Nice Bases for Mixed and Torsion-free Abelian Groups

We prove that the divisible Abelian groups and the global Warfield Abelian groups have a nice basis, that is, they can be represented as a countable ascending union of nice direct sums of cyclic groups. We also show that there exists a mixed Abelian group which does not possess a nice basis as well as we find an unbounded reduced algebraically compact Abelian group which does not satisfy a stronger property. Some related concepts and questions are also considered. This continues our recent investigations in the torsion case published in (Atti Sem. Mat. Fis. Univ. Modena, 2005) and (Ann. Univ. Ferrara - Math., 2007). 2010 Mathematics Subject Classification: 20K10, 20K15, 20K21.