Rings Whose Clean Elements Are Uniquely Strongly Clean

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.