On purifiable torsion-free rank-one subgroups (Q5945557)

The paper deals with arbitrary (mixed) Abelian groups. The torsion-free subgroups of rank \(1\) that are (\(p\)-)purifiable, i.e., contained in a minimal (\(p\)-)pure subgroup (a (\(p\)-)pure hull), are characterized (Theorem~3.2). Theorem~2.9 probably contains what the author describes as ``the structure of pure hulls''. It is a highly technical and lengthy statement describing in great detail the interrelations of \(G\), a \(p\)-purifiable rank-one subgroup \(A\) of \(G\), a \(p\)-pure hull \(H\) of \(A\), and a \(T(H)\)-high subgroup \(N\supseteq A\) of \(H\). In general, \(T(K)\) denotes the maximal torsion subgroup of \(K\). Proposition~2.9 describes the common features of two pure hulls of the same torsion-free rank-one subgroup, still too technical to state here.NEWLINENEWLINENEWLINEA main result is Theorem~3.4. Let \(G\) be an arbitrary Abelian group and \(A\) a torsion-free rank-one subgroup of \(G\). If \(A\) is purifiable in \(G\), then all pure hulls of \(A\) are isomorphic.NEWLINENEWLINENEWLINEThe author next considers the case that a \(T(G)\)-high subgroup of \(G\) is purifiable. The main result is Theorem~4.1. Let \(G\) be an arbitrary Abelian group, \(A\) a purifiable \(T(G)\)-high subgroup of \(G\). Then there is a torsion group \(T\) such that \(G=H\oplus T\) for every pure hull \(H\) of \(A\).NEWLINENEWLINENEWLINEThe author turns to groups \(G\) of torsion-free rank \(1\) that possess \(T(G)\)-high subgroups that are purifiable. Example~5.1 shows that it can happen that some \(T(G)\)-high subgroups are purifiable and others are not. Theorem~5.2 characterizes the groups of torsion-free rank \(1\) in which every \(T(G)\)-high subgroup is purifiable; Theorem~5.5 characterizes the groups \(G\) of torsion-free rank \(1\) that possess purifiable \(T(G)\)-high subgroups. Finally, the author applies his results to give another proof of a splitting theorem for mixed groups of torsion-free rank \(1\) that is due to \textit{A. E. Stratton} [J. Algebra 19, 254-260 (1971; Zbl 0233.20020)].