theories; for example the class of cotorsion-free groups does not have a cogenerating set, and torsion classes generated or cogenerated by a single group do not have small cardinality.
scientific article; zbMATH DE number 3882638
scientific article