2025/08/20 by Bergman, George M.
#16U40 #16U60 (Primary) 16S50 #16U90. (Secondary) #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2508.14396
A longstanding open question is whether every strongly clean ring (ring in which every element is strongly clean, i.e., is the sum of an idempotent and a unit which commute with each other) is Dedekind-finite (has the property that every element with a one-sided inverse is invertible). We give an example of a ring with two 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. We then 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.