发表机构
University of Science, Ho Chi Minh City; Vietnam National University, Ho Chi Minh City; Institute of Mathematics and Informatics, Bulgarian Academy of Sciences; Kharazmi University; School of Mathematics, Institute for Research in Fundamental Sciences (IPM)（胡志明市科学大学; 胡志明市国立大学; 保加利亚科学院数学与信息学研究所; 哈拉兹米大学; 基础科学研究学院数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文引入广义$t$-精细环与广义拟$t$-精细环，用挠单位元与（拟）幂零元分解Jacobson根外元素，刻画其性质并研究矩阵环、群环等上的表现。
AI 中文摘要
精细环与广义精细环已通过涉及单位元和幂零元的加法分解被广泛研究。在本文中，我们发展涉及挠单位元的类似分解。我们引入并研究{\it 广义 $t$-精细环}，在其中，Jacobson根之外的每个元素都可表示为挠单位元与幂零元之和。我们建立了这些环的若干基本性质，并特别证明了当幂零元构成一个子环时，一个环是广义 $t$-精细的当且仅当它是局部环且每个单位元都是挠元。我们进一步研究了该性质在矩阵环、有限阿贝尔群的自同态环以及群环上的表现，获得了若干结构与刻画结果。然后，我们通过将幂零元替换为拟幂零元，引入了更广泛的{\it 广义拟 $t$-精细环}类。我们提供了说明刻画此类环之困难的例子，并研究了其在矩阵环和群环上的表现。
英文摘要
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.
Comments19 pages
