Rings whose Non-Invertible Elements are Strongly Nil-Clean

Abstract

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.