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.