Representing a ring as the endomorphism ring of an abelian group

Abstract

We investigate Baer's realization problem for almost free abelian groups, focusing on the extent to which rings can be represented as endomorphism rings under strong freeness conditions. Building on earlier work that relied on additional set-theoretic principles such as the diamond or strong black boxes, we develop new methods that significantly weaken these assumptions. The main result is obtained in ZFC (assuming a mild cardinal arithmetic configuration): for a strong limit singular cardinal μ with μ⁺ < 2^μ< 2^μ⁺, and for a wide class of cotorsion-free rings, we construct μ⁺-free modules whose endomorphism rings are isomorphic to the given ring. This provides a substantial partial solution to a problem of Göbel and Trlifaj. The key innovation is the integration of κ-frame constructions with Shelah's Super Black Box, enabling a delicate diagonalization that eliminates nontrivial endomorphisms while preserving high degrees of freeness.

Singular Submodules of Abelian Groups over Their Endomorphism Rings
Jiang Yang, Xin Zhang
arXiv18 Sep 2026PreprintarXiv licenceFull textGeometry & topologyarXiv
On the Chow ring of very general abelian varieties and a question of Pirola
Claire Voisin
arXiv30 Aug 2026PreprintCC BY 4.0Full textAlgebra & number theoryarXiv
Symmetric group actions on the coordinate ring of the lower unipotent group and the polynomial ring with quantum parameters
Tatsuya Horiguchi
arXiv23 Sep 2026PreprintarXiv licenceFull textAlgebra & number theory Combinatorics & discrete mathematicsarXiv
Wedderburn decomposition of Twisted skew abelian group algebra
Alvaro Otero Sanchez
arXiv7 Sep 2026PreprintCC BY 4.0Full textAlgebra & number theoryarXiv
Continuous graph homomorphisms of higher dimensional abelian group actions
Ruijun Wang
arXiv30 Sep 2026PreprintCC BY 4.0Full textCombinatorics & discrete mathematics Logic & foundationsarXiv

Cite

BibTeX RIS

BibTeX

@misc{asgharzadeh2026representing,
  title = {{Representing a ring as the endomorphism ring of an abelian group}},
  author = {Mohsen Asgharzadeh and Mohammad Golshani and Saharon Shelah},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.06577},
  eprint = {2609.06577},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06577},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Representing a ring as the endomorphism ring of an abelian group
AU  - Mohsen Asgharzadeh
AU  - Mohammad Golshani
AU  - Saharon Shelah
PY  - 2026
DA  - 2026/09/06
PB  - arXiv
DO  - 10.48550/arxiv.2609.06577
UR  - https://arxiv.org/abs/2609.06577
AB  - We investigate Baer's realization problem for almost free abelian groups, focusing on the extent to which rings can be represented as endomorphism rings under strong freeness conditions. Building on earlier work that relied on additional set-theoretic principles such as the diamond or strong black boxes, we develop new methods that significantly weaken these assumptions. The main result is obtained in ZFC (assuming a mild cardinal arithmetic configuration): for a strong limit singular cardinal μ with μ^+ < 2^μ< 2^μ^+, and for a wide class of cotorsion-free rings, we construct μ^+-free modules whose endomorphism rings are isomorphic to the given ring. This provides a substantial partial solution to a problem of Göbel and Trlifaj. The key innovation is the integration of κ-frame constructions with Shelah's Super Black Box, enabling a delicate diagonalization that eliminates nontrivial endomorphisms while preserving high degrees of freeness.
ER  -

Abstract

Related

Cite