UC Berkeley

A longstanding open question is whether every strongly clean ring is Dedekind-finite (definitions recalled below). We give an example of a ring with a pair of strongly clean elements that are one-sided, but not two-sided inverses of one another, suggesting that the answer to that question may be negative. [Formula: see text] We discuss possible ways of strengthening this result to give a full negative answer. We end with some brief observations on related topics, in particular, uniquely strongly clean rings.