Remarks on rings with u - 1 quasi-nilpotent for each unit u

This paper is concerned with the investigation of those rings [Formula: see text] whose invertible elements are sums [Formula: see text] for an element [Formula: see text] such that [Formula: see text] is invertible for all invertible [Formula: see text] with [Formula: see text]. UJ-rings and UQ-rings (and hence UU-rings) are examples of such rings. We prove that such rings are Dedekind finite, and for semisimple rings [Formula: see text], [Formula: see text] In case [Formula: see text] is also a semipotent ring, it is shown that [Formula: see text] is a Boolean ring.