×

zbMATH — the first resource for mathematics

On direct sums of B\(^{(1)}\)-groups. II. (English) Zbl 1106.20041
A torsionfree abelian group \(G\) is called B\(^{(1)}\)-group if it is a direct sum of rank one pure subgroups, \(G = \langle g_1\rangle ^* +\cdots +\langle g_m\rangle ^*\), where \(g_1+ \cdots +g_m = 0\). The \(m\)-tuple \((g_1 ,\dots ,g_m)\) is called a base of \(G\). Let \(G'\) and \(G''\) be B\(^{(1)}\)-groups with minimum types \(\varrho '\) and \(\varrho ''\), respectively. We say that \(G'\) hooks up to \(G''\) if there is a base type \(t'\) of \(G'\) such that \(t'\leq \varrho '\vee \varrho ''\). Further, \(G'\) and \(G''\) satisfy the hooking condition, if each hooks up to the other. If \(G'\), \(G''\) are indecomposable B\(^{(1)}\)-groups, then \(G=G'\oplus G''\) is a B\(^{(1)}\)-group if and only if \(G'\) and \(G''\) are hooking summands of \(G\) (Theorem 2.2). In the third part the generalized hooking condition is introduced and it is shown that if \(G'\), \(G''\) are B\(^{(1)}\)-groups with \(G''\) indecomposable, then \(G'\oplus G''\) is a B\(^{(1)}\)-group if and only if \(G'\) and \(G''\) satisfy the generalized hooking condition (Theorem 3.3). In the final part some examples are presented.

MSC:
20K15 Torsion-free groups, finite rank
PDF BibTeX XML Cite
Full Text: EMIS EuDML