  We define and explore in-depth the notion of {\it 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 $R$,
the following equivalence hold: $R/J(R)$ is UQ $\iff$ $R$ is UQ having the
property that the set $QN(R)$ of quasinilpotent elements of $R$ coincides with
the Jacobson radical $J(R)$ of $R$.
