Abstract
This paper is devoted to the problem (Glaz, Vinsonhaler, Wickless): When a \(p\)-local torsion-free group is determined by the collection of its splitting rings? For \(p\)-local torsion-free groups with quadratic and cubic minimal splitting fields necessary and sufficient condition has been found according to which \(p\)-local group \(G\) is determined by the collection of its splitting rings.
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REFERENCES
L. Fuchs, Infinite Abelian Groups (Academic, New York, 1970, 1973), Vols. 1, 2.
G. Szekeres, ‘‘Countable Abelian groups without torsion,’’ Duke Math J. 15, 293–306 (1948).
E. L. Lady, ‘‘Splitting fields for torsion-free modules over discrete valuation rings,’’ Int. J. Algebra 49, 261–275 (1977).
S. Glaz, S. Vinsonhaler, and W. Wickless, ‘‘Splitting rings for \(p\)-local torsion-free groups,’’ in Zerodimensional Commutative Rings, Lect. Notes Pure Appl. Math. 171, 223–239 (1995).
S. V. Vershina, ‘‘Indecomposable \(p\)-local torsion-free groups with quadratic and cubic splitting fields,’’ J. Math. Sci. 230, 364–371 (2018).
D. M. Arnold and M. Dugas, ‘‘Indecomposable modules over Nagata valuation domains,’’ Proc. Am. Math. Soc. 122, 689–696 (1994).
D. M. Arnold and M. Dugas, ‘‘Co-purely indecomposable modules over discrete valuation rings,’’ J. Pure Appl. Algebra 161, 1–12 (2001).
P. Zanardo, ‘‘Kurosch invariants for torsion-free modules over Nagata valuation domains,’’ J. Pure Appl. Algebra 82, 195–209 (1992).
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(Submitted by M. M. Arslanov)
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Vershina, S.V. Definability of Local Abelian Groups by Their Splitting Rings. Lobachevskii J Math 41, 1712–1717 (2020). https://doi.org/10.1134/S1995080220090309
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DOI: https://doi.org/10.1134/S1995080220090309