Abstract
Abstract We define the class of CUSC rings, which are rings whose clean 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 CUC rings, defined by Călugăreanu and 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.
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