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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
L. Fuchs, Infinite Abelian Groups, 2 vols. (Academic, New York, 1970–1973; Mir, Moscow, 1974–1977).
E. A. Blagoveshchenskaya and A. V. Mikhalev, “Influence of the Baer–Kaplansky theorem on the development of the theory of groups, rings, and modules,” J. Math. Sci. (N. Y.) 269, 632–696 (2023).
A. Mader, Almost Completely Decomposable Abelian Groups (Gordon and Breach, Amsterdam, 1999), in Ser.: Algebra, Logic and Applications, Vol. 13.
D. Arnold, Finite Rank Torsion Free Abelian Groups and Rings (Springer-Verlag, Berlin, 1982), in Ser.: Lecture Notes in Mathematics, Vol. 931.
E. Blagoveshchenskaya, “Direct decompositions of torsion-free Abelian groups,” Lobachevskii J. Math. 41, 1640–1646 (2020).
Funding
The research was supported by Russian Science Foundation (project no. 22-21-00267).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The authors declare that they have no conflicts of interest.
About this article
Cite this article
Blagoveshchenskaya, E.A., Mikhalev, A.V. Matrix Representations of Endomorphism Rings for Torsion-Free Abelian Groups. Vestnik St.Petersb. Univ.Math. 56, 341–349 (2023). https://doi.org/10.1134/S1063454123030032
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1063454123030032