Quasi-localizations of ℤ

Abstract

Motivated by the categorical notion of localizations applied to the quasi-category of abelian groups, we call a homomorphism α: A → B a quasi-localization of abelian groups if for each ϕ ∈ Hom(A,B) there is an n ∈ ℕ and a unique ψ ∈ End(B) such that nϕ = ψ ∘ α. In this case we call B a quasi-localization of A. In this paper we investigate quasi-localizations of the integers ℤ. While it is well-known that localizations of ℤ are just the E-rings, quasi-localizations of ℤ are much more abundant; an injection α: ℤ → M with M torsion-free, is a quasi-localization if and only if, for R = End(M), one has \(R \subseteq M \subseteq \mathbb{Q} \otimes _\mathbb{Z} R\). We call R the ring of the quasi-localization M. Some old results due to Zassenhaus and Butler show that all rings with free additive groups of finite rank are indeed rings of quasi-localizations of ℤ. We will extend this result and show that there are also rings of infinite rank with this property. While there are many realization results of rings R as endomorphism rings of torsion-free abelian groups M in the literature, the group M is usually not contained in the divisible hull of R+, as is required here. We will use a particular case of a category of left R-modules M with a distinguished family \(\mathcal{F}\) of submodules and thus \((M,\mathcal{F})\). We will restrict our discussion to the case M = R such that \((R,\mathcal{F}) = R\), and in this case we call the family \(\mathcal{F}\) of left ideals E-forcing, not to be confused with the notion of forcing in set theory. We will provide many examples of quasi-localizations M of ℤ, among them those of infinite rank as well as matrix rings for various rings of finite rank.