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Definability of local abelian groups by their splitting rings. (English) Zbl 1460.20012

The author research the problem which formulated by S. Glaz et al. [Lect. Notes Pure Appl. Math. 171, 223–239 (1995; Zbl 0895.20046)]: When a \(p\)-local torsion-free abelian group is determined by the collection of its splitting rings? Let \(\mathcal{L}_p(K)\) be the class of \(p\)-local torsion-free abelian groups with splitting field \(K\). If \(K\) is quadratic over \(\mathbb{Q}\) then \(A\) is determined by the collection of its splitting rings if and only if the \(p\)-rank of \(A\) is equal to \(1\). The author gives necessary and sufficient conditions for definability if \(K\) is cubic over \(\mathbb{Q}\).

MSC:

20K15 Torsion-free groups, finite rank

Citations:

Zbl 0895.20046
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References:

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