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.