### A characterization of countable Butler groups

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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 $({\nu}_{p}:p\in \Pi )$ of cardinals satisfying ${\nu}_{p}\le {2}^{\nu}$ (where Π is the set of all primes), there is a torsion-free abelian group G of size ν such that ${\nu}_{p}$ equals the p-rank of Ext(G,ℤ) for every...

It is proved that if $G$ is a pure ${p}^{\omega +n}$-projective subgroup of the separable abelian $p$-group $A$ for $n\in N\cup \left\{0\right\}$ such that $|A/G|\le {\aleph}_{0}$, then $A$ is ${p}^{\omega +n}$-projective as well. This generalizes results due to Irwin-Snabb-Cutler (CommentṀathU̇nivṠtṖauli, 1986) and the author (Arch. Math. (Brno), 2005).

Torsion-free covers are considered for objects in the category ${q}_{2}.$ Objects in the category ${q}_{2}$ are just maps in $R$-Mod. For $R=\mathbb{Z},$ we find necessary and sufficient conditions for the coGalois group $G(A\u27f6B),$ associated to a torsion-free cover, to be trivial for an object $A\u27f6B$ in ${q}_{2}.$ Our results generalize those of E. Enochs and J. Rado for abelian groups.

In this article we characterize those abelian groups for which the coGalois group (associated to a torsion free cover) is equal to the identity.

Let R be a subring of the rationals. We want to investigate self splitting R-modules G, that is, such that $Ex{t}_{R}(G,G)=0$. For simplicity we will call such modules splitters (see [10]). Also other names like stones are used (see a dictionary in Ringel’s paper [8]). Our investigation continues [5]. In [5] we answered an open problem by constructing a large class of splitters. Classical splitters are free modules and torsion-free, algebraically compact ones. In [5] we concentrated on splitters which are larger...

We present general properties for almost-flat modules and we prove that a self-small right module is almost flat as a left module over its endomorphism ring if and only if the class of $g$-static modules is closed under the kernels.

Let R be a subring of the rational numbers ℚ. We recall from [3] that an R-module G is a splitter if $Ext{\xb9}_{R}(G,G)=0$. In this note we correct the statement of Main Theorem 1.5 in [3] and discuss the existence of non-free splitters of cardinality ℵ₁ under the negation of the special continuum hypothesis CH.

An endomorphism f of an Abelian group A is said to be inessentia! (in the category of Abelian groups) if it can be extended to an endomorphism of any Abelian group which contains A as a subgroup. In this paper we show that f is as above if and only if (f - v idA)(A) is contained in the rnaximal divisible subgroup of A for some v belonging to Z.