On Semi-Nil Clean Rings with Applications

The notion of semi-nil clean rings is investigated; these are rings in which each element can be expressed as a sum of a periodic and a nilpotent element. Among the obtained results, it is shown that if R is a semi-nil clean NI ring, then R is periodic. In addition, every group ring RG of a nilpotent group G over a weakly 2-primal ring R is proved to be semi-nil clean if and only if R is periodic and G is locally finite. Rings in which every unit is a sum of a periodic and a nilpotent element are also studied; such rings are called unit semi-nil clean rings. One of the important results obtained is that if R is an algebraic algebra over a field, then R is unit semi-nil clean if and only if R is periodic. Furthermore, rings in which every non-zero element is a sum of a torsion element and a nilpotent element are investigated. These rings, called t-fine rings, form a proper subclass of the class of all fine rings. One of the main results shows that matrix rings over t-fine rings are themselves t-fine, thereby strengthening the result of Calugareanu-Lam (2016).