扩展广义精细环
Expanding Generalized Fine Rings
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中文总结 AI 辅助
研究广义$\sqrt{J}$-精细环，证明其在全矩阵环下封闭，刻画局部有限群上群环何时为此类环，表明其是2-清洁的，介于广义精细环和2-清洁环之间，通过例子说明概念的复杂行为及边界。
中文摘要 AI 辅助
我们引入并研究了所谓的“广义$\sqrt{J}$-精细环”，其中雅可比根之外的每个元素都是一个单位元和来自集合$\sqrt{J(R)} := \{ x \in R: x^{n} \in J(R) \text{ 对于某些 } n \ge 1 \}$的一个元素的和。这通常扩展了分别由Călugăreanu-Lam（《代数与应用杂志》，2016年）和周（《代数与应用杂志》，2022年）定义的“精细”和“广义精细环”的概念。具体而言，我们证明了此类在任何大小的全矩阵环下是封闭的，并且我们完全刻画了局部有限群上的群环何时是广义$\sqrt{J}$-精细的。我们还表明每个这样的环都是2-清洁的，从而恰当地将其置于广义精细环和2-清洁环之间。还提供了几个例子来说明所引入概念的复杂行为及其众多边界。
英文摘要
We introduce and study the so-called {\it generalized $\sqrt{J}$-fine rings}, where every element outside the Jacobson radical is the sum of a unit and an element from the set $\sqrt{J(R)} := \{ x \in R : x^{n} \in J(R) \text{ for some } n \ge 1 \}$. This commonly extends the notions of {\it fine} and {\it generalized fine rings} defined, respectively, by Călugăreanu-Lam (J. Algebra \& Appl., 2016) and Zhou (J. Algebra \& Appl., 2022). Specifically, we prove that this class is closed under full matrix rings of any size, as well as we completely characterize when group rings over locally finite groups are generalized $\sqrt{J}$-fine. We also show that every such ring is 2-clean, thus properly placing it between generalized fine rings and 2-clean rings. Several examples are also provided to illustrate the complicated behavior of the introduced concept and its numerous boundaries.
