On Strongly Balanced Subgroups of Separable Torsion-Free Abelian Groups

In this note we wish to show how the idea of strong balancedness clarifies the properties of separable abelian groups and helps to (i) reprove some of the classical theorems, (ii) extend well-known results and (iii) construct new separable abelian groups. Specifically, we show that strongly balanced subgroups of separable abelian groups are again separable. This not only extends but also gives a simpler proof of the classical theorem of L. Fuchs on summands of separable groups. Thus separable abelian groups may be characterized as the strongly balanced images of completely decomposable groups. Our point of view enables us to formulate our results for the class of m-separable groups, where m is an arbitrary cardinal.