Let $R$ be a ring (here, rings are always associative, unital, and non-zero). We say that $R$ is a *left duo* ring if $aR \subseteq Ra$ for every $a \in R$. 

> **Question.** Is there a non-commutative, left duo ring whose only unit is the identity?

It is perhaps worth noting that, if the only unit of $R$ is the identity, then $R$ has characteristic $2$ and the Jacobson radical of $R$ is trivial. 

**Edit.** In a previous version of this question, I was asking for the existence of a left duo ring whose only [quasi-regular][1] element is zero: I hadn't realized that this is only possible if the group of units of the ring is trivial (whence the new formulation), which makes the question vaguely reminiscent of an [open problem by M. Henrikson][2].


  [1]: https://en.wikipedia.org/wiki/Quasiregular_element
  [2]: https://mathoverflow.net/a/395800/16537