Rings whose non-units are square-nil clean

Purpose To describe the structure of a special sort of rings whose non-invertible elements are the sum of a nilpotent and a square-idempotent (which commute one another). Specifically, we consider in-depth and characterize in certain aspects the class of so-called strongly NUS-nil clean rings, that are those rings whose non-units are square-nil clean in the sense that they are a sum of a nilpotent and a square-idempotent that commutes with each other. This class of rings lies properly between the classes of strongly nil clean rings and strongly clean rings. Design/methodology/approach We develop an original method of proof based on polynomial expressions. Findings It is proved the valuable criterion that a ring R is strongly NUS-nil clean if, and only if, a4 - a2 ∈ Nil(R) for every a ∉ U(R). In particular, a ring R with only trivial idempotents is strongly NUS-nil clean if, and only if, R is a local ring with nil Jacobson radical. Some special matrix constructions and group ring extensions will provide us with new sources of examples of strongly NUS-nil clean rings. Originality/value We declare that the obtained by us results are absolutely original and do not duplicate other known results.