We complete the characterization of Ext(G,ℤ) for any torsion-free abelian group G assuming Gödel's axiom of constructibility plus there is no weakly compact cardinal. In particular, we prove in (V = L) that, for a singular cardinal ν of uncountable cofinality which is less than the first weakly compact cardinal and for every sequence $(ν_{p}: p ∈ Π)$ of cardinals satisfying $ν_{p} ≤ 2^{ν}$ (where Π is the set of all primes), there is a torsion-free abelian group G of size ν such that $ν_{p}$ equals the p-rank of Ext(G,ℤ) for every prime p and $2^{ν}$ is the torsion-free rank of Ext(G,ℤ).
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Let R be a unital commutative ring and A a unital R-algebra. We introduce the category of E(A,R)-modules which is a natural extension of the category of E-modules. The properties of E(A,R)-modules are studied; in particular we consider the subclass of E(R)-algebras. This subclass is of special interest since it coincides with the class of E-rings in the case R = ℤ. Assuming diamond ⋄, almost-free E(R)-algebras of cardinality κ are constructed for any regular non-weakly compact cardinal κ > ℵ ₀ and suitable R. The set-theoretic hypothesis can be weakened.
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In this paper we improve recent results dealing with cellular covers of R-modules. Cellular covers (sometimes called colocalizations) come up in the context of homotopical localization of topological spaces. They are related to idempotent cotriples, idempotent comonads or coreflectors in category theory. Recall that a homomorphism of R-modules π: G → H is called a cellular cover over H if π induces an isomorphism $π⁎: Hom_{R}(G,G) ≅ Hom_{R}(G,H)$, where π⁎(φ) = πφ for each $φ ∈ Hom_{R}(G,G)$ (where maps are acting on the left). On the one hand, we show that every cotorsion-free R-module of rank $κ < 2^{ℵ₀}$ is realizable as the kernel of some cellular cover G → H where the rank of G is 3κ + 1 (or 3, if κ = 1). The proof is based on Corner's classical idea of how to construct torsion-free abelian groups with prescribed countable endomorphism rings. This complements results by Buckner-Dugas. On the other hand, we prove that every cotorsion-free R-module H that satisfies some rigid conditions admits arbitrarily large cellular covers G → H. This improves results by Fuchs-Göbel and Farjoun-Göbel-Segev-Shelah.
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As is well known, torsion abelian groups are not preserved by localization functors. However, Libman proved that the cardinality of LT is bounded by $|T|^{ℵ₀}$ whenever T is torsion abelian and L is a localization functor. In this paper we study localizations of torsion abelian groups and investigate new examples. In particular we prove that the structure of LT is determined by the structure of the localization of the primary components of T in many cases. Furthermore, we completely characterize the relationship between localizations of abelian p-groups and their basic subgroups.
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