Let $R$ be a ring (here, rings are always associative, unital, and non-zero). An element $r \in R$ is *[quasi-regular][1]* if $1_R + r$ is a unit (so in particular, $0_R$ is quasi-regular); and $R$ 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 quasi-regular element is zero?

It is perhaps worth noting that, if the only quasi-regular element of $R$ is $0_R$, then $R$ has characteristic $2$ (and the Jacobson radical of $R$ is of course trivial).

  [1]: https://en.wikipedia.org/wiki/Quasiregular_element