Quasi-localizations of ℤ

Abstract

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.

Left rigid rings

“…Thus, for any ψ ∈ End(R + , F) we have ψ − ψ(1) = 0 and ψ ∈ R follows. The last part follows from Proposition 4 in [3]. 2…”

“…In the paper [3] we studied torsion-free rings R, 1 ∈ R, such that there exists a left R-module M such that R ⊆ M ⊆ QR and End(M + ) = R. Here M + denotes the additive group of M and we identify the elements of R with the left multiplication they induce on M. Instrumental in the construction of M were families F of left ideals of R. We said that F is an E-forcing family of left ideals of R if End(R + , F) := {ϕ ∈ End(R + ): ϕ(X) ⊆ X} = R. Note that R is an E-ring if and only if the empty set is an E-forcing family of R. We will also say that R is F -rigid if End(R + , F) = R. Moreover, we will call R left rigid, if R is F -rigid for the set F of all left ideals of R. This leads to the very natural Question. What rings are left rigid?…”

An extension of Zassenhaus' theorem on endomorphism rings

“…]), a homomorphism α : A → B is called a quasi-localization of A if for all homomorphisms φ : A → B there exists some natural number n and a unique ψ ∈ End B such that nφ = αψ. It was shown in [1] that an injective homomorphism α : Z → M is a quasilocalization (equivalently, a localization in the quasi-category of torsion-free abelian groups) if and only if there is a Zassenhaus ring R with module M .…”

“…On the one hand, several new examples of Zassenhaus rings of finite as well as infinite rank were presented in [1] and [2]. We will give an example to show that not all rings R such that R + is free of countable rank are Zassenhaus rings.…”

“…Recall that if R is a Zassenhaus ring, then R is right rigid, that is, End(R, R R ) = R as shown (for left rigidity) in [1]; see also [13, Theorem 4. Let R be a ring such that 1 ∈ R and R + is torsion-free and homogeneous of type zero.…”