Nil clean group rings over metacyclic groups

A ring is nil clean if each of its elements is the sum of an idempotent and a nilpotent. Previously in [J. Cui, Y. Li and H. Wang, On nil clean group rings, Commun. Algebra 49(2) (2021) 790–796], the nil clean group rings over dihedral groups and generalized quaternion groups were completely determined. In this paper, we investigate group rings [Formula: see text] over more general metacyclic groups [Formula: see text], where [Formula: see text] We provide a new criterion for a general nil clean group ring based on the upper central series of [Formula: see text] consisting of [Formula: see text]-groups, and whether the group ring over its respective factor group is nil clean. We also determine values of [Formula: see text] and [Formula: see text] (as a prime power) in order for [Formula: see text] to be nil clean, and verify whether their corresponding group rings are indeed nil clean.