A class of rings with quasi-nilpotent unit powers

In this paper, we study rings [Formula: see text] in which every unit [Formula: see text] satisfies [Formula: see text] in the set of quasi-nilpotent elements and [Formula: see text] is fixed. These [Formula: see text]-UQ rings generalize several radical-related classes and exhibit rich structural behavior. We establish lifting and inheritance properties, characterize semi-local [Formula: see text]-UQ rings via residue fields, and show that every [Formula: see text]-UQ ring is Dedekind-finite for some [Formula: see text]. Moreover, we prove that in [Formula: see text]-UQ rings, regularity, [Formula: see text]-regularity with reducedness, and power-idempotency are equivalent, and that exchange and clean properties coincide for some [Formula: see text].