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Abelian groups with one $$\tau$$-adic relation. (English. Russian original) Zbl 0699.20043
Algebra Logic 28, No. 1, 57-73 (1989); translation from Algebra Logika 28, No. 1, 83-104 (1989).
Let $$\tau$$ be a type and $$Q_{\tau}$$ be the algebra of $$\tau$$-adic numbers. If A is a torsionfree group of finite rank r and $$\{x_ 1,...,x_ r\}$$ is any maximal linearly independent subset of A then the natural homomorphism $$A\to A_{\tau}$$ of A into its $$\tau$$-adic completion $$A_{\tau}$$ determines the submodule $\delta (A)=\{(\alpha_ 1,...,\alpha_ r)\in Q^ r_{\tau}| \quad \alpha_ 1x_ 1+...+\alpha_ rx_ r=0\quad in\quad A_{\tau}\}$ of $$\tau$$-relations in $$Q^ r_{\tau}$$. The paper studies the properties of the class $${\mathcal F}$$ of finite rank torsionfree groups A such that the module $$\delta(A)$$ is cyclic. Among other results the groups from $${\mathcal F}$$ are characterized up to quasi-isomorphism (Th.3.7) and pure subgroups of groups from $${\mathcal F}$$ together with the corresponding factor-groups are described (§§ 4, 5).
Reviewer: L.Bican

##### MSC:
 20K15 Torsion-free groups, finite rank 20K27 Subgroups of abelian groups 20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups
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