Rings Whose Non-Invertible Elements are Weakly Nil-Clean

TL;DR

This paper investigates the structure and properties of rings where non-invertible elements are weakly nil-clean, including a detailed analysis of such rings and their group ring counterparts.

Contribution

It provides a comprehensive characterization of rings with non-invertible elements being weakly nil-clean and explores their group ring structures.

Findings

Characterization of rings with non-invertible weakly nil-clean elements

Structural properties of such rings

Description of group rings with these properties

Abstract

In regard to our recent studies of rings with (strongly, weakly) nil-clean-like properties, we explore in-depth both the structural and characterization properties of those rings whose elements that are not units are weakly nil-clean. Group rings of this sort are considered and described as well.