Localizations of torsion-free abelian groups II

A homomorphism α:A→B between abelian groups A,B is called a localization of A if for each φ∈Hom(A,B) there is a unique ψ∈End(B) such that φ=ψ○α. It is well known that if A=Z, then B is an E-ring and α(1) is the identity of B. We investigate localizations of rank-1 groups A=L⊂Q of type τ. It turns out that localizations of L can be surprisingly complicated. If α:L→M is a localization and L is a subring of Q, then M is simply an E-ring that is also an L-module. If L is not a subring, things get more complicated. If M=M(τ), then tensor products of L and E-rings come into play. It is possible that M≠M(τ), and we can say very little in this case. Another topic under consideration are localizations of E-rings. Frequently, localizations of E-rings are E-rings again, but we find examples where this is not the case.