Matrix representations of endomorphism rings for torsion-free abelian groups

Authors

  • Ekaterina А. Blagoveshchenskaya Emperor Alexander I St. Petersburg State Transport University, 9, Moskovskii pr., St. Petersburg, 190031, Russian Federation
  • Alexander V. Mikhalev Lomonosov Moscow State University, 1, Leninskie gory, Moscow, 119991, Russian Federation

DOI:

https://doi.org/10.21638/spbu01.2023.304

Abstract

Non-isomorphic direct decompositions of torsion-free abelian groups are reflected in their endomorphism ring decompositions which admit matrix representations. The set of possible direct decompositions of a special kind matrix rings into direct sums of one-sided indecomposable ideals is described. This leads to the combinatorial constructions of isomorphisms between non-commutative differently decomposable ring structures.

Keywords:

torsion-free abelian groups, endomorphism rings, matrix representations

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References

Литература

1. Фукс Л. Бесконечные абелевы группы, пер. с англ. Москва, Мир, т. 1 (1974), т. 2 (1977).

2. Благовещенская Е. А., Михалев А. В. Влияние теоремы Бэра - Капланского на развитие теории групп, колец и модулей. Фундаментальная и прикладная математика 24 (1), 31-123(2022).

3. Mader A. Almost completely decomposable abelian groups, Gordon and Breach. Algebra, Logic and Applications, vol. 13, Amsterdam (1999).

4. Arnold D. Finite rank Torsion free Abelian Groups and rings, lecture notes. In: Mathematics, vol. 931, Springer Verlag (1982).

5. Blagoveshchenskaya E. Direct Decompositions of Torsion-Free Abelian Groups Lobachevskii. Journal of Mathematics 41, 1640-1646 (2020).

References

1. Fuchs L. Infinite Abelian Groups. Academic Press, vol. 1 (1970), vol. 2 (1973). [Rus. ed.: Fuchs L. Beskonechnye abelevy gruppy. Moscow, Mir Publ., vol. 1 (1974), vol. 2 (1977)]. (In Russian)

2. Blagoveshchenskaya E. A., Mikhalev A. V. Influence of the Baer - Kaplansky Theorem on the Development of the Theory of Groups, Rings, and Modules. Fundamental’naia i prikladnaia matematika 24 (1), 31-123 (2022). (In Russian) [Engl. trans.: Journal of Mathematical Sciences, 269 (5), 632-696, 2023. https://doi.org/10.1007/s10958-023-06306-3].

3. Mader A. Almost completely decomposable abelian groups, Gordon and Breach. Algebra, Logic and Applications, vol. 13, Amsterdam (1999).

4. Arnold D. Finite rank Torsion free Abelian Groups and rings, lecture notes. In: Mathematics, vol. 931, Springer Verlag (1982).

5. Blagoveshchenskaya E. Direct Decompositions of Torsion-Free Abelian Groups Lobachevskii. Journal of Mathematics 41, 1640-1646 (2020).

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Published

2023-09-23

How to Cite

Blagoveshchenskaya E. А., & Mikhalev, A. V. (2023). Matrix representations of endomorphism rings for torsion-free abelian groups. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 10(3), 487–498. https://doi.org/10.21638/spbu01.2023.304

Issue

Vol. 10 No. 3 (2023)

Section

Mathematics

License

Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.