Generalized t-Fine and Quasi t-Fine Rings

Abstract

Fine rings and generalized fine rings have been extensively studied through additive decompositions involving units and nilpotent elements. In this paper, we develop analogous decompositions involving torsion units. We introduce and study {\it generalized t-fine} rings, in which every element outside the Jacobson radical is expressed as a sum of a torsion unit and a nilpotent element. We establish several basic properties of these rings and prove, in particular, that when the nilpotent elements form a subring, a ring is generalized t-fine if and only if it is local and every unit is torsion. We further investigate the behaviour of this property for matrix rings, endomorphism rings of finite abelian groups, and group rings, obtaining several structural and characterization results. We then introduce the broader class of {\it generalized quasi t-fine} rings by replacing nilpotent elements with quasinilpotent elements. We provide examples that illustrate the difficulties in characterizing this class, and investigate its behaviour for matrix rings and group rings.

Fine structure of even Galois deformation rings
Peter Vang Uttenthal
arXiv13 Oct 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv
√Δ-Fine Rings
Peter Danchev, Omid Hasanzadeh, Ahmad Moussavi +1
arXiv11 Oct 2026PreprintCC BY 4.0Full textAlgebra & number theoryarXiv
A Fitting criterion for quasi-Gorenstein local rings
Sora Miyashita
arXiv27 Oct 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv
On Reversible Rings, Their Generalization and Quasi-Idempotent Elements
Vivek Bhabani Lama
arXiv5 Oct 2026PreprintCC BY 4.0Full textAlgebra & number theoryarXiv
Commutativity Criteria and Structural Properties of Prime Near-Rings Admitting Right Generalized Derivations on Specific Ideals
Khaled Aldwoah, Inzamam ul Huque, Hafedh Alnoghashi +3
Axioms22 Oct 2026Peer-reviewedCC BY 4.0Algebra & number theoryCrossref

Cite

BibTeX RIS

BibTeX

@misc{bien2026generalized,
  title = {{Generalized $t$-Fine and Quasi $t$-Fine Rings}},
  author = {Mai Hoang Bien and Peter Vassilev Danchev and Mojtaba Ramezan-Nassab},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.19882},
  eprint = {2609.19882},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.19882},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Generalized t-Fine and Quasi t-Fine Rings
AU  - Mai Hoang Bien
AU  - Peter Vassilev Danchev
AU  - Mojtaba Ramezan-Nassab
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.19882
UR  - https://arxiv.org/abs/2609.19882
AB  - Fine rings and generalized fine rings have been extensively studied through additive decompositions involving units and nilpotent elements. In this paper, we develop analogous decompositions involving torsion units. We introduce and study {\it generalized t-fine} rings, in which every element outside the Jacobson radical is expressed as a sum of a torsion unit and a nilpotent element. We establish several basic properties of these rings and prove, in particular, that when the nilpotent elements form a subring, a ring is generalized t-fine if and only if it is local and every unit is torsion. We further investigate the behaviour of this property for matrix rings, endomorphism rings of finite abelian groups, and group rings, obtaining several structural and characterization results. We then introduce the broader class of {\it generalized quasi t-fine} rings by replacing nilpotent elements with quasinilpotent elements. We provide examples that illustrate the difficulties in characterizing this class, and investigate its behaviour for matrix rings and group rings.
ER  -

Abstract

Related

Cite