Rings Whose Non-Units are Square-Nil Clean

TL;DR

This paper characterizes strongly NUS-nil clean rings, where non-units are sums of nilpotent and square-idempotent elements, providing criteria and examples that situate this class between strongly nil-clean and strongly clean rings.

Contribution

It introduces and thoroughly characterizes strongly NUS-nil clean rings, establishing key criteria and constructing new examples through matrix and group ring extensions.

Findings

A ring is strongly NUS-nil clean iff a^4 - a^2 is nilpotent for all non-unit elements.

Rings with only trivial idempotents are strongly NUS-nil clean iff they are local with nil Jacobson radical.

New examples of NUS-nil clean rings are provided via matrix constructions and group ring extensions.

Abstract

We consider in-depth and characterize in certain aspects the class of so-called {\it strongly NUS-nil clean rings}, that are those rings whose non-units are {\it 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. In fact, it is proved the valuable criterion that a ring $R$ is strongly NUS-nil clean if, and only if, $a^4-a^2\in Nil(R)$ for every $a\not\in U(R)$. 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 NUS-nil clean rings.