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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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