发表机构
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences; Tarbiat Modares University（保加利亚科学院数学与信息学研究所; 塔比阿特·莫达雷斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文引入推广经典精细环的√Δ-精细环，证明其为单环、矩阵环仍为该类环，刻画半局部此类环为单Artin环，研究群环相关条件并提出开放问题。
AI 中文摘要
我们引入并研究了所谓的√Δ-精细环，这是一类新型环，它推广了Călugăreanu与Lam于2016年在《J. Algebra & Appl.》中提出的经典精细环，要求对环R中的每个非零元素r，都可表示为r = u + a，其中u是单位，a属于√Δ(R)。我们证明了每个此类环都是单环，每个交换√Δ-精细环都是不可分解的，最值得注意的是，对任意n ≥ 1，√Δ-精细环R上的矩阵环Mₙ(R)仍是√Δ-精细环。由此，我们将所有半局部√Δ-精细环刻画为恰好是单Artin环的那些环。我们还研究了群环，给出了群环为√Δ-精细环或广义精细环的条件，其中广义精细环是Zhou于2022年在《J. Algebra & Appl.》中引入的一类环，最后以一个困难的开放问题结束本研究，即询问每个√Δ-精细环是否一定是精细环。
英文摘要
We introduce and study the so-termed {\it $\sqrtΔ$-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 $r \in R$ can be written as $r = u + a$, where $u$ is a unit and $a \in \sqrt{Δ(R)}$. We establish that every such ring is simple, every abelian $\sqrtΔ$-fine ring is indecomposable, and most notably, the matrix ring $M_n(R)$ over a $\sqrtΔ$-fine ring $R$ is again $\sqrtΔ$-fine for every $n \ge 1$. As a consequence, we characterize all semi-local $\sqrtΔ$-fine rings as those rings which are precisely the simple Artinian rings. We also examine group rings, providing conditions under which they are either $\sqrtΔ$-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 $\sqrtΔ$-fine ring is necessarily fine.
Comments13 pages
