Co-local subgroups of Abelian groups. II.
This paper continues earlier work on co-local subgroups of Abelian groups by \textit{J. Buckner} and \textit{M. Dugas} [Lect. Notes Pure Appl. Math. 249, 29-37 (2006; see the preceding review Zbl 1008.20054)]. Recall that a subgroup \(K\) of an Abelian group \(A\) is called co-local if the natural map \(\Hom(A,A)\to\Hom(A,A/K)\) is an isomorphism. This notion is a dual to the notion of localization which has been in the focus of research in various areas of mathematics. While Buckner and Dugas mainly considered co-local subgroups of torsion-free groups, the author now shifts his attention to co-local subgroups of mixed groups which are non-splitting. As for torsion-free groups it is shown that one may restrict oneself to reduced groups and in that case a co-local subgroup \(K\) of a reduced mixed group \(A\) is a pure cotorsion-free subgroup of \(A\). This fact implies that \((K+t(A))/t(A)\) is a pure subgroup of \(A/t(A)\), where \(t(A)\) is the torsion part of \(A\). The main purpose of the paper is to explore the connection between \(K\) and \((K+t(A))/t(A)\). On the one hand it is proved that, if \(A/t(A)=D/t(A)\oplus H/t(A)\) with \(D/t(A)\) the maximal divisible part of \(A/t(A)\), then \(K\cap D=0\) and one may assume that \(K\subseteq H\). However, using a Black Box argument examples are constructed which show that \(K\) need not be a co-local subgroup of \(H\). On the other hand, the author shows that all co-local subgroups of torsion-free groups are induced by co-local subgroups of non-splitting mixed Abelian groups. In fact, he proves that for any co-local subgroup \(K\) of a torsion-free group \(G\) there is a reduced torsion group \(T\) and a non-splitting mixed Abelian group \(A\) with \(t(A)=T\) and \(A/t(A)=G\) such that \(K=(K'+t(A))/t(A)\) for some co-local subgroup \(K'\) of \(A\).
scientific article; zbMATH DE number 5078560