A B S T R A C T Some years ago A. L. S. Corner proved that every countable and cotorsion-free ring can be realized as the endomorphism ring of some torsion-free abelian group. This result has many interesting consequences for abelian groups. Using a set-theoretic axiom V k ., which follows for instance from V = L, we can drop the countability condition in Corner's theorem.

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Localizations of objects play an important role in category theory, homology, and elsewhere.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.  2004 Elsevier Inc. All rights reserved. Keywords: Local abelian groups IntroductionLocalization functors are a well-known notion in category theory. If C is any category, then a localization functor is a pair (L, a) such that L : C → C is a covariant functor and a : id C → L is a natural transformation such that a L(X) = L(a X ) for all objects X of C and a L(X) : L(X) → L(L(X)) is an isomorphism. The morphism a X : X → L(X) is called the co-augmentation morphism of X. If a X is an isomorphism, then X is called L-local. Moreover, if α : X → Y is a morphism such that L(α) : L(X) → L(Y ) is an isomorphism, then α is called an L-equivalence. If α : X → Y is an L-equivalence and A is L-local, then α ⊥ A, i.e., for each f ∈ Hom(X, A) there is a unique ϕ ∈ Hom(Y, A) such that f = ϕ • α.