Strongly clean ring elements that are one-sided inverses

TL;DR

This paper investigates whether all strongly clean rings are Dedekind-finite by providing examples of elements with one-sided inverses that are not two-sided, suggesting the answer may be negative.

Contribution

It presents the first known example of strongly clean elements with one-sided inverses that are not two-sided, challenging previous assumptions.

Findings

Example of strongly clean elements with one-sided inverses but not two-sided

Discussion on possible extensions to fully answer the open question

Observations on related concepts like uniquely strongly clean rings

Abstract

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.