Rings Whose Non-Invertible Elements Are Nil-Clean

TL;DR

This paper characterizes rings where non-invertible elements can be expressed as a sum of an idempotent and a nilpotent, providing structural insights and extending previous research in the area.

Contribution

It offers a comprehensive description of rings with non-invertible elements that are nil-clean, advancing the understanding of their structural properties.

Findings

Complete characterization of such rings.

Structural results for this class of rings.

Extension of previous work by Diesl and Karimi-Mansoub et al.

Abstract

We systematically study those rings whose non-units are a sum of an idempotent and a nilpotent. Some crucial characteristic properties are completely described as well as some structural results for this class of rings are obtained. This work somewhat continues two publications on the subject due to Diesl (J. Algebra, 2013) and Karimi-Mansoub et al. (Contemp. Math., 2018).