Rings with u − 1 quasinilpotent for each unit u

In this paper, we define and explore in-depth the notion of UQ rings by showing their important properties and by comparing their behavior with that of the well-known classes of UU rings and JU rings, respectively. Specifically, among the other established results, we prove that UQ rings are always Dedekind finite (often named directly finite) as well as that, for semipotent rings [Formula: see text], the following equivalence hold: [Formula: see text] is [Formula: see text] is UJ having the property that the set [Formula: see text] of quasinilpotent elements of [Formula: see text] coincides with the Jacobson radical [Formula: see text] of [Formula: see text].