On Semi-Nil Clean Rings with Applications

Abstract: We investigate the notion of \textit{semi-nil clean} rings, defined as those rings in which each element can be expressed as a sum of a periodic and a nilpotent element. Among our results, we show that if $R$ is a semi-nil clean NI ring, then $R$ is periodic. Additionally, we demonstrate that every group ring $RG$ of a nilpotent group $G$ over a weakly 2-primal ring $R$ is semi-nil clean if, and only if, $R$ is periodic and $G$ is locally finite. Moreover, we also study those rings in which every unit is a sum of a periodic and a nilpotent element, calling them \textit{unit semi-nil clean} rings. As a remarkable result, we show that if $R$ is an algebraic algebra over a field, then $R$ is unit semi-nil clean if, and only if, $R$ is periodic. Besides, we explore those rings in which non-zero elements are a sum of a torsion element and a nilpotent element, naming them \textit{t-fine} rings, which constitute a proper subclass of the class of all fine rings. One of the main results is that matrix rings over t-fine rings are again t-fine rings.