  We consider in-depth and characterize in certain aspects those rings whose
non-units are strongly nil-clean in the sense that they are a sum of commuting
nilpotent and idempotent. In addition, we examine those rings in which the
non-units are uniquely nil-clean in the sense that they are a sum of a
nilpotent and an unique idempotent. In fact, we succeeded to prove that these
two classes of rings can completely be characterized in terms of already
well-studied and fully described sorts of rings.
