Rigid ℵ₁-Free Abelian Groups with Prescribed Factors and Their Role in the Theory of Cellular Covers

Abstract

In Rodríguez and Strüngmann (J. Algebra Appl. 14, 2016) Rodríguez and the second author gave a new method to construct cellular exact sequences of abelian groups with prescribed torsion-free kernels and co-kernels. In particular, the method was applied to the class of $\aleph _{1}$-free abelian groups in order to complement results from Rodríguez–Strüngmann (Mediterr. J. Math. 6:139–150, 2010) and Göbel–Rodríguez–Strüngmann (Fundam. Math. 217:211–231, 2012). However, $\aleph _{1}$-free abelian groups G with trivial dual but $\mathop{\mathrm{Hom}}\nolimits (G,R)\neq \{0\}$ for all rational groups $R \subseteq \mathbb{Q}$ not isomorphic to $\mathbb{Z}$ had to be excluded. Here we give two constructions of such groups, e.g., using Shelah’s Black Box prediction principle.