On Semi-Nil Clean Rings with Applications

We investigate the notion of 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 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 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.