Rings whose non-invertible elements are strongly weakly nil-clean

The target of this work is to give a new insight in the theory of strongly weakly nil-clean rings, recently defined by Kosan and Zhou [On weakly nil-clean rings, Front. Math. China 11(4) (2016) 949–955] and further explored in detail by Chen–Sheibani [Strongly weakly nil-clean rings, J. Algebra Appl. 16 (2017)]. Indeed, we consider those rings whose non-units are strongly weakly nil-clean and succeed to establish that this class of rings is strongly [Formula: see text]-regular and, even something more, that it possesses a complete characterization in terms of the Jacobson radical and sections of the [Formula: see text] full matrix ring. Additionally, some extensions like Morita context rings and groups rings are also studied in this directory.