Rings Whose Non-Invertible Elements are Strongly Weakly Nil-Clean

TL;DR

This work introduces generalized strongly weakly nil-clean (GSWNC) rings, characterizing rings whose non-invertible elements admit a strongly weakly nil-clean decomposition and establishing that such rings are strongly π-regular with nil Jacobson radical quotients. It provides a robust framework linking GSWNC to existing concepts like GSNC and strong weakly clean rings, and develops extensive closure properties under matrix, Morita-context, and skew constructions, including a key nonexistence result for $M_n(R)$ with $n\ge3$. The paper then extends the theory to group rings $RG$, presenting necessary and sufficient conditions involving the augmentation ideal $\Delta(RG)$ and the structure of $G$ (especially $2$- and $3$-groups) to determine when $RG$ is GSWNC or GSNC. These results yield precise classifications for semilocal and Artinian quotients $R/J(R)$ and illuminate how ring and group-theoretic properties interact under generalized nil-clean decompositions, with implications for Morita contexts and formal matrix rings.

Abstract

The target of the present work is to give a new insight in the theory of {\it strongly weakly nil-clean} rings, recently defined by Kosan and Zhou in the Front. Math. China (2016) and further explored in detail by Chen-Sheibani in the J. Algebra Appl. (2017). Indeed, we consider those rings whose non-units are strongly weakly nil-clean and succeed to establish that this class of rings is strongly $π$-regular and, even something more, that it possesses a complete characterization in terms of the Jacobson radical and sections of the $2\times 2$ full matrix ring. Additionally, some extensions like Morita context rings and groups rings are also studied in this directory.

Theorems & Definitions (100)

• Definition 2.1
• Lemma 2.2
• proof
• Corollary 2.3
• Lemma 2.4
• proof
• Proposition 2.5
• proof
• Proposition 2.6
• proof