Localizations of torsion-free abelian groups

Localizations of objects play an important role in category theory, homology, and elsewhere. A (homo)morphism α :A→B is a localization of A if for each f :A→B there is a unique :B→B extending f. In this paper we will investigate localizations of (co)torsion-free abelian groups and show that they exist in abundance. We will present several methods for constructing localizations. We will also show that free abelian groups of infinite rank have localizations that are not direct sums of E-rings.