We study rings R in which every unit u ∈ U (R) satisfies u n –1 in the set of quasi-nilpotent elements and n ≥ 2 is fixed. These n-UQ rings generalize several radical-related classes and exhibit rich structural behavior. We establish lifting and inheritance properties, characterize semi-local (2k – 1)-UQ rings via residue fields, and show that every (2k – 1)-UQ ring is Dedekind-finite for some k = 3, 4, 6. Moreover, we prove that in (2n – 1)-UQ rings, regularity, π-regularity with reducedness, and power-idempotency are equivalent, and that exchange and clean properties coincide for some n = 3, 4, 6.
Nguyen Quoc Tien (Fri,) studied this question.