Rings Whose Non-Invertible Elements Are Uniquely Strongly Clean

TL;DR

This paper introduces GUSC rings, a new class of rings where non-invertible elements are uniquely strongly clean, generalizing previous classes like USC and GUC rings, and explores their properties in detail.

Contribution

It defines GUSC rings and demonstrates how they extend existing classes of rings with unique clean properties, enriching the algebraic theory.

Findings

GUSC rings generalize USC and GUC rings.

Non-invertible elements in GUSC rings are uniquely strongly clean.

The paper provides detailed properties and characterizations of GUSC rings.

Abstract

We define and explore in details the class of GUSC rings, that are those rings whose non-invertible elements are uniquely strongly clean. These rings are a common generalization of the so-called USC rings, introduced by Chen-Wang-Zhou in J. Pure & Appl. Algebra (2009), which are rings whose elements are uniquely strongly clean. These rings also generalize the so-called GUC rings, defined by Guo-Jiang in Bull. Transilvania Univ. Bra\c{s}ov (2023), which are rings whose non-invertible elements are uniquely clean.