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PreprintCC BY 4.0arXiv

Finite Non-Commutative Centrally Essential Rings of Minimal Order

  • Adel Abyzov
  • Oleg Lyubimtsev
  • Askar Tuganbaev
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Abstract

A ring R with non-zero identity element is said to be centrally essential if for any non-zero element a of R, there exist non-zero central elements x and y such that ax = y. In the paper, we prove that the minimal order of a finite non-commutative centrally essential ring is equal to $64$. A partial classification of such rings is presented, and the existence of non-isomorphic non-commutative centrally essential rings of minimal order is proved.

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Cite

BibTeXRIS
BibTeX
@misc{abyzov2026finite,
  title = {{Finite Non-Commutative Centrally Essential Rings of Minimal Order}},
  author = {Adel Abyzov and Oleg Lyubimtsev and Askar Tuganbaev},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.36174},
  eprint = {2610.36174},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.36174},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Finite Non-Commutative Centrally Essential Rings of Minimal Order
AU  - Adel Abyzov
AU  - Oleg Lyubimtsev
AU  - Askar Tuganbaev
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.36174
UR  - https://arxiv.org/abs/2610.36174
AB  - A ring R with non-zero identity element is said to be centrally essential if for any non-zero element a of R, there exist non-zero central elements x and y such that ax = y. In the paper, we prove that the minimal order of a finite non-commutative centrally essential ring is equal to $64$. A partial classification of such rings is presented, and the existence of non-isomorphic non-commutative centrally essential rings of minimal order is proved.
ER  -