Motivated by the categorical notion of localizations applied to the quasicategory of abelian groups, we call a homomorphism α: A → B a quasilocalization 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 ⊆ M ⊆ Q⊗ Z R. We call R the ring of the quasilocalization M . Some old results due to Zassenhaus and Butler show that all rings with free additive groups of finite rank are indeed rings of quasilocalizations 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 F of submodules and thus End(M, F ) = {ψ ∈ End(M ) : ψ(X) ⊆ X for all X ∈ F}. We will restrict our discussion to the case M = R such that End(R, F ) = R, and in this case we call the family 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.

Given any ring R with 1 ∈ R and any family F of left ideals of R, we definethe family of all left ideals of R. We will present numerous examples of rings that are left rigid or not left rigid but End(R + , F ) is a left rigid ring.

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