Tight subgroups in torsion-free Abelian groups

LetX be a torsion-free abelian group. We study the class of all completely decomposable subgroups ofX which are maximal with respect to inclusion. These groups are called tight subgroups ofX and we state sufficient conditions on a subgroup to be tight. In particular we consider tight subgroups of bounded completely decomposable groups. For those we show that every regulating subgroup is tight and we characterize the tight subgroups of finite index in almost completely decomposable groups. The second author was supported by a MINERVA fellowship.