Rings Whose Clean Elements Are Uniquely Strongly Clean

TL;DR

This paper introduces CUSC rings, a new class where clean elements are uniquely strongly clean, generalizing USC and CUC rings, and explores their properties and relationships.

Contribution

It defines CUSC rings and establishes their relationship with USC and CUC rings, expanding the understanding of clean element properties in ring theory.

Findings

A ring is USC if and only if it is CUSC and potent.

Relationships between CUSC and CUC rings are characterized.

The paper generalizes existing classes of rings with unique clean element properties.

Abstract

We define the class of {\it CUSC} rings, that are those rings whose clean elements are uniquely strongly clean. These rings are a common generalization of the so-called {\it USC} rings, introduced by Chen-Wang-Zhou in J. Pure \& Applied Algebra (2009), which are rings whose elements are uniquely strongly clean. These rings also generalize the so-called {\it CUC} rings, defined by Calugareanu-Zhou in Mediterranean J. Math. (2023), which are rings whose clean elements are uniquely clean. We establish that a ring is USC if, and only if, it is simultaneously CUSC and potent. Some other interesting relationships with CUC rings are obtained as well.