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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
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