√Δ-Fine Rings
Abstract
We introduce and study the so-termed {\it √Δ-fine rings}, a new class of rings that generalizes the classical {\it fine rings} introduced by Călugăreanu-Lam in J. Algebra \& Appl. (2016) by requiring that every nonzero element r ∈ R can be written as r = u + a, where u is a unit and a ∈ √(Δ(R)). We establish that every such ring is simple, every abelian √Δ-fine ring is indecomposable, and most notably, the matrix ring Mn(R) over a √Δ-fine ring R is again √Δ-fine for every n ≥ 1. As a consequence, we characterize all semi-local √Δ-fine rings as those rings which are precisely the simple Artinian rings. We also examine group rings, providing conditions under which they are either √Δ-fine or generalized fine, where the latter class was introduced by Zhou in J. Algebra \& Appl. (2022), and conclude our work with the difficult open question asking of whether each √Δ-fine ring is necessarily fine.
Related
BibTeX
@misc{danchev2026δfine,
title = {{$\sqrtΔ$-Fine Rings}},
author = {Peter Danchev and Omid Hasanzadeh and Ahmad Moussavi and Arash Javan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.30734},
eprint = {2608.30734},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30734},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - √Δ-Fine Rings
AU - Peter Danchev
AU - Omid Hasanzadeh
AU - Ahmad Moussavi
AU - Arash Javan
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2608.30734
UR - https://arxiv.org/abs/2608.30734
AB - We introduce and study the so-termed {\it √Δ-fine rings}, a new class of rings that generalizes the classical {\it fine rings} introduced by Călugăreanu-Lam in J. Algebra \& Appl. (2016) by requiring that every nonzero element r ∈ R can be written as r = u + a, where u is a unit and a ∈ √(Δ(R)). We establish that every such ring is simple, every abelian √Δ-fine ring is indecomposable, and most notably, the matrix ring M_n(R) over a √Δ-fine ring R is again √Δ-fine for every n ≥ 1. As a consequence, we characterize all semi-local √Δ-fine rings as those rings which are precisely the simple Artinian rings. We also examine group rings, providing conditions under which they are either √Δ-fine or generalized fine, where the latter class was introduced by Zhou in J. Algebra \& Appl. (2022), and conclude our work with the difficult open question asking of whether each √Δ-fine ring is necessarily fine.
ER -